N Van den Bergh

Using Maple and the Web to grade mathematics tests

We present AIM, a web-based system designed to administer graded tests with mathematical content. Its main features are: use of Maple as the engine and implementation language; several methods of giving partial credit; various feedback mechanisms; randomisation of quizzes and questions; versatility in question and quiz design; extensive grade reporting and monitoring capabilities; ability to collect surveys; web interface for both teacher and student. AIM can be used to administer graded tests, homeworks or ungraded self-assessment exercises. A case study using vector calculus was conducted and more courses are being planned for September 2000, including linear algebra, ODEs and precalculus. Preparation is also under way to use AIM to mark a part of the final exam for linear algebra in February 2001.

Viability criteria for the theories of gravity and finsler spaces

It is commonly held that the Riemannian geometry adopted as the theoretical framework within which observations and experiments, concerning the ‘correct’ theory of gravity, are analyzed is the most general ‘viable’ geometry consistent with observed phenomena. This viewpoint is further strengthened by the belief that the projective and the conformal structures of the space-time together with an additional assumption concerning the constancy of the norm of vectors under parallel transport would uniquely determine its underlying geometry to be Riemannian. We show here that a more general geometrical framework due to Finsler can be made compatible with these structures and still remain non-Riemannian. The potential importance of this result in connection with developing and testing alternative theories of gravity is briefly discussed.

Finsler spaces and the underlying geometry of space-time

Abstract It is commonly believed that the projective and the conformal structures of the space-time together with an extra assumption concerning the constancy of the norm of vectors under parallel transport would uniquely determine its underlying geometry to be riemannian. Here we show that the Finsler geometrical framework can be made compatible with the above substructures and still remain non-riemannian. This can be of importance in developing alternative theories of gravity.

Exact solutions for the spherically symmetric vacuum field in the general scalar tensor theory

A conformai technique is given for the generation of exact solutions for the spherically symmetric vacuum field in the general Bergmann-Wagoner-Nordtvedt scalar-tensory theory with vanishing cosmological constant. We discuss in particular the solution for Schwingers theory and for models withφn coupling or with curvature coupling. It appears that all theories with vanishing cosmological term lead to the presence of naked singularities.

General solutions for a static isotropic metric in the Brans-Dicke gravitational theory

The complete set of vacuum solutions for the metric tensor of a static spherically symmetric field is given, some of these solutions showing the remarkable feature of not agreeing-even in first order-with the classically well-known weak-field solutions of the Brans-Dicke (B.D.) equations. The existence of a particular two-parameter family of solutions raises severe doubts about the so-called Machian aspect of B.D. theory.

The shear-free perfect fluid conjecture

There is mounting evidence that general relativistic shear-free perfect fluids, obeying a barotropic equation of state with , are necessarily irrotational or non-expanding. This conjecture has been demonstrated in a number of particular cases, but a general proof is still lacking. In the tetrad-based approach two particular cases require a special treatment, namely = constant and = constant. A proof is given that the conjecture holds for each of these. In addition, a formalism is presented enabling one to deal with the more general case of a -law equation of state + constant.

Conformally Ricci‐flat perfect fluids. II

Classes of inhomogeneous perfect fluid solutions can be obtained by requiring that the associated Weyl tensor corresponds to a nonflat vacuum solution of Einstein’s field equations. It is shown how one derives from this assumption useful information on the Newman–Penrose variables. Some particular classes of shear‐free perfect fluid solutions are discussed, which all turn out to be locally rotationally symmetric.

Irrotational and conformally Ricci-flat perfect fluids

When a space-time, containing an irrotational perfect fluid withw + p ≠ 0, is conformally Ricci-flat, three possibilities arise: (a) When the gradient of the conformal scalar field is aligned with the fluid velocity, the solution is conformally flat; (b) when the gradient is orthogonal to the fluid velocity, solutions are either shearfree, nonexpanding and (pseudo-) spherically or plane-symmetric, or they are conformally related to a particular new vacuum solution admitting a three-dimensional group of motions of Bianchi type VIo on a timelike hypersurface; (c) in the general case solutions are (pseudo) spherically or plane-symmetric and have nonvanishing expansion.

Exact solutions for nonstatic perfect fluid spheres with shear and an equation of state

It is shown that fairly general assumptions about the relevant physical and mathematical quantities in the study of nonstatic and spherically symmetric perfect fluid configurations usually single out the Friedmann-Lemaître solutions as the only physically plausible ones. In addition three new classes of exact solutions having shear and acceleration are presented, as well as a generalization of Wessons stiff fluid solution.

Shear-free perfect fluids with solenoidal magnetic curvature and a γ-law equation of state

We show that shear-free perfect fluids obeying an equation of state p = (γ − 1)μ are non-rotating or non-expanding under the assumption that the spatial divergence of the magnetic part of the Weyl tensor is zero.