We derive the fully extended supersymmetry algebra carried by Dbranes in a massless type IIA superspace vacuum. We find that the extended algebra contains not only topological charges that probe the presence of compact spacetime dimensions but also pieces that measure non-trivial configurations of the gauge field on the worldvolume of the brane. Furthermore there are terms that measure the coupling of the nontriviality of the worldvolume regarded as a U(1)-bundle of the gauge field to possible compact spacetime dimensions. In particular, the extended algebra carried by the D-2-brane can contain the charge of a Dirac monopole of the gauge field. In the course of this work we derive a set of generalized Gamma-matrix identities that include the ones presently known for the IIA case. -In the first part of the paper we give an introduction to the basic notions of Noether current algebras and charge algebras; furthermore we find a Theorem that describes in a general context how the presence of a gauge field on the worldvolume of an embedded object transforming under the symmetry group on the target space alters the algebra of the Noether charges, which otherwise would be the same as the algebra of the symmetry group. This is a phenomenon recently found by Sorokin and Townsend in the case of the M-5-brane, but here we show that it holds quite generally, and in particular also in the case of D-branes.

A method to reconstruct weakly anisotropic inhomogeneous dielectric tensors inside a transparent medium is proposed. The mathematical theory of integral geometry is cast into a workable framework that allows the full determination of dielectric tensor fields by scalar Radon inversions of the polarization transformation data obtained from six planar tomographic scanning cycles. Furthermore, a careful derivation of the usual equations of integrated photoelasticity in terms of heuristic length scales of the material inhomogeneity and anisotropy is provided, resulting in a self-contained account about the reconstruction of arbitrary three-dimensional, weakly anisotropic dielectric tensor fields.

The Poincaré Equivalence Theorem states that any optical element which contains no absorbing components can be replaced by an equivalent optical model which consists of one linear retarder and one rotator only, both of which are uniquely determined. This has many useful applications in the field of Optics of Polarized Light. In particular, it arises naturally in attempts to reconstruct spatially varying refractive tensors or dielectric tensors from measurements of the change of state of polarization of light beams passing through the medium, a field which is known as Tensor Tomography. A special case is Photoelasticity, where the internal stress of a transparent material may be reconstructed from knowledge of the local optical tensors by using the stress-optical laws. -We present a rigorous approach to the Poincaré Equivalence Theorem by explicitly proving a matrix decomposition theorem, from which the Poincaré Equivalence Theorem follows as a corollary. To make the paper self-contained we supplement a brief account of the Jones matrix formalism, at least as far as linear retarders and rotators are concerned. We point out the connection between the parameters of the Poincaré-equivalent model to previously introduced notions of the Characteristic Parameters of an optical model in the engineering literature. Finally, we briefly illustrate how characteristic parameters and Poincaré-equivalent models naturally arise in Photoelasticity.

We give a purely algebro-geometric proof of the fact that every nonsingular projective curve can be defined over a finite extension of its moduli field. This extends a result by Wolfart [7] to curves over fields of arbitrary characteristic. Introduction.To a nonsingular projective variety X over an algebraically closed field C, one can associate two interesting kinds of subfields of C: the fields of definition D(X), which are subfields of C such that X = X 0 × C with a scheme X 0 over D(X), and the moduli field M(X). The somewhat technical definition of M(X) is recalled below (see Definition 3); if X is a curve, M(X) can be interpreted as the residue field of the point corresponding to X in the moduli scheme of curves.From the definition it is easily seen that M(X) is contained in any D(X). In the opposite direction, J. Wolfart proved in [7] that if char(C) = 0 and X is a curve, then D(X) can be chosen as a finite algebraic extension of M(X) (and in fact, is even equal to M(X) in many cases). Another, more recent and stronger result in this direction can be found in [5].In this note we give a purely algebro-geometric version of Wolfart's proof, thereby extending his result to curves over algebraically closed fields of arbitrary characteristic. It should be noted that most of our arguments are valid for nonsingular varieties of any dimension. Only the use of the coarse moduli space M g at the end of the proof of Theorem 5 relies essentially on the fact that X is a curve. So our main result can immediately be generalized to classes of varieties for which coarse moduli spaces are known to exist (such as abelian varieties, surfaces, . . .). But the result should be true more generally, as can be seen e.g. from A. Weil's original paper [6], and we plan to come back to these questions in a subsequent paper.This paper grew out of the Diplomarbeit of the first author. We would like to thank B. Köck, who served as co-adviser and gave us numerous valuable hints and comments. Thanks also go to S. Kühnlein for many valuable discussions.Throughout this article let C be an algebraically closed field and let k be its prime field. By a curve we denote a nonsingular projective integral separated scheme of finite type over

Abstract. We explain the main concepts centered around Sharafutdinov's ray transform, its kernel, and the extent to which it can be inverted. It is shown how the ray transform emerges naturally in any attempt to reconstruct optical and stress tensors within a photoelastic medium from measurements on the state of polarization of light beams passing through the strained medium. The problem of reconstruction of stress tensors is crucially related to the fact that the ray transform has a nontrivial kernel; the latter is described by a theorem for which we provide a new proof which is simpler and shorter as in Sharafutdinov's original work, as we limit our scope to tensors which are relevant to Photoelasticity. We explain how the kernel of the ray transform is related to the decomposition of tensor fields into longitudinal and transverse components. The merits of the ray transform as a tool for tensor reconstruction are studied by walking through an explicit example of reconstructing the σ33-component of the stress tensor in a cylindrical photoelastic specimen. In order to make the paper self-contained we provide a derivation of the basic equations of Integrated Photoelasticity which describe how the presence of stress within a photoelastic medium influences the passage of polarized light through the material.