We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored -categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored -categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored -categories, all morphisms are nuclear, and in the tensored -category of Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps. We also introduce two new examples of tensored -categories, in which integration plays the role of composition. In the first, mor...

Summary. Monoidal dagger categories play a central role in the abstract quantum mechanics of Abramsky and Coecke. The authors show that a great deal of elementary quantum mechanics can be carried out in these categories; for example, the Born rule emerges naturally. In this paper, we construct a category of tame formal distributions with coefficients in a commutative associative algebra and show that it is a dagger category. This gives access to a broad new class of models, with the abstract scalars in the sense of Abramsky being the elements of the algebra. We will also consider a subcategory of local formal distributions, based on the ideas of Kac. Locality has been of fundamental significance in various formulations of quantum field theory. Thus our work may provide the possibility of extending the abstract framework to QFT. We also show that these categories of formal distributions are monoidal and contain a nuclear ideal, a weak form of adjunction appropriate for analyzing categories

This paper deals with the electrical conductivity problem in geophysics. It is formulated as an elliptic boundary value problem of second order for a large class of bounded and unbounded domains. A special boundary condition, the so called "Complete Electrode Model", is used. Poincar'e inequalities are formulated and proved in the context of weighted Sobolev spaces, leading to existence and uniqueness statements for the boundary value problem. In addition, a parameter-to-solution operator arising from the inverse conductivity problem in medicine (EIT) and geophysics is investigated mathematically and is shown to be smooth and analytic. 1 INTRODUCTION 3 1 Introduction 1.1 Inverse problems in geophysics A major application of geophysical methods is the exploration of the earth's interior by means of measurements on the boundary. There are many methods currently in use, based on the observation of a variety of physical effects: for example, geomagnetics, seismic approaches and geoelec...

1 Introduction The equations governing the transport and measurement of light energy can be writtenin two equivalent forms, depending on whether we solve for radiance or importance. Most current global illumination algorithms take advantage of this duality. For exam-ple, importance is often used to guide mesh refinement in finite-element approaches, and traditional ray tracing is the dual of particle tracing, which simulates the emissionand scattering of photons.

Abstract Non-symmetric scattering is far more common in computer graphics thanis generally recognized, and can occur even when the underlying scattering model is physically correct. For example, we show that non-symmetry oc-curs whenever light is refracted, and also whenever shading normals are used (e.g. due to interpolation of normals in a triangle mesh, or bump mapping [5]).We examine the implications of non-symmetric scattering for light transport theory. We extend the work of Arvo et al. [4] into a complete frameworkfor light, importance, and particle transport with non-symmetric kernels. We show that physically valid scattering models are not always symmetric, andderive the condition for arbitrary models to obey Helmholtz reciprocity. By rewriting the transport operators in terms of optical invariants, we obtain anew framework where symmetry and reciprocity are the same. We also consider the practical consequences for global illumination al-gorithms. The problem is that many implementations indirectly assume symmetry, by using the same scattering rules for light and importance, or particlesand viewing rays. This can lead to incorrect results for physically valid models. It can also cause different rendering algorithms to converge to differentsolutions (whether the model is physically valid or not), and it can cause shading artifacts. If the non-symmetry is recognized and handled correctly, theseproblems can easily be avoided. 1 Introduction The equations governing the transport and measurement of light energy can be writ-ten in two equivalent forms, depending on whether we solve for radiance or importance. Most current global illumination algorithms take advantage of this duality.For example, importance is often used to guide mesh refinement in finite-element approaches, and traditional ray tracing is the dual of particle tracing, which simu-lates the emission and scattering of photons. 1

Quantum constraints of the type Q|ψphys 〉 = 0 can be straightforwardly implemented in cases where Q is a self-adjoint operator for which zero is an eigenvalue. In that case, the physical Hilbert space is obtained by projecting onto the kernel of Q, i.e. Hphys = ker Q = ker Q ∗. It is, however, nontrivial to identify and project onto Hphys when zero is not in the point spectrum but instead is in the continuous spectrum of Q, because then ker Q = ∅. Here, we observe that the topology of the underlying Hilbert space can be harmlessly modified in the direction perpendicular to the constraint surface in such a way that Q becomes non-self-adjoint. This procedure then allows us to conveniently obtain Hphys as the proper Hilbert subspace Hphys = ker Q ∗ on which one can project as usual. In the simplest case, the necessary change of topology amounts to passing from an L 2 Hilbert space to a Sobolev space. 1 1