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Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... 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 comp ..."
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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 HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
Dagger categories and formal distributions
"... 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 cat ..."
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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
UFIFTHEP0024 quantph/0009072 On the Implementation of Constraints through Projection Operators
, 2000
"... Quantum constraints of the type Qψphys 〉 = 0 can be straightforwardly implemented in cases where Q is a selfadjoint 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, nont ..."
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Quantum constraints of the type Qψphys 〉 = 0 can be straightforwardly implemented in cases where Q is a selfadjoint 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 nonselfadjoint. 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
Nonsymmetric Scattering in Light Transport Algorithms
"... 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 example, importance is often used to ..."
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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 example, importance is often used to guide mesh refinement in finiteelement approaches, and traditional ray tracing is the dual of particle tracing, which simulates the emissionand scattering of photons.
Nonsymmetric Scattering inLight Transport Algorithms
"... Abstract Nonsymmetric 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 nonsymmetry occurs whenever light is refracted, and also whenever shading normals are used ..."
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Abstract Nonsymmetric 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 nonsymmetry occurs 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 nonsymmetric scattering for light transport theory. We extend the work of Arvo et al. [4] into a complete frameworkfor light, importance, and particle transport with nonsymmetric 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 algorithms. 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 nonsymmetry is recognized and handled correctly, theseproblems can easily be avoided. 1 Introduction The equations governing the transport and measurement of light energy can be written 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 finiteelement approaches, and traditional ray tracing is the dual of particle tracing, which simulates the emission and scattering of photons. 1
DOI:10.5829/idosi.wasj.22.am.12.2013 Tensorial approximate identities for vector valued functions
"... Abstract: In this article, we introduce a tensorial kernel for vector valued functions which is compactly supported on a bounded region V in R 3. It is proved that the convolutions of the derived kernels with a vector valued function f having continuous first derivatives converges to f, i.e. the def ..."
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Abstract: In this article, we introduce a tensorial kernel for vector valued functions which is compactly supported on a bounded region V in R 3. It is proved that the convolutions of the derived kernels with a vector valued function f having continuous first derivatives converges to f, i.e. the defining property of an approximate identity, is proved. Further, some numerical tests are shown. Key words: Green’s function, weak convergence, approximate identity, tensorial kernel.
CONSTRUCTIVE APPROXIMATION © 2001 SpringerVerlag New York Inc. Approximation in L p(R d) from a Space Spanned by the Scattered Shifts of a Radial Basis Function
"... Abstract. A new multivariate approximation scheme on Rd using scattered translates of the “shifted ” surface spline function is developed. The scheme is shown to provide spectral L papproximation orders with 1 ≤ p ≤∞, i.e., approximation orders that depend on the smoothness of the approximands. In ..."
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Abstract. A new multivariate approximation scheme on Rd using scattered translates of the “shifted ” surface spline function is developed. The scheme is shown to provide spectral L papproximation orders with 1 ≤ p ≤∞, i.e., approximation orders that depend on the smoothness of the approximands. In addition, it applies to noisy data as well as noiseless data. A numerical example is presented with a comparison between the new scheme and the surface spline interpolation method. 1.
Geoelectrical conductivity Problems on unbounded Domains
, 1998
"... 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' ..."
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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 parametertosolution 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...