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Spin(9)-invariant valuations
(2013)
The first aim of this thesis is to give a Hadwiger-type theorem for the exceptional Lie group Spin(9). The space of Spin(9)-invariant k-homogeneous valuations is studied through the construction of an exact sequence involving some spaces of differential forms. We present then a description of the spin representation using the properties of the 8-dimensional division algebra of the octonions. Using this description as well as representation-theoretic formulas, we can compute the dimensions of the spaces of differential forms appearing in the exact sequence. Hence we obtain the dimensions of the spaces of k-homogeneous Spin(9)-invariant valuations for k=0,1,...,16.
In the second part of this work, we construct one new element for a basis of one of these spaces. It is clear, that the k-th intrinsic volume is also Spin(9)-invariant. The last chapter of this work presents the construction of a new 2-homogeneous Spin(9)-invariant valuation. On a Riemannian manifold (M,g), we construct a valuation by integrating the curvature tensor over the disc bundle. We associate to this valuation on M a family of valuations on the tangent spaces. We show that these valuations are even and homogeneous of degree 2. Moreover, since the valuation on M is invariant under the action of the isometry group of M, the induced valuation on the tangent space in a point p in M is invariant under the action of the stabilisator of p for all p in M. In the special case where M is the octonionic projective plane, this construction yields an even, homogeneous of degree 2, Spin(9)-invariant valuation, whose Klain function is not constant, i.e. which is linearly independent of the second intrinsic volume.
Understanding the diverging opinions of academic experts, stakeholders and the public is important for effective conservation management. This is especially so when a consensus is needed for action to minimize future risks but the knowledge upon which to base this action is uncertain or missing. How to manage non-native, invasive species (NIS) is an interesting case in point: the issue has long been controversial among stakeholders, but publicly visible, major disagreement among experts is recent. To characterize the multitude of experts’ understanding and valuation of non-native, NIS we performed structured qualitative interviews with 26 academic experts, 13 of whom were invasion biologists and 13 landscape experts. Within both groups, thinking varied widely, not only about basic concepts (e.g., non-native, invasive) but also about their valuation of effects of NIS. The divergent opinions among experts, regarding both the overall severity of the problem in Europe and its importance for ecosystem services, contrasted strongly with the apparent consensus that emerges from scientific synthesis articles and policy documents. We postulate that the observed heterogeneity of expert judgments is related to three major factors: (1) diverging conceptual understandings, (2) lack of empirical information and high scientific uncertainties due to complexities and contingencies of invasion processes, and (3) missing deliberation of values. Based on theory from science studies, we interpret the notion of an NIS as a boundary object, i.e., concepts that have a similar but not identical meaning to different groups of experts and stakeholders. This interpretative flexibility of a concept can facilitate interaction across diverse groups but bears the risk of introducing misunderstandings. An alternative to seeking consensus on exact definitions and risk assessments would be for invasive species experts to acknowledge uncertainties and engage transparently with stakeholders and the public in deliberations about conflicting opinions, taking the role of honest brokers of policy alternatives rather than of issue advocates.
The pricing of digital art
(2023)
The intersection of recent advancements in generative artificial intelligence and blockchain technology has propelled digital art into the spotlight. Digital art pricing recognizes that owners derive utility beyond the artwork’s inherent value. We incorporate the consumption utility associated with digital art and model the stochastic discount factor and risk premiums. Furthermore, we conduct a calibration analysis to analyze the effects of shifts in the real and digital economy. Higher returns are required in a digital market upswing due to increased exposure to systematic risk and digital art prices are especially responsive to fluctuations in business cycles within digital markets.