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The wave equation generalizing the Dirac operator to the Z3-graded case is introduced, whose diagonalization leads to a sixth-order equation. It intertwines not only quark and anti-quark state as well as the u and d quarks, but also the three colors, and is therefore invariant under the product group Z2 × Z2 × Z3. The solutions of this equation cannot propagate because their exponents always contain non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry and of the SU(2) × U(1) that arise automatically in this model, leading to the full bosonic gauge sector of the Standard Model. [ABSTRACT FROM AUTHOR]

Key message: We review causes of synergies in mixed-species stands, develop guiding hypotheses for revealing their mechanisms and present a rainfall exclusion experiment along with a transect (KROOF) for exploring drought effects. Abstract: While monocultures have dominated forest research and practice in the past, in face of growing resource scarcity and climate change, mixed-species stands are on the advance. Long-term observations show that mixed-species stands frequently over-yield monocultures, and they further suggest that the over-yielding is often higher on poor than on fertile sites and in low-growth than in high-growth years. However, the underlying causes have not yet been clarified. We start with a review of, among others, hydraulic redistribution, complementary eco-physiological traits, and ectomycorrhizal networks as possible causes behind the observed productivity gains in mixed-species stands. Then, we develop guiding hypotheses for further exploration of synergies in mixed-species stands. Finally, we introduce into the concept and model approach of the rainfall exclusion experiment for examining the role of water supply in mixed versus pure forest stands of spruce and beech. At the Kranzberg site, six plots are equipped with water retention roofs, which only close during rain events. The remaining six plots serve as non-roofed control. Together with the rainfall exclusion experiment, an ecological gradient with five sites extending through SE-Germany forms the 'Kranzberg Roof Experiment' (in short KROOF). Kranzberg Forest is a part of this gradient from moist to dry conditions, with each site providing mixed and pure associations each of spruce and beech. The ecophysiological approach will be complemented by tree ring analysis and modelling of productivity of the tree associations under intense drought. [ABSTRACT FROM AUTHOR]