Hamilton Jacobi Equations with Obstacles

We consider a problem in the theory of optimal control proposed for the first time by Bressan. We characterize the associated minimum time function using tools from geometric measure theory and we obtain, as a corollary, an existence theorem for a related variational proble

Optimization of Protein Quality of Plant-Based Foods Through Digitalized Product Development

With the increasing availability of plant-based protein products that should serve as alternatives to animal-based protein products, it is necessary to develop not only environmentally friendly but also nutritious foods. Especially the protein content and quality are of concern in these products. The algorithm of NutriOpt was developed using linear programming to support the development of food products with a balanced amino acid profile while considering digestibility. The current version contains a database with 84 plant protein sources from different food groups (legumes, cereals, nuts, seeds) and with different grades of purification (flours, concentrates, isolates) from which NutriOpt can create mixtures with high protein quality while complying with constraints such as protein content, number of ingredients, and weight of the mixture. The program was tested through different case studies based on commercial plant-based drinks. It was possible to obtain formulations with a Protein Digestibility Corrected Amino Acid Score (PDCAAS) over 100 with ingredients and quantities potentially suitable for plant-based analogs. Our model can help to develop the second generation of plant-based product alternatives that can really be used as an alternative on long-term consumption. Further, there is still a great potential of expansion of the program for example to use press cakes or even to model whole menus or diets in the future

SBV regularity for Hamilton-Jacobi equations in Rn\mathbb R^n

In this paper we study the regularity of viscosity solutions to the following Hamilton-Jacobi equations $$\partial_t u + H(D_{x} u)=0 \qquad \textrm{in} \Omega\subset \mathbb R\times \mathbb R^{n} .$$ In particular, under the assumption that the Hamiltonian $H\in C^2(\mathbb R^n)$ is uniformly convex, we prove that $D_{x}u$ and $\partial_t u$ belong to the class $SBV_{loc}(\Omega)$.Comment: 15 page

Temporal Dynamics of Root Reinforcement in European Spruce Forests

The quantification of post-disturbance root reinforcement (RR) recovery dynamics is of paramount importance for the optimisation of forest ecosystem services and natural hazards risk management in mountain regions. In this work we analyse the long-term root reinforcement dynamic of spruce forests combining data of the Swiss National Forest Inventory with data on root distribution and root mechanical properties. The results show that root reinforcement recovery depends primarily on stand altitude and slope inclination. The maximum root reinforcement recovery rate is reached at circa 100 years. RR increases continuously with different rates for stand ages over 200 years. These results shows that RR in spruce stands varies considerably depending on the local conditions and that its recovery after disturbances requires decades. The new method applied in this study allowed for the first time to quantify the long term dynamics of RR in spruce stands supporting new quantitative approaches for the analysis of shallow landslides disposition in different disturbance regimes of forests

SBV regularity of entropy solutions for a class of genuinely nonlinear scalar balance laws with non-convex flux function

In this work we study the regularity of entropy solutions of the genuinely nonlinear scalar balance laws We assume that the source term g ∈ C1(ℝ × ℝ × ℝ+), that the flux function f ∈ C2(ℝ × ℝ × ℝ+) and that {ui ∈ ℝ : fuu(ui,x,t) = 0} is at most countable for every fixed (x,t) ∈ Ω. Our main result, which is a unification of two proposed intermediate theorems, states that BV entropy solutions of such equations belong to SBVloc(Ω). Moreover, using the theory of generalized characteristics we prove that for entropy solutions of balance laws with convex flux function, there exists a constant C > 0 such that: where C can be chosen uniformly for (x +h,t), (x,t) in any compact subset of Ω

Hamilton-Jacobi equations with obstacles

We consider a problem in the theory of optimal control proposed for the first time by Bressan. We characterize the associated minimum time function using tools from geometric measure theory and we obtain, as a corollary, an existence theorem for a related variational problem

Correction: Flepp et al. Temporal Dynamics of Root Reinforcement in European Spruce Forests. Forests 2021, 12, 815

The authors wish to make the following corrections to this paper [...

Comparisons of the kinematics and deep structures of the Zagros and Himalaya and of the Iranian and Tibetan plateaus and geodynamic implications

International audienceWe compare the geologic histories, the deep structures, and the present-day kinematics of deformation of the Himalaya and the adjacent Tibetan Plateau with those of the Zagros and Iranian Plateau to test geodynamic processes of continental collision. Shortly after India and Arabia collided with Eurasia, horizontal shortening manifested itself by folding and thrust faulting of sedimentary rock detached from India's and Arabia's underlying crystalline basement. Subsequently, slip on thrust faults stacked slices of India's basement to build the Himalaya on India's northern margin. Such faulting has not yet developed in the Zagros, where collision is more recent and Arabia penetrates into Eurasia more slowly than India does, so that postcollision convergence with Eurasia is less. The greater elevation, thicker crust, and more marked heterogeneity of the upper mantle beneath the Tibetan than beneath the Iranian Plateau also reflect a more advanced stage of development. Moreover, while thrust or reverse faulting and crustal shortening continue on the margins of both plateaus, normal faulting, suggesting horizontal extension and crustal thinning, occurs within Tibet but not in Iran. Hence, the balance of forces that built the high Tibetan Plateau must have changed, apparently some time since ∼15 Ma. Removal of Tibetan mantle lithosphere could have altered that balance. If mantle lithosphere beneath the Iranian Plateau has been removed, however, the change in force balance has been too small to initiate normal faulting. Low seismic wave speeds in the uppermost mantle just beneath the Moho of both plateaus suggest (to us) that lithosphere beneath both is thin, consistent with late Cenozoic removal of it, but alternative explanations might account for these low speeds. Despite its apparently thin, and hence presumably weak, mantle lithosphere, much of central Iran undergoes little deformation. It illustrates how a crustal block can behave rigidly not necessarily because it is strong but because deviatoric stresses can be small. Whereas differences between the two regions clearly depend on the amount that Arabia and India have penetrated into Eurasia, which scales with both the dates of collision and rates of convergence, we see no differences in the operative processes that depend on the present-day rates of convergence