On a conjecture of R. E. Miles about the convex hull of random points

Denoty byp _d+i (B _d ,d+m) the probability that the convex hull ofd+m points chosen independently and uniformly from ad-dimensional ballB _d possessesd+i(i=1,...,m) vertices. We prove Mile's conjecture that, given any integerm, p _d+m (B _d ,d+m)»1 asd». This is obvious form=1 and was shown by Kingman form=2 and by Miles form=3. Further, a related result by Miles is generalized, and several consequences are deduced.Dedicated to Professor E. Halwaka on the occasion of his seventieth     

Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations

High-dimensional partial differential equations (PDEs) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work, we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (1) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (2) a merged formulation of the PDE and the 2BSDE problem, (3) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (4) a stochastic gradient descent-type optimization procedure. Numerical results obtained using TensorFlow in Python illustrate the efficiency and the accuracy of the method in the cases of a 100-dimensional Black–Scholes–Barenblatt equation, a 100-dimensional Hamilton–Jacobi–Bellman equation, and a nonlinear expectation of a 100-dimensional G-Brownian motion.

     

Back to the future: A retrospective assessment of model‐based scenarios for the management of the shrimp fishery in French Guiana facing global change

While the number of models dedicated to predicting the consequences of alternative resource management strategies has increased, instances in which authors look back at past predictions to learn from discrepancies between these and observed developments are scarce. In the past decades, the French Guiana shrimp fishery has experienced shrimp market globalization and decreasing levels of shrimp recruitment due to environmental changes. In 2006, a bio‐economic model of this fishery was developed to simulate its possible responses to economic and environmental scenarios up to 2016. Here, we compare here these predictions to the observed trajectories. While the number of active vessels corresponds to that which was predicted, the estimated shrimp stock does not. Important driving factors had not been anticipated, including a general strike, natural disasters, and the end of the global financial crisis. These results show the importance of participative approaches involving stakeholders in the co‐construction and shared representation of scenarios. Recommendations for resource managers

• Effective fisheries resources management and a fortiori, the capacity of the fisheries to adapt to global change, requires understanding of both ecological and economics dynamics.
• The temporal trajectory of the trawling shrimp fisheries has been well monitored, and the decline of both stock and fleet is understood regarding ecological and economic changes: Changes in the environmental conditions of shrimp recruitment, and oil price increase and selling price decrease.
• However, our bio‐economic modeling work showed that, even with a good understanding of the dynamics explaining past trajectories, unpredictable events (strike, natural disasters…) have acted as other key driving factors altering the capacity of the model to represent possible futures.
• These results led us to recommend a better integration of the expertise of social and political scientists in developing models of bio‐economic systems to increase the quality of scenario predictions, and to argue for more participative approaches involving the stakeholders.

     

DEM-FEM coupled numerical investigation of granular materials to increase crashworthiness of double-hull vessels

In this contribution, a novel design strategy is investigated for improving the crashworthiness of double-hull vessels, which involves usage of granular materials. For numerical simulation, a framework based on coupling of the Discrete Element and the Finite Element Method is used to study load bearing capacity of granular materials during uniaxial compression. In addition, the effect of Young's modulus of particles is also investigated with respect to the load transferability. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Unexpected Formation of a 2,2′-Di( N -ethyl-acetamino)substituted Diphenyl Disulfane on Oxidation of 3-Ethyl-2-methylbenzothiazolium Tetrafluoroborate

Attempts at the oxidation of 3-ethyl-2-methylbenzothiazolium salt 2 with a variety of oxidizing reagents did not lead to the desired isochiral S -oxide 3 or achiral S , S -dioxide 4 , in some cases, however, unexpectedly to the ring-opened dimeric 2,2'-di( N -ethyl-acetamino)substituted diphenyl disulfane 5 , the molecular structure of which was confirmed by x-ray analysis. The synthesis of 2-methylbenzothiazole- S , S -dioxide 14 , reported by Zincke et al. in 1915, turned out to be not reproducible. \centerline{\epsfbox{:art:fig-01.jpg}}     

Short time deposition of TiO2 nanoparticles on SiC as photocatalysts for the degradation of organic dyes

The deposition of TiO₂ nanoparticles on SiC was carried out by mechanical milling under different conditions. SiC–TiO₂ samples were used as photocatalysts for the degradation of organic dyes such as methylene blue and rhodamine B. A short time deposition of TiO₂ nanoparticles was observed during mechanical milling (2 min at 200 rpm) to cover the SiC particles. The presence of SiC and TiO₂ (anatase and rutile) was confirmed by means of X-ray diffraction after thermal treatment at 450 °C. The deposition of TiO₂ on SiC was corroborated by scanning electron microscopy analysis; the thickness of the thin layer of TiO₂ deposited on SiC increases as the proportion of TiO₂ increases. The energy band gap values obtained for these compounds were around 3.0 eV. SiC–TiO₂ photocatalysts prepared by mechanical milling exhibited better activity under UV-light irradiation for the degradation of methylene blue and rhodamine B than commercial TiO₂ powder (titania P25).