We study the rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvaturelike quantities for polyhedral surfaces are introduced and are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived from the cosine law. They can be considered as 2-dimensional counterparts of the Schlaefli formula.

The irrationality exponent of an irrational number $\xi$, which measures the approximation rate of $\xi$ by rationals, is in general extremely difficult to compute explicitly, unless we know the continued fraction expansion of $\xi$. Results obtained so far are rather fragmentary and often treated case by case. In this work, we shall unify all the known results on the subject by showing that the irrationality exponents of large classes of automatic numbers and Mahler numbers (which are transcendental) are exactly equal to $2$. Our classes contain the Thue--Morse--Mahler numbers, the sum of the reciprocals of the Fermat numbers, the regular paperfolding numbers, which have been previously considered respectively by Bugeaud, Coons, and Guo, Wu and Wen, but also new classes such as the Stern numbers and so on. Among other ingredients, our proofs use results on Hankel determinants obtained recently by Han.

Recent consumer interest in controlling and preventing chronic diseases through improved diet has promoted research on the bioactive components of agricultural products. Wheat is an important agricultural and dietary commodity worldwide with known antioxidant properties concentrated mostly in the bran fraction. The objective of this study was to determine the relative contributions of genotype (G) and growing environment (E) to hard winter wheat bran antioxidant properties, as well as correlations of these properties to growing conditions. Bran samples of 20 hard winter wheat varieties grown in two locations were examined for their free radical scavenging capacities against DPPH, ABTS cation, peroxyl (ORAC), and superoxide anion radicals and chelating properties, as well as their total phenolics and phenolic acid compositions. Results showed significant differences for all antioxidant properties tested and multiple significant correlations between these properties. A factorial designed analysis of variance for these data and pooled previously published data showed similar results for four of the six antioxidant properties, indicating that G effects were considerably larger than E effects for chelating capacity and DPPH radical scavenging properties, whereas E was much stronger than G for ABTS cation radical scavenging capacity and total phenolics, although small interaction effects (G × E) were significant for all antioxidant properties analyzed. Results also showed significant correlations between temperature stress or solar radiation and some antioxidant properties. These results indicate that each antioxidant property of hard winter wheat bran is influenced differently by genotype and growing conditions.

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Mean field games (MFG) and mean field control (MFC) are critical classes of multi-agent models for efficient analysis of massive populations of interacting agents. Their areas of application span topics in economics, finance, game theory, industrial engineering, crowd motion, and more. In this paper, we provide a flexible machine learning framework for the numerical solution of potential MFG and MFC models. State of-the-art numerical methods for solving such problems utilize spatial discretization that leads to a curse-of-dimensionality. We approximately solve high-dimensional problems by combining Lagrangian and Eulerian viewpoints and leveraging recent advances from machine learning. More precisely, we work with a Lagrangian formulation of the problem and enforce the underlying Hamilton-Jacobi-Bellman (HJB) equation that is derived from the Eulerian formulation. Finally, a tailored neural network parameterization of the MFG/MFC solution helps us avoid any spatial discretization. Our numerical results include the approximate solution of 100-dimensional instances of optimal transport and crowd motion problems on a standard work station and a validation using a Eulerian solver in two dimensions. These results open the door to much-anticipated applications of MFG and MFC models that were beyond reach with existing numerical methods.