Capable Lie algebras with the derived subalgebra of dimension 2 over an arbitrary field

In this paper, we classify all capable nilpotent Lie algebras with the derived subalgebra of dimension 2 over an arbitrary field. Moreover, the explicit structure of such Lie algebras of class 3 is given.     

A calibrated model of local food system of Hawaii: What are the economic implications of the state's food goals and policies?

We develop a fully calibrated positive mathematical programming model for Hawaii's local food systems—which captures the production and the consumer sides of the market. Then we use the model to assess two proposed policies—a general excise tax (GET) exemption on locally produced foods, and an investment in agricultural infrastructure. For the GET exemption case, our results indicate an economic gain of $118 per $100 cost. On the other hand, an investment in 1,200 acres of land injected to support local production may generate an economic gain of up to $357 per $100 annual cost of the investment. However, these estimates should be considered preliminary, and thus viewed with caution. Although the model is used to capture Hawaii's local food systems, we believe that our model is generalizable and can be adopted by other economies to assess their respective food localization policies. Recommendations for Resource Managers

• Local food policies need to be based on quantitative terms instead of mere armchair speculation because often their potential outcomes may vary significantly.

• The current modeling framework demonstrates the potential of using positive mathematical programming (PMP) in capturing the intricacies of local food systems. However, this exploratory exercise should be viewed as preliminary in nature and the ensuing results were taken with caution because many important factors such as labor availability may have been left out.

• Thus, further model refinements are necessary to better capture the complexities of local food systems such as farm heterogeneity, availability of farm labor, water availability, and interisland transportation of farm products in the case of Hawaii.

     

Closed‐form approximations for spread options in Lévy markets

We provide new closed‐form approximations for the pricing of spread options in three specific instances of exponential Lévy markets, ie, when log‐returns are modeled as Brownian motions (Black‐Scholes model), variance gamma processes (VG model), or normal inverse Gaussian processes (NIG model). For the specific case of exchange options (spread options with zero strike), we generalize the well‐known Margrabe formula (1978) that is valid in a Black‐Scholes model to the VG model under a homogeneity assumption.     

Weighted estimates for the multilinear maximal function on the upper half‐spaces

For a general dyadic grid, we give a Calderón–Zygmund type decomposition, which is the principle fact about the multilinear maximal function on the upper half‐spaces. Using the decomposition, we study the boundedness of . We obtain a natural extension to the multilinear setting of Muckenhoupt's weak‐type characterization. We also partially obtain characterizations of Muckenhoupt's strong‐type inequalities with one weight. Assuming the reverse Hölder's condition, we get a multilinear analogue of Sawyer's two weight theorem. Moreover, we also get Hytönen–Pérez type weighted estimates.     

Fractional powers of the noncommutative Fourier's law by the S‐spectrum approach

Let e_?, for ? = 1,2,3, be orthogonal unit vectors in and let be a bounded open set with smooth boundary ?Ω. Denoting by a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: into the conservation of energy law, here a, b, are given functions. With the S‐spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, the fractional powers of T exist in the sense of the S‐spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory.     

Existence of strong minimizers for the Griffith static fracture model in dimension two

We consider the Griffith fracture model in two spatial dimensions, and prove existence of strong minimizers, with closed jump set and continuously differentiable deformation fields. One key ingredient, which is the object of the present paper, is a generalization to the vectorial situation of the decay estimate by De Giorgi, Carriero, and Leaci. This is based on replacing the coarea formula by a method to approximate $SB{D}^p$ functions with small jump set by Sobolev functions, and is restricted to two dimensions. The other two ingredients will appear in companion papers and consist respectively in regularity results for vectorial elliptic problems of the elasticity type and in a method to approximate in energy $GSB{D}^p$ functions by $SB{V}^p$ ones.