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Saurabh Bansal and James S. Dyer study a common problem in the commercial agribusiness market, where farmers have a preference for a farm input such as a seed based on a fit with their geographical location but are also willing to accept a closely related substitute. Such consumer-driven choices may not be adequately represented by traditional models that maximize the profit of a firm that seeks to make substitutions while maximizing its profit. They use a set of recent results for evaluations of moments over polyhedra to determine the exact inventory levels a firm should keep of substitutable products. Using proprietary data from a large firm in this domain, they highlight the role of geographical and climate-related factors that affect product substitution in the agribusiness industry and identify specific regions in the United States where product substitution is a source of substantial revenue for firms. In this paper, we consider the problem in which a firm offers a portfolio of products (agricultural seeds) to multiple customer segments comprising farmers under aggressive fill-rate constraints, and some, but not all, customers will accept a substitute to their preferred choice. This business situation is not adequately represented by traditional inventory-management models, where a firm initiates a substitution based on its monetary considerations. By exploiting some recent results on polyhedral expectations, we develop a decomposition-based approach to determine optimal inventory levels for the firm's seed portfolio under aggressive fill-rate targets. The approach provides an exact solution that is implementable in managerial-friendly environments and permits a what-if analysis for real-time decision support. Subsequently we extend the technical development to establish: (i) a simple computable bound on the value of substitution, (ii) a procedure for determining implied penalty costs for substitutable products, and (iii) comparative static results for the product portfolio. We also discuss the implementation of the technical development at a Fortune 100 firm that has resulted in significant monetary savings. Finally, we provide geography- and climate-specific managerial insights for managing seed substitution by end-users. [ABSTRACT FROM AUTHOR]

Apexite (IMA2015-002), NaMg(PO4)⋅9H2O, is a new mineral from the Apex mine, Lander County, Nevada, U.S.A., where it occurs as a low-temperature secondary mineral on massive quartz matrix in association with andersonite, calcite, čejkaite, gaylussite, and goethite. Apexite forms colorless needles up to 0.5 µm in length. The streak is white. Crystals are transparent and have vitreous to satiny luster. The Mohs hardness is about 2, the tenacity is brittle, the fracture is curved, and crystals exhibit one perfect cleavage on {100}. The measured density is 1.74(1) g/cm³ and the calculated density is 1.741 g/cm³. Electron microprobe analyses provided: Na2O 9.26, MgO 14.42, P2O5 23.31, H2O 53.01 (structure), total 100.00 wt% (normalized). The empirical formula (based on 13 O apfu) is: Na0.91Mg1.09P1.00O13.00H17.91. Apexite is triclinic, P..., a = 6.9296(7), b = 11.9767(13), c = 14.9436(19) Å, α = 92.109(6), β = 102.884(7), γ = 105.171(7)°, V = 1160.9(2) ų, and Z = 4. The eight strongest lines in the X-ray powder diffraction pattern are [dobs in Å(I)(hkl)]: 14.63(35)(001); 5.11(61)(021,...1,110,...20); 4.68(75)(0...2,...2,1...1,0...3); 4.301(96)(102,013,022,...13); 4.008(44) (...3,1...2); 2.876(46)(040); 2.762(100)(...3,2...1,0...4,...04,015); and 2.507(30)(212,025,...3). Apexite is a new struvite-type phase with a unique structure (R1 = 4.44% for 1401 Fo > 4σF) consisting of four components: (1) a [Na2Mg4(H2O)14]10+ heteropolyhedral cation cluster; (2) a trans edge-sharing chain of Na(H2O)6 octahedra; (3) an isolated PO4 group; and (4) an isolated H2O group. The structural components are linked to one another only via hydrogen bonds. Its structure is related to that of hazenite. [ABSTRACT FROM AUTHOR]

Abstract: A US Federal election in which candidates from two major political parties compete for the votes of those undecided voters in a state who usually do not vote in US elections is considered. A mathematical model for evaluating the expectation of the margin of votes to be received from such voters by either candidate as a result of the election campaigns of all the competing candidates is proposed. On the basis of this model, finding the estimation under consideration is reducible to finding the minimum of the maximin function of the difference of two bilinear functions with one and the same first vector argument whose second vector arguments belong to a polyhedron of connected variables (strategies of the candidates), and this minimum is sought on another polyhedron. [Copyright &y& Elsevier]

The article discusses the rotational symmetries of the regular polyhedra which can be incorporated into a high school geometry classroom in the U.S. where students will have an opportunity to further their knowledge about two-dimensional rotations through the study of three-dimensional rotations. A summary of the properties of the five regular polyhedra, including the duality between the regular polyhedra is presented. This activity also provides students the opportunity to connect algebra concepts such as 1-1 mapping and algebraic transformations.

Focuses on the spatial understanding of children in the U.S. Creation of polyhedrons; Description of the polyhedrons; Introduction of closed structures concept.

Features activities that can help students see the connections between polyhedra and tessellations, an arrangement of shapes that will cover an entire plane. Link between soccer fields and mathematics; Tessellations and the measures of the interior angles of polygons; Tessellation and regular polyhedra; Regular polyhedra and semiregular polyhedra; Grass tessellation of the Silverdome in Detroit, Michigan.

Debreu's fundamental result on the existence of utility functions that are continuous with respect to the order topology requires his Gap Theorem. A new proof of this result is given. [ABSTRACT FROM AUTHOR]