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Let q be a prime number. The number of subgroups of order qk in an abelian group G of order qn and type λ is a polynomial in q, [kλ′]q. In 1987, Lynne Butler showed that the first difference, [kλ′] - [k - 1λ′], has nonnegative coefficients as a polynomial in q, when 2k ≤ |λ|. We generalize the first difference to the rth difference, and give conditions for the nonnegativity of its coefficients. © 1995.
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Let X and Y be varieties over a field k; π : X → Y is a quadruple cover of Y if π*script O signX is a locally free, rank 4 script O signY-algebra. If char k ≠ 2, we see that π*script O signX splits as script O signY ⊕ε where ε is a locally free rank 3 sheaf over script O signY, which is locally the "trace zero" module. For each y ∈ Y, we therefore have a rank 4 associative, commutative algebra over script O signY,y. We find that these algebras are parametrized by an affine cone over the Grassmanian G(2, 6) with vertex corresponding to the algebra k[x, y, z]/(x, y, z)2. We then show that a quadruple cover with trace zero module ε over a variety Y is determined by a totally decomposable section η ∈ H0(∧2S2 E* ⊗ ∧3 E). We then examine the case in which the section η has no zeros. Here, each rank 4 algebra may be associated to a pencil of conics. As a special case of this, we look at the work of G. Casnati and T. Ekedahl on Gorenstein covers, and we show that their analysis is the subcase where the pencil of conics has length 4 base locus. Finally, we study the case in which the trace zero module ε is split. In this context, Galois covers, which are covers induced by the action of a group of order 4 on the covering variety X, are also studied.
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Kirillov and Reshetikhin introduced rigged configurations as a new way to calculate the entries of the Kostka matrix. Macdonald defined the two-parameter Kostka matrix whose entries generalize. We use rigged configurations and a formula of Stembridge to provide a combinatorial interpretation of in the case where is a partition with no more than two columns. In particular, we show that in this case, has nonnegative coefficients. © 1995 American Mathematical Society.
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This paper presents a fresh approach to a general education mathematics course. The basic idea is to turn the customary mathematics class on its head by focusing on applications first, through a reading of articles from magazines and newspapers, and then turning to the technical mathematical details. A description of the topics that were covered in my course, along with references to various publications, is given. A course such as this one is important because it conveys how mathematics is serving the goals of society. © 1994 Taylor and Francis Group, LLC.
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One of the problems which many companies face is the distribution of inventory from one or more centrally located distribution centers. The retail, industry, and in particular the women's clothing industry, faces the additional problem that not only does the merchandise have to be distributed but it must be distributed in such a way that every store gets a reasonable spread of colors and sizes. This research attempts to optimize the multi-criteria allocation objective by developing a computer algorithm. In order to be implemented any algorithm developed must be computationally efficient due to the size of the industry and other system related programming constraints. The algorithm which was developed not only provided a solution which was a marked improvement regarding the color spread of the merchandise but was also efficient enough to be immediately implemented on a national basis. © 1992.
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