Narrative Strategies in Benedikte Naubert's Neue Volksmarchen der Deutschen

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The slopes determined by n points in the plane

Let $m_{12}$, $m_{13}$, ..., $m_{n-1,n}$ be the slopes of the $\binom{n}{2}$ lines connecting $n$ points in general position in the plane. The ideal $I_n$ of all algebraic relations among the $m_{ij}$ defines a configuration space called the {\em slope variety of the complete graph}. We prove that $I_n$ is reduced and Cohen-Macaulay, give an explicit Gr\"obner basis for it, and compute its Hilbert series combinatorially. We proceed chiefly by studying the associated Stanley-Reisner simplicial complex, which has an intricate recursive structure. In addition, we are able to answer many questions about the geometry of the slope variety by translating them into purely combinatorial problems concerning enumeration of trees.Comment: 36 pages; final published versio

Geometry of graph varieties

A picture P of a graph G = (V,E) consists of a point P(v) for each vertex v in V and a line P(e) for each edge e in E, all lying in the projective plane over a field k and subject to containment conditions corresponding to incidence in G. A graph variety is an algebraic set whose points parametrize pictures of G. We consider three kinds of graph varieties: the picture space X(G) of all pictures, the picture variety V(G), an irreducible component of X(G) of dimension 2|V|, defined as the closure of the set of pictures on which all the P(v) are distinct, and the slope variety S(G), obtained by forgetting all data except the slopes of the lines P(e). We use combinatorial techniques (in particular, the theory of combinatorial rigidity) to obtain the following geometric and algebraic information on these varieties: (1) a description and combinatorial interpretation of equations defining each variety set-theoretically; (2) a description of the irreducible components of X(G); and (3) a proof that V(G) and S(G) are Cohen-Macaulay when G satisfies a sparsity condition, rigidity independence. In addition, our techniques yield a new proof of the equality of two matroids studied in rigidity theory.Comment: 19 pages. To be published in Transactions of the AM

Lignosulfonate as a Strength Additive for Non-Wood Paperboard

Recycle mills that use old corrugated cardboard (OCC) in their furnish experience difficulties in maintaining the quality of the paperboard produced. Recycle mills using the OCC collect their OCC from many parts of the world. Countries such as China and Japan use rice fibers in the production of corrugated cardboard. Other countries use straw as a fiber source. The end result is that OCC in the United States contains a portion of non-wood fibers as well as the typical wood fibers. Paperboard containing these non-wood fibers typically has lower strength properties than paperboard produced from pure wood fibers. Literature suggests that lignosulfonate compounds can be used as a strength agent for recycled wood fiber paperboards. Calcium lignosulfonate is readily available and is not costly and would prove to be an ideal strength agent for use in recycled paperboard. The objective of this project was to test calcium lignosulfonate as a strength agent in improving the runnability and strength properties on paperboard made from wheat straw paperboard and/or paperboard containing a mixture of wheat straw and wood fibers. Handsheets (120g/m2 ) were prepared on a Noble and Wood handsheet machine. The handsheets from each furnish were then immersed in a bath of calcium lignosulfonate followed by an immersion in kymene. Calcium lignosulfonate levels were varied in the bath in order to control the amount of calcium lignosulfonate applied to each handsheet. The results show that as far as recycled pulp is concerned, CaLS at 10% is definitely beneficial compared with no CaLS in all strength properties. In the case of straw paperboard, 10% CaLS definitely gives higher strength properties compared with no CaLS (except for burst and Scott bond). Higher CaLS levels (10% or 20%) may be justified only in the case of ring crush. As for mixed fiber paperboard, CaLS seems to yield better strength properties (except in the case of Scott bond and burst). While 10% CaLS still seems to be sufficient, 20% seems to result in better crushing resistance and stiffness. The conclusion of this project is that 10% CaLS yields better strength properties in most of the cases and can be the starting point for further refinement studies

The Possible Detection of Dark Energy on Earth Using Atom Interferometry

This paper describes the concept and the beginning of an experimental investigation of whether it is possible to directly detect dark energy density on earth using atom interferometry. The concept is to null out the gravitational force using a double interferometer. This research provides a non-astronomical path for research on dark energy. The application of this method to other hypothetical weak forces and fields is also discussed. In the the final section I discuss the advantages of carrying out a dark energy density search in a satellite in earth orbit where more precise nulling of gravitational forces can be achieved