18,841 research outputs found
The slopes determined by n points in the plane
Let , , ..., be the slopes of the
lines connecting points in general position in the plane. The ideal
of all algebraic relations among the defines a configuration space
called the {\em slope variety of the complete graph}. We prove that 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
Evidence-Based Practice and the Use of Information in State Agency Decision-Making
The contemporary policy environment makes persistent demands on agency officials to use the best information available when making decisions about policies, programs, and practices. State and federal legislation calls on agencies to incorporate evidence-based practices in their programs. Using data from a 2008 survey of state agency administrators, we examine the extent to which state government agencies draw upon various sources of information to guide their decisions about programmatic operations. Our findings reveal the extent to which agencies rely on, or weight, scientific studies and formal evaluations compared to other sources. The paper offers new insights into the use of information, particularly scientific evidence, in state agency decision-making, offering the first systematic look at how widely such information is used.
Nonconcentration in partially rectangular billiards
In specific types of partially rectangular billiards we estimate the mass of
an eigenfunction of energy in the region outside the rectangular set in the
high-energy limit. We use the adiabatic ansatz to compare the Dirichlet energy
form with a second quadratic form for which separation of variables applies.
This allows us to use sharp one-dimensional control estimates and to derive the
bound assuming that is not resonating with the Dirichlet spectrum of the
rectangular part.Comment: 28 pages, 1 figure; Typos corrected, error in final proof corrected
and techniques strengthene
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