Browsing by Subject "splines"
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Item Homological algebra and problems in combinatorics and geometry(Texas A&M University, 2007-09-17) Tohaneanu, Stefan OvidiuThis dissertation uses methods from homological algebra and computational commutative algebra to study four problems. We use Hilbert function computations and classical homology theory and combinatorics to answer questions with a more applied mathematics content: splines approximation, hyperplane arrangements, configuration spaces and coding theory. In Chapter II we study a problem in approximation theory. Alfeld and Schumaker give a formula for the dimension of the space of piecewise polynomial functions (splines) of degree d and smoothness r. Schenck and Stiller conjectured that this formula holds for all d 2r + 1. In this chapter we show that there exists a simplicial complex such that for any r, the dimension of the spline space in degree d = 2r is not given by this formula. Chapter III is dedicated to formal hyperplane arrangements. This notion was introduced by Falk and Randell and generalized to formality by Brandt and Terao. In this chapter we prove a criteria for formal arrangements, using a complex constructed from vector spaces introduced by Brandt and Terao. As an application, we give a simple description of formality of graphic arrangements in terms of the homology of the flag complex of the graph. Chapter IV approaches the problem of studying configuration of smooth rational curves in P2. Since an irreducible conic in P2 is a P1 (so a line) it is natural to ask if classical results about line arrangements in P2, such as addition-deletion type theorem, Yoshinaga criterion or Terao's conjecture verify for such configurations. In this chapter we answer these questions. The addition-deletion theorem that we find takes in consideration the fine local geometry of singularities. The results of this chapter are joint work with H. Schenck. In Chapter V we study a problem in algebraic coding theory. Gold, Little and Schenck find a lower bound for the minimal distance of a complete intersection evaluation codes. Since complete intersections are Gorenstein, we show a similar bound for the minimal distance depending on the socle degree of the reduced zero-dimensional Gorenstein scheme. The results of this chapter are a work in progress.Item Piecewise polynomial functions on a planar region: boundary constraints and polyhedral subdivisions(Texas A&M University, 2006-08-16) McDonald, Terry LynnSplines are piecewise polynomial functions of a given order of smoothness r on a triangulated region (or polyhedrally subdivided region) of Rd. The set of splines of degree at most k forms a vector space Crk() Moreover, a nice way to study Cr k()is to embed n Rd+1, and form the cone b of with the origin. It turns out that the set of splines on b is a graded module Cr b() over the polynomial ring R[x1; : : : ; xd+1], and the dimension of Cr k() is the dimension o This dissertation follows the works of Billera and Rose, as well as Schenck and Stillman, who each approached the study of splines from the viewpoint of homological and commutative algebra. They both defined chain complexes of modules such that Cr(b) appeared as the top homology module. First, we analyze the effects of gluing planar simplicial complexes. Suppose 1, 2, and = 1 [ 2 are all planar simplicial complexes which triangulate pseudomanifolds. When 1 \ 2 is also a planar simplicial complex, we use the Mayer-Vietoris sequence to obtain a natural relationship between the spline modules Cr(b), Cr (c1), Cr(c2), and Cr( \ 1 \ 2). Next, given a simplicial complex , we study splines which also vanish on the boundary of. The set of all such splines is denoted by Cr(b). In this case, we will discover a formula relating the Hilbert polynomials of Cr(cb) and Cr (b). Finally, we consider splines which are defined on a polygonally subdivided region of the plane. By adding only edges to to form a simplicial subdivision , we will be able to find bounds for the dimensions of the vector spaces Cr k() for k 0. In particular, these bounds will be given in terms of the dimensions of the vector spaces Cr k() and geometrical data of both and . This dissertation concludes with some thoughts on future research questions and an appendix describing the Macaulay2 package SplineCode, which allows the study of the Hilbert polynomials of the spline modules.