Theses and Dissertations

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Michael DiPasquale, Ph.D.


Loosely speaking, splines are piece-wise polynomial functions which are continuously differentiable of order r. The individual polynomial segments are degree less than or equal to d. Splines are typically associated with approximation theory, computer-aided graphic design, and partial differential equations. Due to the wide variety of applications, mathematicians became interested in better understanding the behavior of the vector spaces containing splines in various setting and values r and d. In particular, many focused on formulating the dimension of these vector spaces. In the 1980s, commutative algebra was first applied to the study of spline spaces. In our work, we use tools from commutative algebra to study of bivariate splines on curved divisions of the plane radiating cyclically from a single vertex. We focus on a particular subdivision where prior algebraic methodology cannot be applied to compute the dimension. Our primary tools include Grobner bases and colon ideals. We make a number of conjectures related to the dimension of splines on this particular subdivision. We are close to proving some of these conjectures but leave this for future work.

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