Algebraic geometric methods in studying splines

Sipahi Ös, Neslihan
In this thesis, our main objects of interest are piecewise polynomial functions (splines). For a polyhedral complex $\Delta$ in $\mathbb{\R}^n$, $C^{r}(\Delta)$ denotes the set of piecewise polynomial functions defined on $\Delta$. Determining the dimension of the space of splines with polynomials having degree at most $k$, denoted by $C^r_k(\Delta)$, is an important problem, which has many applications. In this thesis, we first give an exposition on splines and introduce different algebraic geometric methods used to compute the dimension of splines both on polyhedral and simplicial complexes. Then we generalize the important result of Mcdonald and Schenck \cite{McdSch} on planar splines on a polyhedral complex. Also, by using the method in \cite{GeraSch}, we make generalizations on the dimension of the spaces of splines on simplicial complexes in dimension three. This generalizaton includes simplicial complexes having no interior points, and octahedrons with one interior point. In the latter case, we make some generalizations by considering the number of linearly independent interior planes.


On maximal curves and linearized permutation polynomials over finite fields
Özbudak, Ferruh (Elsevier BV, 2001-08-08)
The purpose of this paper is to construct maximal curves over large finite fields using linearized permutation polynomials. We also study linearized permutation polynomials under finite field extensions.
Algebraic Nahm transform for parabolic Higgs bundles on P-1
Aker, Kursat; Szabo, Szilard (2014-01-01)
We formulate the Nahm transform in the context of parabolic Higgs bundles on P-1 and extend its scope in completely algebraic terms. This transform requires parabolic Higgs bundles to satisfy an admissibility condition and allows Higgs fields to have poles of arbitrary order and arbitrary behavior. Our methods are constructive in nature and examples are provided. The extended Nahm transform is established as an algebraic duality between moduli spaces of parabolic Higgs bundles. The guiding principle behind ...
Functional calculus on Noetherian schemes
Dosi, Anar (2015-01-01)
The present note is devoted to the functional calculus problem for sections of a quasi-coherent sheaf on a Noetherian scheme. We prove scheme-theoretic analogs of the known results on the multivariable holomorphic functional calculus over Frechet modules which are mainly due to of J. Taylor and M. Putinar. The generalization of the Taylor joint spectrum considered in the paper leads to subvarieties of an algebraic variety over an algebraically closed field. In particular, every algebraic variety is represen...
Low-power and area-efficient finite field arithmetic architecture based on irreducible all-one polynomials
Mohaghegh, Shima; Muhtaroğlu, Ali; Electrical and Electronics Engineering (2020-9)
This thesis presents a low-power and area-efficient finite field multiplier based on irreducible all-one polynomials (AOP). The proposed organization implements the AOP multiplication algorithm in three stages, which are reduction network, AND network (multiplication), and three input XOR tree (accumulation), while state-of-the-art implementations distribute reduction, multiplication and accumulation operations in a systolic array. The optimization reduces the overall number of sequential elements and provi...
Symbolic polynomial interpolation using Mathematica
Yazıcı, Adnan; Ergenc, T (2004-01-01)
This paper discusses teaching polynomial interpolation with the help of Mathematica. The symbolic power of Mathematica is utilized to prove a theorem for the error term in Lagrange interpolating formula. Derivation of the Lagrange formula is provided symbolically and numerically. Runge phenomenon is also illustrated. A simple and efficient symbolic derivation of cubic splines is also provided.
Citation Formats
N. Sipahi Ös, “Algebraic geometric methods in studying splines,” Ph.D. - Doctoral Program, Middle East Technical University, 2013.