Basis criteria for generalized spline modules via determinant

GENERALIZED PRINCIPAL IDEAL THEOREM FOR MODULES

The Generalized Principal Ideal Theorem is one of the cornerstones of dimension theory for Noetherian rings. For an R-module M, we identify certain submodules of M that play a role analogous to that of prime ideals in the ring R. Using this definition, we extend the Generalized Principal Ideal Theorem to modules.

An algorithm for constructing a basis for Cr-spline modules over polynomial rings

Let 2 be a polyhedral complex embedded in the euclidean space Ed and Sr(2), r ≥ 0, denote the set of all Cr-splines on 2. Then Sr(2) is an R-module where R = E[x1, . . . , xd] is the ring of polynomials in several variables. In this paper we state and prove the existence of an algorithm to write down a free basis for the above R-module in terms of obvious linear forms defining common faces of m...

Generalized B-spline functions ‎method‎‎ for solving optimal control problems

‎In this paper we introduce a numerical approach that solves optimal control problems (OCPs) ‎using collocation methods‎. ‎This approach is based upon B-spline functions‎. ‎The derivative matrices between any two families of B-spline functions are utilized to‎ ‎reduce the solution of OCPs to the solution of nonlinear optimization problems‎. ‎Numerical experiments confirm our heoretical findings‎.

Generalized Derivations on Modules

Stability of the B-spline basis via knot insertion

We derive the stability inequality ‖C‖ 6 γ ‖∑i cibi‖ for the B-splines bi from the formula for knot insertion. The key observation is that knot removal increases the norm of the B-spline coefficients C = {ci}i∈Z at most by a constant factor, which is independent of the knot sequence. As a consequence, stability for splines follows from the stability of the Bernstein basis.  2000 Elsevier Scien...