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Full-Text Articles in Physical Sciences and Mathematics
Generalized Inclusion-Exclusion, Mike W. Ghesquiere
Generalized Inclusion-Exclusion, Mike W. Ghesquiere
Electronic Thesis and Dissertation Repository
Sets are a foundational structure within mathematics and are commonly used as a building block for more complex structures. Just above this we have functions and sequences before an explosion of increasingly specialized structures. We propose a re-hanging of the tree with hybrid sets (that is, signed multi-sets), as well hybrid functions (functions with hybrid set domains) joining the ranks of sequences and functions. More than just an aesthetic change, this allows symbolic manipulation of structures in ways that might otherwise be cumbersome or inefficient. In particular, we will consider simplifying the product and sum of two piecewise functions or …
Algorithmic Contributions To The Theory Of Regular Chains, Wei Pan
Algorithmic Contributions To The Theory Of Regular Chains, Wei Pan
Electronic Thesis and Dissertation Repository
Regular chains, introduced about twenty years ago, have emerged as one of the major
tools for solving polynomial systems symbolically. In this thesis, we focus on different
algorithmic aspects of the theory of regular chains, from theoretical questions to high-
performance implementation issues.
The inclusion test for saturated ideals is a fundamental problem in this theory.
By studying the primitivity of regular chains, we show that a regular chain generates
its saturated ideal if and only if it is primitive. As a result, a family of inclusion tests
can be detected very efficiently.
The algorithm to compute the regular GCDs …