Generalized Inclusion-Exclusion

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 block matrices, integrating over hybrid set domains and the convolution of two piecewise interval functions

Abstract Matrix Arithmetic

... matrices, i.e., matrices of symbolic dimension with underspecified components. We define a simple basis function that enables the representation of abstract matrices composed of arbitrary regions in a single term that supports matrix addition and multiplication by regular arithmetic on terms. This can, in particular, be exploited to obtain general arithmetic closure properties for classes of structured matrices. We also describe an approach using alternative basis functions that allow more compact expressions and admit additional arithmetic simplifications