2015/04/30 by Graeme W. Milton, Milton, Graeme W. · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Algebraic Geometry (math.AG) #Composite Material Mechanics #FOS: Mathematics #Numerical methods in engineering #math.AG
paper · pdf · doi:10.48550/arxiv.1504.08061
This has 60 pages and 8 figures. It will be Chapter 7 of the book "Extending the Theory of Composites to other Areas of Science", edited by Graeme W. Milton
openalex publication_date 2015/04/30 · arxiv created 2016/02/21 · arxiv updated 2016/02/23 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
A natural connection between rational functions of several real or complex variables, and subspace collections is explored. A new class of function, superfunctions, are introduced which are the counterpart to functions at the level of subspace collections. Operations on subspace collections are found to correspond to various operations on rational functions, such as addition, multiplication and substitution. It is established that every rational matrix valued function which is homogeneous of degree 1 can be generated from an appropriate, but not necessarily unique, subspace collection: the mapping from subspace collections to rational functions is onto, but not one to one. For some applications superfunctions may be more important than functions, as they incorporate more information about the physical problem, yet can be manipulated in much the same way as functions. Previously subspace collections had been introduced when there was an inner product on the vector (or Hilbert) space, and appropriate subspaces were mutually orthogonal. In that setting certain normalization and reduction operations on subspace collections led to a continued fraction expansion of the associated function, which allowed one to bound the function in terms of a set of weight matrices and normalization matrices that are derived from series expansions. Here we also initiate the theory of normalization and reduction operations, appropriate when there is no inner product on the space.