Decompositions of a $T_1$ space

DECOMPOSITIONS OF A Ti SPACE

Introduction. Several authors have proved theorems of the type: The "structure" of a certain class of transformations defined on a "suitable" space A to a fixed "suitable" space B "determines" the space A. As examples, we have: Banach [3, p. 170, see also 6, 7, 13] i The Banach space of all real, continuous functions defined on a compact metric space A "determines" A. Eidelheit [5, see also 2, ...

Maintenance of Geometric Representations through Space Decompositions

The ability to transform between distinct geometric representations is the key to success of multiple-representation modeling systems. But the existing theory of geometric mod-eling does not directly address or support construction, conversion, and comparison of geometric representations. A study of classical problems of CSG $ b-rep conversions, CSG optimization, and other representation conver...

Hierarchical Space Decompositions for Low-Density Scenes

Combinatorial Cellular Decompositions for the Space of Complex Coefficient Polynomials

We describe a classification of degree n complex coefficient polynomials with respect to combinatorial patterns that arise from the two real algebraic curves obtained as the zero sets for their real and imaginary part. In particular, we work out explicitly this classification for degree 3 polynomials, and other special families of polynomials. This work extends to the singular case similar cons...

Product Space Decompositions for Continuous Representations of Brain Connectivity

We develop a method for the decomposition of structural brain connectivity estimates into locally coherent components, leveraging a non-parametric Bayesian hierarchical mixture model with tangent Gaussian components. This model provides a mechanism to share information across subjects while still including explicit mixture distributions of connections for each subject. It further uses mixture c...