Quasi extended Chebyshev spaces (QEC-spaces for short) represent the largest class of spaces of sufficient regularity which are suitable for design [1]. In particular, they contain the well-known extended Chebyshev spaces as a special instance (and therefore can represent transcendental functions) as well as the so called variable degree polynomial spaces [2], that are very effective in handlin...

In this paper we study the classifying theory of principal bundles in the parametrized setting, motivated by recent interest in higher gauge theory. Using simplicial techniques, we construct a product-preserving classifying space functor for groups in the category of spaces over a fixed space B. Additionally, we prove that the fiberwise geometric realization functor sends a large class of simpl...

We present a Bayesian approach to comparison of geometric shapes with applications to classification of the molecular structures of proteins. Our approach involves the use of distributions defined on transformation invariant shape spaces and the specification of prior distributions on bipartite matchings. Here we emphasize the computational aspects of posterior inference arising from such model...

The highly influential framework of conceptual spaces provides a geometric way of representing knowledge. Instances are represented by points in a high-dimensional space and concepts are represented by convex regions in this space. After pointing out a problem with the convexity requirement, we propose a formalization of conceptual spaces based on fuzzy star-shaped sets. Our formalization uses ...

The geometric aspects of mirror symmetry are reviewed, with an eye towards future developments. Given a mirror pair (X, Y ) of Calabi–Yau threefolds, the best-understood mirror statements relate certain small corners of the moduli spaces of X and of Y . We will indicate how one might go beyond such statements, and relate the moduli spaces more globally. In fact, in the boldest version of mirror...

Using techniques from computational differential geometry, we present a new approach to the algorithmic study of planar curves, with applications to shape and image analysis. The main idea is to construct spaces of curves satisfying constraints suited to specific problems and exploit the geometric structure of these spaces to quantify and analyze properties of contours and solve optimization an...

In this paper we shall construct master spaces for certain coupled vector bundle problems over a fixed projective variety X. These master spaces, which are moduli spaces for oriented torsion free coherent sheaves coupled with morphisms into a fixed reference sheaf E 0 , have the structure of polarized projective varieties endowed with a natural C *-action. The fixed point set of this action con...