Covariant Functors and Shapes in the Category of Compacts

In this paper, we consider covariant functors F : Comp → Comp acting in category of shape-preserving compact sets [2], infinite compact sets, and shape equivalence [9]. Also we study action of compact functors and shape properties of the compact space X consisting of connected components ОX of the compact X as well as shape identity ShX = ShY of infinite compacts X and Y for the space P (X) of probability measures and its subspaces.

Beyond Alternating Permutations: Pattern Avoidance in Young Diagrams and Tableaux

We investigate pattern avoidance in alternating permutations and generalizations thereof. First, we study pattern avoidance in an alternating analogue of Young diagrams. In particular, we extend Babson-West's notion of shape-Wilf equivalence to apply to alternating permutations and so generalize results of Backelin-West-Xin and Ouchterlony to alternating permutations. Second, we study pattern avoidance in the more general context of permutations with restricted ascents and descents. We consider a question of Lewis regarding permutations that are the reading words of thickened staircase Young tableaux, that is, permutations that have $k-1$ ascents followed by a descent, followed by $k-1$ ascents, et cetera. We determine the relative sizes of the sets of pattern-avoiding $(k-1)$-ascent permutations in terms of the forbidden pattern. Furthermore, inequalities in the sizes of sets of pattern-avoiding permutations in this context arise from further extensions of shape-equivalence type enumerations. This paper is the first of a two-paper series presenting the work of Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux (arXiv:1301.6796v1). The second in the series is Ascent-descent Young diagrams and pattern avoidance in alternating permutations (by the second author, submitted).

Localization between Curved Shell Plate and Its Unfolded Shape in Different Coordinate Systems for Ship-Hull Plate Forming

This paper describes a mathematical formulation for the efficient localization of 3D surfaces including free-form surfaces and flat surfaces. An important application of this paper is to register flat surface calculated from unfolding process with a curved surface extracted from ship CAD prior to the multipoint press forming works. The mathematical formulation handles the registration and comparison of two free surfaces represented by sparse points based on the iterative closest point (ICP) algorithm and localization that can be applicable to ship-hull plate forming. The ICP algorithm gives an adequate set of initial translation and rotation for surface objects with little correspondence through the minimization of mean square distance metric. Comparison of surfaces is explained in order to determine a corresponding set which gives the optimized press stroke between unfold surface and referential object surface. It thereby allows the optimized press works in ship-hull forming. The combination of registration and comparison is applied to decide the shape equivalence of correspondent surfaces as well as to estimate the transform matrix between point sets where similarity is low. Experimental results show the capabilities of the registration on unfolding surface and curved surface.

A Tuning Mechanism for Feature Recognition

In this paper a flexible feature recognition mechanism that can be tuned by the user in order to meet the specific application requirements, is presented. The recognition process is performed at different levels of detail and is based on topological, morphological and geometric reasoning. Invariant shape characteristics are studied and used for defining shape equivalence relations among features, on which the tuning mechanism provided to the user is based. The system supports the recognition of both classes and specific instances of features.