This paper introduces a new family of models of intensional Martin-L "of type theory. We use constructive ordered algebra in toposes. Identity types in the models are given by a notion of Moore path. By considering a particular gros topos, we show that there is such a model that is non-truncated, i.e. contains non-trivial structure at all dimensions. In other words, in this model a type in a nested sequence of identity types can contain more than one element, no matter how great the degree of nesting. Although inspired by existing non-truncated models of type theory based on simplicial and on cubical sets, the notion of model presented here is notable for avoiding any form of Kan filling condition in the semantics of types.
Spitters [Section 3] SpittersB:cubstt uses a somewhat different formulation of Moore path in the cubical topos @cite_2 . His notion is the reflexive-transitive closure of the usual path types given by the bounded interval. For better properties and to get a closer relationship with our version, one would like to quotient these cubical Spitters-Moore'' path objects up to degenerate paths; but the undecidability of degeneracy seems to stop one being able to do that while retaining the (uniform) Kan-fibrancy of such path objects. Here we can side-step such issues, since notably our models manage to avoid using a notion of Kan fibrancy at all.
@cite_2
2964029175 2613622891 2002195055 2043690242
Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky's univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly provable in the theory. This paper describes a constructive semantics, expressed in a presheaf topos with suitable structure inspired by cubical sets, of some higher inductive types. It also extends cubical type theory by a syntax for the higher inductive types of spheres, torus, suspensions, truncations, and pushouts. All of these types are justified by the semantics and have judgmental computation rules for all constructors, including the higher dimensional ones, and the universes are closed under these type formers. Using the Minimal Model Program, any degeneration of K-trivial varieties can be arranged to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities. In the hyper-K "ahler setting, we can then deduce a finiteness statement for monodromy acting on @math , once one knows that one component of the central fiber is not uniruled. Independently of this, using deep results from the geometry of hyper-K "ahler manifolds, we prove that a finite monodromy projective degeneration of hyper-K "ahler manifolds has a smooth filling (after base change and birational modifications). As a consequence of these two results, we prove a generalization of Huybrechts' theorem about birational versus deformation equivalence, allowing singular central fibers. As an application, we give simple proofs for the deformation type of certain geometric constructions of hyper-K "ahler manifolds (e.g. Debarre--Voisin or Laza--Sacc a--Voisin). In a slightly different direction, we establish some basic properties (dimension and rational homology type) for the dual complex of a Kulikov type degeneration of hyper-K "ahler manifolds. A spatio-temporal descriptor for representation and recognition of time-varying textures is proposed (binarized statis- tical image features on three orthogonal planes (BSIF-TOP)) in this paper. The descriptor, similar in spirit to the well known local binary patterns on three orthogonal planes approach, estimates histograms of binary coded image sequences on three orthogonal planes corresponding to spatial spatio-temporal dimensions. However, unlike some other methods which generate the code in a heuristic fashion, binary code generation in the BSIF-TOP approach is realized by filtering operations on different regions of spatial spatio-temporal support and by binarizing the filter responses. The filters are learnt via independent component analysis on each of three planes after preprocessing using a whitening transformation. By extending the BSIF-TOP descriptor to a multiresolution scheme, the descriptor is able to capture the spatio-temporal content of an image sequence at multiple scales, improving its representation capacity. In the evaluations on the UCLA, Dyntex, and Dyntex dynamic texture databases, the proposed method achieves very good performance compared to existing approaches. This paper proposes a method for bin-picking for objects without assuming the precise geometrical model of objects. We consider the case where the shape of objects are not uniform but are similarly approximated by cylinders. By using the point cloud of a single object, we extract the probabilistic properties with respect to the difference between an object and a cylinder and consider applying the probabilistic properties to the pick-and-place motion planner of an object stacked on a table. By using the probabilistic properties, we can also realize the contact state where a finger maintain contact with the target object while avoiding contact with other objects. We further consider approximating the region occupied by fingers by a rectangular parallelepiped. The pick-and-place motion is planned by using a set of regions in combination with the probabilistic properties. Finally, the effectiveness of the proposed method is confirmed by some numerical examples and experimental result.