Instant topological relationships hidden in the reality

In most applications of general topology, topology usually is not the first, primary structure, but the information which finally leads to the construction of the certain, for some purpose required topology, is filtered by more or less thick filter of the other mathematical structures. This fact has two main consequences: (1) Most important applied constructions may be done in the primary structure, bypassing the topology. (2) Some topologically important information from the reality may be lost (filtered out by the other, front-end mathematical structures). Thus some natural and direct connection between topology and the reality could be useful. In this contribution we will discuss a pointless topological structure which directly reflects relationship between various locations which are glued together by possible presence of a physical object or a virtual ``observer"

Stability and statistical inferences in the space of topological spatial relationships

Modelling topological properties of the spatial relationship between objects, known as the extit{topological relationship}, represents a fundamental research problem in many domains including Artificial Intelligence (AI) and Geographical Information Science (GIS). Real world data is generally finite and exhibits uncertainty. Therefore, when attempting to model topological relationships from such data it is useful to do so in a manner which is both extit{stable} and facilitates extit{statistical inferences}. Current models of the topological relationships do not exhibit either of these properties. We propose a novel model of topological relationships between objects in the Euclidean plane which encodes topological information regarding connected components and holes. Specifically, a representation of the persistent homology, known as a persistence scale space, is used. This representation forms a Banach space that is stable and, as a consequence of the fact that it obeys the strong law of large numbers and the central limit theorem, facilitates statistical inferences. The utility of this model is demonstrated through a number of experiments

Recursive Algorithms for Distributed Forests of Octrees

The forest-of-octrees approach to parallel adaptive mesh refinement and coarsening (AMR) has recently been demonstrated in the context of a number of large-scale PDE-based applications. Although linear octrees, which store only leaf octants, have an underlying tree structure by definition, it is not often exploited in previously published mesh-related algorithms. This is because the branches are not explicitly stored, and because the topological relationships in meshes, such as the adjacency between cells, introduce dependencies that do not respect the octree hierarchy. In this work we combine hierarchical and topological relationships between octree branches to design efficient recursive algorithms. We present three important algorithms with recursive implementations. The first is a parallel search for leaves matching any of a set of multiple search criteria. The second is a ghost layer construction algorithm that handles arbitrarily refined octrees that are not covered by previous algorithms, which require a 2:1 condition between neighboring leaves. The third is a universal mesh topology iterator. This iterator visits every cell in a domain partition, as well as every interface (face, edge and corner) between these cells. The iterator calculates the local topological information for every interface that it visits, taking into account the nonconforming interfaces that increase the complexity of describing the local topology. To demonstrate the utility of the topology iterator, we use it to compute the numbering and encoding of higher-order $C^0$ nodal basis functions. We analyze the complexity of the new recursive algorithms theoretically, and assess their performance, both in terms of single-processor efficiency and in terms of parallel scalability, demonstrating good weak and strong scaling up to 458k cores of the JUQUEEN supercomputer.Comment: 35 pages, 15 figures, 3 table

Requirements for Topology in 3D GIS

Topology and its various benefits are well understood within the context of 2D Geographical Information Systems. However, requirements in three-dimensional (3D) applications have yet to be defined, with factors such as lack of users' familiarity with the potential of such systems impeding this process. In this paper, we identify and review a number of requirements for topology in 3D applications. The review utilises existing topological frameworks and data models as a starting point. Three key areas were studied for the purposes of requirements identification, namely existing 2D topological systems, requirements for visualisation in 3D and requirements for 3D analysis supported by topology. This was followed by analysis of application areas such as earth sciences and urban modelling which are traditionally associated with GIS, as well as others including medical, biological and chemical science. Requirements for topological functionality in 3D were then grouped and categorised. The paper concludes by suggesting that these requirements can be used as a basis for the implementation of topology in 3D. It is the aim of this review to serve as a focus for further discussion and identification of additional applications that would benefit from 3D topology. © 2006 The Authors. Journal compilation © 2006 Blackwell Publishing Ltd

Voronoi-Based Region Approximation for Geographical Information Retrieval with Gazetteers

Gazetteers and geographical thesauri can be regarded as parsimonious spatial models that associate geographical location with place names and encode some semantic relations between the names. They are of particular value in processing information retrieval requests in which the user employs place names to specify geographical context. Typically the geometric locational data in a gazetteer are confined to a simple footprint in the form of a centroid or a minimum bounding rectangle, both of which can be used to link to a map but are of limited value in determining spatial relationships. Here we describe a Voronoi diagram method for generating approximate regional extents from sets of centroids that are respectively inside and external to a region. The resulting approximations provide measures of areal extent and can be used to assist in answering geographical queries by evaluating spatial relationships such as distance, direction and common boundary length. Preliminary experimental evaluations of the method have been performed in the context of a semantic modelling system that combines the centroid data with hierarchical and adjacency relations between the associated place names

Quantum channels as a categorical completion

We propose a categorical foundation for the connection between pure and mixed states in quantum information and quantum computation. The foundation is based on distributive monoidal categories. First, we prove that the category of all quantum channels is a canonical completion of the category of pure quantum operations (with ancilla preparations). More precisely, we prove that the category of completely positive trace-preserving maps between finite-dimensional C*-algebras is a canonical completion of the category of finite-dimensional vector spaces and isometries. Second, we extend our result to give a foundation to the topological relationships between quantum channels. We do this by generalizing our categorical foundation to the topologically-enriched setting. In particular, we show that the operator norm topology on quantum channels is the canonical topology induced by the norm topology on isometries.Comment: 12 pages + ref, accepted at LICS 201

An Investigation into the Semantics of English Topological Prepositions

This paper describes a psycholinguistic experiment that investigates whether the applicability of the topological spatial prepositions at , on or in to describe the spatial configuration between two objects is related to the topological relationships between objects being describe

The spatiotemporal representation of dance and music gestures using topological gesture analysis (TGA)

SPATIOTEMPORAL GESTURES IN MUSIC AND DANCE HAVE been approached using both qualitative and quantitative research methods. Applying quantitative methods has offered new perspectives but imposed several constraints such as artificial metric systems, weak links with qualitative information, and incomplete accounts of variability. In this study, we tackle these problems using concepts from topology to analyze gestural relationships in space. The Topological Gesture Analysis (TGA) relies on the projection of musical cues onto gesture trajectories, which generates point clouds in a three-dimensional space. Point clouds can be interpreted as topologies equipped with musical qualities, which gives us an idea about the relationships between gesture, space, and music. Using this method, we investigate the relationships between musical meter, dance style, and expertise in two popular dances (samba and Charleston). The results show how musical meter is encoded in the dancer's space and how relevant information about styles and expertise can be revealed by means of simple topological relationships