Showing papers on "Topological space published in 1994"

Deriving the Composition of Binary Topological Relations

Abstract: A new formalism is presented to derive knowledge about the composition of two binary topological relations over a common object. The formalism is based on a topological data model and compares the nine empty and non-empty intersections of interiors, boundaries and exteriors between two objects. Based upon the transitivity of set inclusion, the intersections of the composed topological relations are derived. These intersections are then matched with the intersections of the eight fundamental topological relations, giving an interpretation to the composition of topological relations. The result of this study is the composition table of the eight binary topological relations that exist between n-dimensional point sets with a co-dimension of 0. While the combined topological relations are unique for some compositions, more than half of all possible compositions are disjunctions of possible relations. Geometric prototypes are shown for the two-dimensional case. The composition table enables topological reasoning at the conceptual level of relations, rather than having to calculate all relations from the representation of the spatial objects. Its practical value is that it can serve as in a computational model for an assessment of whether a set of topological predicates is consistent or not and in spatial query processing when no explicit information about spatial relations is available.

Triangulating topological spaces

Abstract: Given a subspace 𝒳 ⊆ Rd and a finite set S⊆Rd, we introduce the Delaunay simplicial complex, D𝒳, restricted by 𝒳. Its simplices are spanned by subsets T⊆S for which the common intersection of Voronoi cells meets 𝒳 in a non-empty set. By the nerve theorem,⋃D𝒳 and 𝒳 are homotopy equivalent if all such sets are contractible. This paper shows that ⋃D𝒳 and 𝒳 are homeomorphic if the sets can be further subdivided in a certain way so they form a regular CW complex.

Mountain pass theorems and global homeomorphism theorems

Abstract: We show that mountain-pass theorems can be used to derive global homeomorphism theorems. Two new mountain-pass theorems are proved, generalizing the “smooth” mountain-pass theorem, one applying in locally compact topological spaces, using Hofer’s concept of mountain-pass point, and another applying in complete metric spaces, using a generalized notion of critical point similar to the one introduced by Ioffe and Schwartzman. These are used to prove global homeomorphism theorems for certain topological and metric spaces, generalizing known global homeomorphism theorems for mappings between Banach spaces.

Discrete differential calculus graphs, topologies and gauge theory

Abstract: Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a ‘‘reduction’’ of the ‘‘universal differential algebra’’ and this allows a systematic exploration of differential algebras on a given set. Associated with a differential algebra is a (di)graph where two vertices are connected by at most two (antiparallel) arrows. The interpretation of such a graph as a ‘‘Hasse diagram’’ determining a (locally finite) topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure. It is shown that field theories, and in particular gauge theories, can be formulated on a discrete set in close analogy with the continuum case. The framework presented generalizes ordinary lattice theory which is recovered from an oriented (hypercubic) lattice graph. It also includes, e.g., the two‐point space used by Connes and Lott (and others) in models of elementary particle physics. The formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an ‘‘internal’’ discrete space (a la Connes et al.) to models of dimensionally reduced gauge fields. Furthermore, a ‘‘symmetric lattice’’ is also studied which (in a certain continuum limit) turns out to be related to a ‘‘noncommutative differential calculus’’ on manifolds.

Discrete differential calculus, graphs, topologies and gauge theory

Abstract: Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic exploration of differential algebras on a given set. Associated with a differential algebra is a (di)graph where two vertices are connected by at most two (antiparallel) arrows. The interpretation of such a graph as a `Hasse diagram' determining a (locally finite) topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure. It is shown that field theories, and in particular gauge theories, can be formulated on a discrete set in close analogy with the continuum case. The framework presented generalizes ordinary lattice theory which is recovered from an oriented (hypercubic) lattice graph. It also includes, e.g., the two-point space used by Connes and Lott (and others) in models of elementary particle physics. The formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an `internal' discrete space ({\`a} la Connes et al.) to models of dimensionally reduced gauge fields. Furthermore, also a `symmetric lattice' is studied which (in a certain continuum limit) turns out to be related to a `noncommutative differential calculus' on manifolds.

Partition Theorems for Spaces of Variable Words

Abstract: Furstenberg and Katznelson applied methods of topological dynamics to Ramsey theory, obtaining a density version of the Hales-Jewett partition theorem. Inspired by their methods, but using spaces of ultrafilters instead of their metric spaces, we prove a generalization of a theorem of Carlson about variable words. We extend this result to partitions of finite or infinite sequences of variable words, and we apply these extensions to strengthen a partition theorem of Furstenberg and Katznelson about combinatorial subspaces of the set of words.

Knots and Surfaces

Abstract: 1. Knots, Links, and Diagrams 2. Knot and Link Polynomials 3. Topological Spaces 4. Surfaces 5. The Arithmetic of Knots 6. Presentations of Groups 7. Graphs and Trees 8. Alexanders Matrices and Alexander Polynomials 9. The Fundamental Group 10. Van Kampen's Theorem 11. Applications of Van Kampen Theorem 12. Covering Spaces Bibliography Index

A smooth compactification of the space of transfer functions with fixed McMillan degree

Abstract: It is a classical result of Clark that the space of all proper or strictly properp ×m transfer functions of a fixed McMillan degreed has, in a natural way, the structure of a noncompact, smooth manifold. There is a natural embedding of this space into the set of allp × (m+p) autoregressive systems of degree at mostd. Extending the topology in a natural way we will show that this enlarged topological space is compact. Finally we describe a homogenization process which produces a smooth compactification.

Connectedness of efficient solution sets for set-valued maps in normed spaces

Abstract: In vector optimization, the topological properties of the set of efficient solutions are of interest. Several authors have studied this topic for point-valued functions. In this paper, we study the connectedness of the efficient solution sets in convex vector optimization for set-valued maps in normed spaces.

Dimension on Discrete Spaces

Abstract: In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by means of axioms, and the axioms are based on an obvious geometrical background. This work presents some discrete models of n-dimensional Euclidean spaces, n-dimensional spheres, a torus and a projective plane. It explains how to construct new discrete spaces and describes in this connection several three-dimensional closed surfaces with some topological singularities It also analyzes the topology of (3+1)-spacetime. We are also discussing the question by R. Sorkin [19] about how to derive the system of simplicial complexes from a system of open covering of a topological space S.

Dimension on discrete spaces

Abstract: In this paper we develop some combinatorial models for continuous spaces. We study the approximations of continuous spaces by graphs, molecular spaces, and coordinate matrices. We define the dimension on a discrete space by means of axioms based on an obvious geometrical background. This work presents some discrete models ofn-dimensional Euclidean spaces,n-dimensional spheres, a torus, and a projective plane. It explains how to construct new discrete spaces and describes in this connection several three-dimensional closed surfaces with some topological singularities. It also analyzes the topology of (3+1)-space-time. We are also discussing the question by R. Sorkin about how to derive the system of simplicial complexes from a system of open coverings of a topological space.

Variants of openness

Abstract: Algebraic conditions on frame homomorphisms representing various types of openness requirements on continuous maps are investigated. It turns out that several of these can be expressed in terms of formulas involving pseudocomplements. A full classification of the latter is presented which shows that they group into five equivalence classes and establishes the logical connections between them. Among the relation of our algebraic conditions to continuous maps between topological spaces, we establish that the coincidence of the algebraic and topological notion of openness is equivalent to the separation axiomT D for the domain space.

Continuity and Computability of Relations

Abstract: The main theorem of the theory of effectivity ( cf. Kreitz and Weihrauch [KWl], [Wl]) states that in admissibly represented topological spaces a function is continuous iff it has a continuous representation. Hence continuity is a necessary condition for computability. We investigate an extended model of computability in order to compute relations. From another point of view these relations are nondeterministic operations or set-valued functions. We show that for a special class of topological spaces (including the complete separable metric ones) and for a certain notion of continuity for relations the main theorem can be extended too.

A New Concept for Digital Geometry

Abstract: A concept for geometry in a topological space with finitely many elements without the use of infinitesimals is presented. The notions of congruence, collinearity, convexity, digital lines, perimeter, area, volume, etc. are defined. The classical notion of continuous mappings is transferred (without changes) onto finite spaces. A slightly more general notion of connectivity preserving mappings is introduced. Applications for shape analysis are demonstrated.

Local Theory of Complex Spaces

Abstract: A fundamental tenet of contemporary Complex Analysis is that geometric properties of complex spaces and algebraic properties of their structure sheaves are living in happy symbiosis. This introductory chapter is a rambling through basic notions and results of Local Complex Analysis based on local function theory, local algebra and sheaves. There are many advantages to develop the theory in a general context. Howerver, as in algebraic geometry, one has to burden oneself with a considerable load of technical luggage. Sheaves are a powerful and verstile tool, they provide the natural way of keeping track of continuous variations of local algebraic data on topological spaces. The revolutionary slogan of the fifties “il faut faiseautiser” is a truism long since.

Topology learning solved by extended objects: A neural network model

Abstract: It is shown that local, extended objects of a metrical topological space shape the receptive fields of competitive neurons to local filters. Self-organized topology learning is then solved with the help of Hebbian learning together with extended objects that provide unique information about neighborhood relations. A topographical map is deduced and is used to speed up further adaptation in a changing environment with the help of Kohonen-type learning that teaches the neighbors of winning neurons as well.

The complex of maximal lattice free simplices

Abstract: The simplicial complex K(A) is defined to be the collection of simplices, and their proper subsimplices, representing maximal lattice free bodies of the form (x: Ax⩽b), with A a fixed generic (n +1 ) × n matrix. The topological space associated with K(A) is shown to be homeomorphic to ℝ n , and the space obtained by identifying lattice translates of these simplices is homeorphic to the n-torus.

New minimax inequality with applications to existence theorems of equilibrium points

Abstract: A new minimax inequality is proved on a set which is the union of an increasing sequence of compact convex sets in a topological vector space. As applications, several existence theorems of equilibrium points for different games are obtained.