Hausdorff connectifications

[EN] Disconnectedness in topological space is analyzed to obtain Hausdorff connectifications of that topological space. Hausdorff connectifications are obtained by some direct constructions and by some partitions of connectifications. Also lattice structure is included in the collection of all Hausdorff connectifications.Ramkumar, S. (2014). Hausdorff connectifications. Applied General Topology. 15(1):1-9. doi:http://dx.doi.org/10.4995/agt.2014.2019.SWORD19151Alas, O. T., Tkačenko, M. G., Tkachuk, V. V., & Wilson, R. G. (1996). Connectifying some spaces. Topology and its Applications, 71(3), 203-215. doi:10.1016/0166-8641(95)00012-7J. J. Charatonik, One point connectifications of subspaces of generalized graphs, Kyungpook Math. J. 41 (2001), 335-340.J. J. Charatonik, On one point connectifications of spaces, Kyungpook Math. J. 43 (2003), 149-156.Fedeli, A., & Le Donne, A. (1999). Dense embeddings in pathwise connected spaces. Topology and its Applications, 96(1), 15-22. doi:10.1016/s0166-8641(98)00016-9K. D. Jr. Magill, The lattice of compactificatons of a locally compact space, Proc. London Math. Soc. 18, (1968), 231-244.J. R. Munkres, Topology, second edi., Prentice Hall of India, New Delhi, 2000.Porter, J. R., & Woods, R. G. (1996). Subspaces of connected spaces. Topology and its Applications, 68(2), 113-131. doi:10.1016/0166-8641(95)00057-7Porter, J. R., & Woods, R. G. (1988). Extensions and Absolutes of Hausdorff Spaces. doi:10.1007/978-1-4612-3712-9S.w, . "Watson and R.G. Wilson, Embeddings in connected spaces, Houston J. Math. 19, no. 3 (1993), 469-481. . Print

Some mixed Hodge structure on l^2-cohomology of covering of K\"ahler manifolds

We give methods to compute l^2-cohomology groups of a covering manifolds obtained by removing pullback of a (normal crossing) divisor to a covering of a compact K\"ahler manifold. We prove that in suitable quotient categories, these groups admit natural mixed Hodge structure whose graded pieces are given by expected Gysin maps.Comment: 40 pages. This revised version will be published in Mathematische Annale

Cosmic censorship of smooth structures

It is observed that on many 4-manifolds there is a unique smooth structure underlying a globally hyperbolic Lorentz metric. For instance, every contractible smooth 4-manifold admitting a globally hyperbolic Lorentz metric is diffeomorphic to the standard $\R^4$. Similarly, a smooth 4-manifold homeomorphic to the product of a closed oriented 3-manifold $N$ and $\R$ and admitting a globally hyperbolic Lorentz metric is in fact diffeomorphic to $N\times \R$. Thus one may speak of a censorship imposed by the global hyperbolicty assumption on the possible smooth structures on $(3+1)$-dimensional spacetimes.Comment: 5 pages; V.2 - title changed, minor edits, references adde

A continuous rating method for preferential voting. The complete case

A method is given for quantitatively rating the social acceptance of different options which are the matter of a complete preferential vote. Completeness means that every voter expresses a comparison (a preference or a tie) about each pair of options. The proposed method is proved to have certain desirable properties, which include: the continuity of the rates with respect to the data, a decomposition property that characterizes certain situations opposite to a tie, the Condorcet-Smith principle, and a property of clone consistency. One can view this rating method as a complement for the ranking method introduced in 1997 by Markus Schulze. It is also related to certain methods of one-dimensional scaling or cluster analysis.Comment: This is part one of a revised version of arxiv:0810.2263. Version 3 is the result of certain modifications, both in the statement of the problem and in the concluding remarks, that enhance the results of the paper; the results themselves remain unchange

Persistent Homology Over Directed Acyclic Graphs

We define persistent homology groups over any set of spaces which have inclusions defined so that the corresponding directed graph between the spaces is acyclic, as well as along any subgraph of this directed graph. This method simultaneously generalizes standard persistent homology, zigzag persistence and multidimensional persistence to arbitrary directed acyclic graphs, and it also allows the study of more general families of topological spaces or point-cloud data. We give an algorithm to compute the persistent homology groups simultaneously for all subgraphs which contain a single source and a single sink in $O(n^4)$ arithmetic operations, where $n$ is the number of vertices in the graph. We then demonstrate as an application of these tools a method to overlay two distinct filtrations of the same underlying space, which allows us to detect the most significant barcodes using considerably fewer points than standard persistence.Comment: Revised versio

Parallel Mapper

The construction of Mapper has emerged in the last decade as a powerful and effective topological data analysis tool that approximates and generalizes other topological summaries, such as the Reeb graph, the contour tree, split, and joint trees. In this paper, we study the parallel analysis of the construction of Mapper. We give a provably correct parallel algorithm to execute Mapper on multiple processors and discuss the performance results that compare our approach to a reference sequential Mapper implementation. We report the performance experiments that demonstrate the efficiency of our method

Relative Convex Hull Determination from Convex Hulls in the Plane

A new algorithm for the determination of the relative convex hull in the plane of a simple polygon A with respect to another simple polygon B which contains A, is proposed. The relative convex hull is also known as geodesic convex hull, and the problem of its determination in the plane is equivalent to find the shortest curve among all Jordan curves lying in the difference set of B and A and encircling A. Algorithms solving this problem known from Computational Geometry are based on the triangulation or similar decomposition of that difference set. The algorithm presented here does not use such decomposition, but it supposes that A and B are given as ordered sequences of vertices. The algorithm is based on convex hull calculations of A and B and of smaller polygons and polylines, it produces the output list of vertices of the relative convex hull from the sequence of vertices of the convex hull of A.Comment: 15 pages, 4 figures, Conference paper published. We corrected two typing errors in Definition 2: $I_S$ has to be defined based on $O_S$, and $I_E$ has to be defined based on $O_E$ (not just using $O$). These errors appeared in the text of the original conference paper, which also contained the pseudocode of an algorithm where $I_S$ and $I_E$ appeared as correctly define

Ergodic Jacobi matrices and conformal maps

We study structural properties of the Lyapunov exponent $\gamma$ and the density of states $k$ for ergodic (or just invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function $w=-\gamma+i\pi k$ as a conformal map between certain domains. This idea goes back to Marchenko and Ostrovskii, who used this device in their analysis of the periodic problem

Simplicial Complex based Point Correspondence between Images warped onto Manifolds

Recent increase in the availability of warped images projected onto a manifold (e.g., omnidirectional spherical images), coupled with the success of higher-order assignment methods, has sparked an interest in the search for improved higher-order matching algorithms on warped images due to projection. Although currently, several existing methods "flatten" such 3D images to use planar graph / hypergraph matching methods, they still suffer from severe distortions and other undesired artifacts, which result in inaccurate matching. Alternatively, current planar methods cannot be trivially extended to effectively match points on images warped onto manifolds. Hence, matching on these warped images persists as a formidable challenge. In this paper, we pose the assignment problem as finding a bijective map between two graph induced simplicial complexes, which are higher-order analogues of graphs. We propose a constrained quadratic assignment problem (QAP) that matches each p-skeleton of the simplicial complexes, iterating from the highest to the lowest dimension. The accuracy and robustness of our approach are illustrated on both synthetic and real-world spherical / warped (projected) images with known ground-truth correspondences. We significantly outperform existing state-of-the-art spherical matching methods on a diverse set of datasets.Comment: Accepted at ECCV 202