Abstract
This chapter introduces computational proximity. Basically, computational proximity (CP) is an algorithmic approach to finding nonempty sets of points that are either close to each other or far apart. The methods used by CP to find either near sets or remote sets result from the study of structures called proximity spaces.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For more about this, see C. Stover and E.W. Weisstein, Point, http://mathworld.wolfram.com/Point.html.
- 2.
Many thanks to Anna Di Concilio for pointing this out.
- 3.
References
Di Maio, G., Naimpally, S.A., Meccariello, E.: Theory and applications of proximity, nearness and uniformity. Seconda Università di Napoli, Napoli (2009), 264 pp., MR1269778
Naimpally, S., Peters, J., Wolski, M.: Foreword [near set theory and applications]. Math. Comput. Sci. 7(1), 1–2 (2013)
Peters, J.: Proximal relator spaces. Filomat (2016). Accepted
Peters, J., Guadagni, C.: Strongly proximal continuity and strong connectedness, pp. 1–11 (2015). arXiv:1504.02740
Peters, J.: Topology of digital images - visual pattern discovery in proximity spaces, intelligent systems reference library, vol. 63. Springer, Berlin (2014). Xv + 411 pp. Zentralblatt MATH Zbl 1295 68010
Peters, J., Guadagni, C.: Strong proximities on smooth manifolds and Voronoï diagrams. Adv. Math.: Sci. J. 4(2), 91–107 (2015)
Naimpally, S., Peters, J.: Topology with applications. Topological spaces via near and far. World Scientific, Singapore (2013). Xv + 277 pp. Am. Math. Soc. MR3075111
Dochviri, I., Peters, J.: Topological sorting of finitely many near sets. Math. Comput. Sci. 1–6 (2015). Communicated
Naimpally, S.: Proximity Spaces. Cambridge University Press, Cambridge (1970). X + 128 pp., ISBN 978-0-521-09183-1
Lodato, M.: On topologically induced generalized proximity relations. Ph.D. thesis. Rutgers University (1962). Supervisor: S. Leader
Guadagni, C.: Bornological convergences on local proximity spaces and \(\omega _{\mu }\)-metric spaces. Ph.D. thesis, Università degli Studi di Salerno, Salerno (2015). Supervisor: A. Di Concilio, 79 pp
Di Concilio, A.: Proximity: a powerful tool in extension theory, functions spaces, hyperspaces, boolean algebras and point-free geometry. In: Mynard, F., Pearl, E. (eds.) Beyond Topology. AMS Contemporary Mathematics, vol. 486, pp. 89–114. American Mathematical Society, Providence (2009)
Peters, J.: Local near sets: pattern discovery in proximity spaces. Math. Comput. Sci. 7(1), 87–106 (2013). doi:10.1007/s11786-013-0143-z, MR3043920
Hettiarachchi, R., Peters, J.: Multi-manifold LLE learning in pattern recognition. Pattern Recognit. Elsevier 48(9), 2947–2960 (2015)
Hettiarachchi, R., Peters, J.: Multi-manifold skin classifier for feature space voronoï region-based skin segmentation. Image Vis. Comput. (2015). Communicated
İnan, E., Öztürk, M.: Near groups on nearness approximation spaces. Hacet. J. Math. Stat. 41(4), 545–558 (2012)
Peters, J., İnan, M.A., Öztürk, E.: Spatial and descriptive isometries in proximity spaces. Gen. Math. Notes 21(2), 125–134 (2014)
Peters, J., Öztürk, M., Uçkun, M.: Klee-Phelps convex groupoids, pp. 1–5 (2014). arXiv:1411.0934, Mathematica Slovaca 2015. Accepted
Edelsbrunner, H., Harer, J.: Computational Topology. An Introduction. American Mathematical Society, Providence (2010). Xii + 110 pp., MR2572029
Peters, J.: Visibility in proximal Delaunay meshes and strongly near Wallman proximity. Adv. Math.: Sci. J. 4(1), 41–47 (2015)
Di Concilio, A., Gerla, G.: Quasi-metric spaces and point-free geometry. Math. Struct. Comput. Sci. 16(1), 115–137 (2006). MR2220893
Di Concilio, A.: Point-free geometries: proximities and quasi-metrics. Math. Comput. Sci. 7(1), 31–42 (2013). MR3043916
Beer, G., Lucchetti, R.: Weak topologies for the closed subsets of a metrizable space. Trans. Am. Math. Soc. 335(2), 805–822 (1993)
Aurenhammer, F.: Voronoi diagrams–a survey of a fundamental geometric data structure. ACM Comput. Surv. 23(3), 345–405 (1991)
Peters, J., Ramanna, S.: Proximal three-way decisions: theory and applications in social networks. Knowl.-Based Syst. 1–12 (2014). http://dx.doi.org/10.1016/j.knosys.2015.07.021 (in press)
Pták, P., Kropatsch, W.: Nearness in digital images and proximity spaces. LNCS. In: Proceedings of the 9th International Conference on Discrete Geometry, vol. 1953, pp. 69–77 (2000)
Beer, G., Di Concilio, A., Di Maio, G., Naimpally, S., Pareek, C., Peters, J.: Somashekhar Naimpally, 1931–2014. Topol. Appl. 188, 97–109 (2015). doi:10.1016/j.topol.2015.03.010, MR3339114
Di Maio, G., Naimpally, S.: Preface, theory and applications of proximity, nearness and uniformity, Quaderni di Matematica [Mathematics Series], vol. 22. Department of Mathematics, Seconda Universit di Napoli, Caserta (2008). viii + 364 pp., MR2760945
Plato: the allegory of the cave, in Republic, VII, 514a,2–517a, 7. Stanford University, Stanford (c300BC, 2015). Translated by T. Sheehan
Frege, G.: Foundations of arithmetic, Translated by J.L. Austin. Blackwell, Oxford (1953)
Whitehead, A.: An Inquiry into the Principles of Natural Knowledge. Cambridge University Press, Cambridge (1920)
Whitehead, A.: Process and Reality. Macmillan, London (1929)
Hooke, R.: Micrographia: or, some physiological descriptions of minute bodies made by magnifying glasses. With observations and inquiries thereupon. Martyn and J. Allestry, London (1665). 270 pp
Prince, S.: Computer Vision. Models, Learning, and Inference. Cambridge University Press, Cambridge (2012). Xvii + 580 pp
Favorskaya, M., Jain, L.C.: Computer Vision in Control Systems-1. Mathematical Theory. Springer, Berlin (2015). Xvii + 371 pp
Arkowitz, M.: Introduction to Homotopy Theory. Springer, New York (2011). Xiv + 344 pp. ISBN: 978-1-4419-7328-3, MR2814476
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002). Xii + 544 pp. ISBN: 0-521-79160-X; 0-521-79540-0, MR1867354
Whitehead, J.: Simplicial spaces, nuclei and m-groups. Proc. Lond. Math. Soc. 45, 243–327 (1939)
Krantz, S.: Essentials of topology with applications. CRC Press, Boca Raton (2010). Xvi + 404 pp. ISBN: 978-1-4200-8974-5, MR2554895
Euclid: The Thirteen Books of Euclid’s Elements, 2nd ed. Dover Publications, New York (c300BC, 1956). Translated by T.L. Heath, from the text by Heiberg, xi + 432pp; i + 436pp; i + 546 pp, MR0075873
Ronse, C.: Regular open or closed sets. Philips Research Laboratory Series, Brussels WD59, pp. 1–8 (1990)
Peters, J., Naimpally, S.: Applications of near sets. Notices Am. Math. Soc. 59(4), 536–542 (2012). http://dx.doi.org/10.1090/noti817, MR2951956
Edelsbrunner, H.: Geometry and Topology of Mesh Generation. Cambridge University Press, Cambridge (2001). 209 pp
Deritei, D., Lázár, Z., Papp, I., Járai-Szabó, F., Sumi, R., Varga, L., Regan, E., Ercsey-Ravasz, M.: Community detection by graph voronoi diagrams. New J. Phys. 16, 1–17 (2014)
Riesz, F.: Stetigkeitsbegriff und abstrakte mengenlehre. Atti del IV Congresso Internazionale dei Matematici, vol. II, pp. 182–109 (1908)
C̆ech, E.: Topological Spaces. Wiley, London (1966). Fr seminar, Brno, 1936–1939; rev. ed. Z. Frolik, M. Katĕtov
Efremovič, V.: The geometry of proximity I (in Russian). Mat. Sb. (N.S.) 31(73)(1), 189–200 (1952)
Peters, J.: Near sets. Special theory about nearness of objects. Fundamenta Informaticae 75, 407–433 (2007). MR2293708
Peters, J., Guadagni, C.: Strongly near proximity and hyperspace topology, pp. 1–6 (2015). arXiv:1502.05913
Wallman, H.: Lattices and topological spaces. Ann. Math. 39(1), 112–126 (1938)
Euclid: Elements. Alexandria (300 B.C.). English translation by R. Fitzpatrick, from Euclidis Elementa Latin text by B.G. Teubneri, 1883–1885 and the Greek text by J.L. Heiberg, 1883–1885
Klette, R., Rosenfeld, A.: Digital Geometry. Geometric Methods for Digital Picture Analysis. Morgan-Kaufmann Publishers, Amsterdam (2004)
Du, Q., Faber, V., Gunzburger, M.: Centroidal voronoi tessellations: applications and algorithms. SIAM Rev. 41(4), 637–676 (1999). MR
Boissonnat, J.D., Wormser, C., Yvinec, M.: Curved Voronoi diagrams. In: Boissonnat, J.D., Teillaud, E. (eds.) Effective Computational Geometry for Curves and Surfaces, pp. 67–116. Springer, New York (2006)
Kovalevsky, V.: Finite topology as applied to image analysis. Comput. Vis. Graph. Image Process. 46, 141–161 (1989)
Grünbaum, B., Shephard, G.: Tilings and Patterns. W.H. Freeman and Co., New York (1987). Xii + 700 pp., MR0857454
Romesburg, H.: Cluster Analysis for Researchers. Lulu Press, North Carolina (2004). Xii + 334 pp., MR3155265
Peters, J.: Near sets. General theory about nearness of sets. Appl. Math. Sci. 1(53), 2609–2629 (2007)
Peters, J.: Near sets: an introduction. Math. Comput. Sci. 7(1), 3–9 (2013). doi:10.1007/s11786-013-0149-6, MR3043914
Kuratowski, C.: Topologie I. Panstwowe Wydawnictwo Naukowe, Warsaw (1958). XIII + 494 pp
Kuratowski, K.: Introduction to Set Theory and Topology, 2nd edn. Pergamon Press, Oxford (1962, 1972). 349 pp
Naimpally, S., Warrack, B.: Proximity Spaces. Cambridge Tract in Mathematics, vol. 59. Cambridge University Press, Cambridge (1970). X + 128 pp., Paperback (2008)
Peters, J., Guadagni, C.: Strongly far proximity and hyperspace topology, pp. 1–6 (2015). arXiv:1502.02771
Peters, J., Guadagni, C.: Strongly hit and far miss hypertopology and hit and strongly far miss hypertopology, pp. 1–8 (2015). arXiv:1503.02587
Solan, V.: Introduction to the axiomatic theory of convexity [Russian with English and French summaries]. Shtiintsa, Kishinev (1984). 224 pp., MR0779643
Zelins’kyi, Y.: Generalized convex envelopes of sets and the problem of shadow. J. Math. Sci. 211(5), 710–717 (2015)
Kay, D., Womble, E.: Automatic convexity theory and relationships between the carathèodory, helly and radon numbers. Pac. J. Math. 38(2), 471–485 (1971)
Tuz, V.: Axiomatic convexity theory [Russian]. Rossiïskaya Akademiya Nauk. Matematicheskie Zametki [Math. Notes Math. Notes] 20(5), 761–770 (1976)
Rocchi, N.: Parliamo Di Insiemi. Instituto Didattico Editoriale Felsineo. Bologna (1969). 316 pp
Bourbaki, N.: Elements of Mathematics. General Topology, Part 1. Hermann and Addison-Wesley, Paris and Reading (1966). I-vii, 437 pp
Edelsbrunner, H.: A Short Course in Computational Geometry and Topology. Springer, Berlin (2014). 110 pp
Frank, N., Hart, S.: A dynamical system using the Voronoi tessellation. Am. Math. Monthly 117(2), 92–112 (2010)
Weisstein, E.: Voronoi diagram. Wolfram MathWorld (2015). http://mathworld.wolfram.com/VoronoiDiagram.html
Liebling, T., Pourin, L.: Voronoi diagrams and Delaunay triangulations: Ubiquitous siamese twins. Documenta Mathematica Extra volume: Optimization stories, 419–431 (2012). MR2991503
Okabe, A., Boots, B., Sugihara, K., Chiu, S.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, Chichester (2000). Xvi + 671 pp. ISBN: 0-471-98635-6, MR1770006
Stover, C.: Point Process. Wolfram MathWorld (2015). http://mathworld.wolfram.com/PointProcess.html
Weisstein, E.: Poissonprocess. Wolfram MathWorld (2015). http://mathworld.wolfram.com/PoissonProcess.html
Floriani, L.D., Spagnuolo, M.: Shape Analysis and Structuring. Springer, Berlin (2008). Xiv + 296 pp. ISBN 978-3-540-33264-0
Hilhorst, H.: Statistical properties of planar Voronoi tessellations. Eur. Phys. J. B 64, 437–441 (2008)
Anton, F., Mioc, D., Gold, C.: The Voronoi diagram of circles and its application to the visualization of the growth of particles. In: Gavrilova, M., Tan, C.K. (eds.) Transactions on Computational Science III, pp. 20–54. Springer, Berlin (2009). MR2912541
Surendran, S., Chitraprasad, D., Kaimal, M.: Voronoi diagrams-based geometric approach to social network analysis. In: Krishnan, G.S.S., et al. (eds.) Computational Intelligence, Cyber Security and Computational Models, Advances in Intelligent Systems and Computing, vol. 246 pp. 359–369. Springer, India (2014)
Liu, H.: Dynamic concept cartography for social networks. Master’s thesis, School of Information Technologies (2007)
Peters, J.: Proximal Voronoï regions, convex polygons, and leader uniform topology. Adv. Math.: Sci. J. 4(1), 1–5 (2015)
Munkres, J.: Topology, 2nd edn. Prentice-Hall, Englewood Cliffs (2000). Xvi + 537 pp., 1st edn. in 1975, MR0464128
Krantz, S.: A Guide to Topology. The Mathematical Association of America, Washington (2009). Ix + 107 pp
Leader, S.: On clusters in proximity spaces. Fundamenta Mathematicae 47, 205–213 (1959)
Naimpally, S.: Proximity Approach to Problems in Topology and Analysis. Oldenbourg Verlag, Munich (2009). 73 pp., ISBN 978-3-486-58917-7, MR2526304
Mozzochi, C., Gagrat, M., Naimpally, S.: Symmetric Generalized Topological Structures. Exposition Press, Hicksville (1976). I + 73 pp
Di Concilio, A., Naimpally, S.: Proximal convergence. Monatsh. Math. 103, 93–102 (1987)
Willard, S.: General Topology. Dover Publications Inc, Mineola (1970). Xii + 369 pp, ISBN: 0-486-43479-6 54-02, MR0264581
Install, M., Weisstein, E.: Connected set. Mathworld. A Wolfram Web Resource (2015). http://mathworld.wolfram.com/ConnectedSet.html
Leader, S.: Extensions based on proximity and boundedness. Math. Z. 108, 137–144 (1969)
Leader, S.: Local proximity spaces. Mathematische Annalen 169, 275–281 (1967)
Sierpiński, W.: Sur une courbe dont tout point est un point de ramification. C.R.A.S. 160, 302–305 (1915)
Wolfram, S.: A New Kind of Science. Wolfram Media Inc, Champaign (2002). Xiv + 1197 pp., MR1920418
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Peters, J.F. (2016). Computational Proximity. In: Computational Proximity. Intelligent Systems Reference Library, vol 102. Springer, Cham. https://doi.org/10.1007/978-3-319-30262-1_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-30262-1_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-30260-7
Online ISBN: 978-3-319-30262-1
eBook Packages: EngineeringEngineering (R0)