Temporal Proximity of 1-cycles in CW Spaces, Time-Varying Cell Complexes

Abstract

This paper introduces approximate temporal proximities of 1-cycle cell complexes in a spacetime view of a planar Whitehead CW space. Divergence of the vector field of a 1-cycle provides a natural basis for an approximate Temporal Proximity (aTP) of time-varying 1-cycles useful in the detection, characterization, analysis, and measurement of the closeness of changing geometric realizations of simplicial complexes in a J.H.C Whitehead CW topological space. A practical application of aTP is given in terms of the temporal closeness of 1-cycle shapes in sequences of video frames. A main result in this paper is that every pair of cell complexes with the same descriptions over the same temporal interval have two properties, namely, (i) persistence and (ii) approximate temporal closeness.

Keywords

• CW space
• 1-Cycle
• Cell Complex
• Divergence
• Temporal Proximity

References

1. [1] Bronshtein I.N., Semendyayev K.A., Musiol G., Mühlig H., Handbook of Mathematics, 6th Ed., Springer, 2015.
2. [2] İnan E., Öztürk M.A., Near groups on nearness approximation spaces, Hacettepe Journal of Mathematics and Statistics, 41(4), 545-558, 2012.
3. [3] İnan E., Öztürk M.A., Erratum and notes for near groups on nearness approximation spaces, Hacettepe Journal of Mathematics and Statistics, 43(2), 279-281, 2014.
4. [4] Haider M.S., Peters J.F., Temporal proximities. Self-similar, temporally close shapes, Chaos, Solitons and Fractals, 151(111237), 1-10, 2021.
5. [5] Hausdorff F., Grundzüge der Mengenlehre, Veit and Company, 1914.
6. [6] Hausdorff F., Set Theory, AMS Chelsea Publishing, Providence, RI, 1957.
7. [7] İnan E., Approximately rings in proximal relator spaces, Turkish Journal of Mathematics, 43(6), 2941-2953, 2019.
8. [8] Lefschetz S., Introduction to Topology, Princeton Mathematical Series, Vol. 11, Princeton University Press, 1949.

9. [9] Matthews P.C., Vector Calculus, Springer Undergraduate Mathematics Series, Springer-Verlag London, 1998.
10. [10] Munkres J.R., Elements of Algebraic Topology, 2nd Ed., Perseus Publishing, 1984.
11. [11] Myškis A.D., Advanced Mathematics for Engineers, MIR Publishers, 1975.
12. [12] Naimpally S.A., Peters J.F., Topology with Applications. Topological Spaces via Near and Far, World Scientific, 2013.
13. [13] Naimpally S.A., Warrack B.D., Proximity Spaces, Cambridge Tract in Mathematics No. 59, Cambridge University Press, 1970.
14. [14] Öztürk M.A., Uçkun M., İnan E., Near groups of weak cosets on nearness approximation spaces, Fundamenta Informaticae, 133(4), 433-448, 2014.
15. [15] Peters J.F., Near sets: An introduction, Mathematics in Computer Science, 7(1), 3-9, 2013.
16. [16] Peters J.F., Computational Proximity. Excursions in the Topology of Digital Images, Intelligent Systems Reference Library 102, Springer, 2016.
17. [17] Peters J.F., Proximal relator spaces, Filomat, 30(2), 469-472, 2016.
18. [18] Peters J.F., Proximal planar shapes. Correspondence between triangulated shapes and nerve complexes, Bulletin of the Allahabad Mathematical Society, 33, 113-137, 2018.
19. [19] Peters J.F., Vortex nerves and their proximities. Nerve betti numbers and descriptive proximity, Bulletin of The Allahabad Mathematical Society, 34(2), 263-276, 2019.
20. [20] Peters J.F., Computational Geometry, Topology and Physics of Digital Images with Applications. Shape Complexes, Optical Vortex Nerves and Proximities, Springer Nature, 2020.
21. [21] Peters J.F., Ribbon complexes and their approximate descriptive proximities. Ribbon and vortex nerves, Betti numbers and planar divisions, Bulletin of the Allahabad Mathematical Society, 35, 1-13, 2020.
22. [22] Peters J.F., Amiable and almost amiable fixed sets. Extension of the brouwer fixed point theorem, Glasnik Matematički, 56(76), 175-194, 2021.
23. [23] Peters J.F., Homotopic nerve complexes with free group presentations, In Int. Online Conf. Algebraic and Geometric Methods of Analysis, 25-28 May 2021, Ukraine, pages 110-111.
24. [24] Peters J.F., İnan E., Öztürk M.A., Spatial and descriptive isometries in proximity spaces, General Mathematics Notes, 21(2), 125-134, 2014.
25. [25] Peters J.F., Öztürk M.A., Uçkun M., Klee-phelps convex groupoids, Mathematica Slovaca, 67(2), 397-400, 2017.
26. [26] Peters J.F., Ramanna S., Treasure trove at banacha. Set patterns in descriptive proximity spaces, Fundamenta Informaticae, 127(1-4), 357-367, 2013.
27. [27] Rotman J.J., An introduction to the Theory of Groups, 4th Ed., Springer-Verlag, 1995.
28. [28] Smirnov J.M., On proximity spaces, Matematicheskii Sbornik (N.S.), 31(73), 543–574, 1952 (English translation: Amer. Math. Soc. Trans. Ser., 2, 38, 5-35, 1964).
29. [29] Switzer R.M., Algebraic Topology–Homology and Homotopy, Springer, 2002.
30. [30] Uçkun M., Approximately Γ-rings in proximal relator spaces, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(2), 1780-1796, 2019.
31. [31] Uçkun M., Genç A., Near-rings on nearness approximation spaces, Turkish Journal of Mathematics, 45(1), 549-565, 2021.
32. [32] Uçkun M., İnan E., Approximately Γ-semigroups in proximal relator spaces, Applicable Algebra in Engineering, Communication and Computing, 30, 299-311, 2018.
33. [33] Čech E., Topological Spaces, John Wiley and Sons Ltd., 1966.
34. [34] Whitehead J.H.C., Simplicial spaces, nuclei and m-groups, Proceedings of the London Mathematical Society, 45, 243-327, 1939.
35. [35] Whitehead J.H.C., Combinatorial homotopy I, Bulletin of the American Mathematical Society, 55(3), 213–245, 1949.