Proximal vortex cycles and vortex nerve structures. Non-concentric, nesting, possibly overlapping homology cell complexes

James F. Peters

doi:10.33187/jmsm.425066

Araştırma Makalesi

Yıl 2018, Cilt: 1 Sayı: 2, 56 - 72, 30.09.2018

James F. Peters

https://doi.org/10.33187/jmsm.425066

Cited By: 4

Öz

Kaynakça

[1] M.Z. Ahmad and J.F. Peters, Proximal C˘ech complexes in approximating digital image object shapes. Theory and application, Theory and Applications of Math. & Comp. Sci. 7 (2017), no. 2, 81–123, MR3769444.
[2] V.A. Baikov, R.R. Gilmanov, I.A. Taimanov, and A.A. Yakovlev, Topological characteristics of oil and gass reservoirs and their applications, Integrative Machine Learning, LNAI 10344 (A. Halzinger et. al., ed.), Springer, Berlin, 2017, pp. 182–193.
[3] D. Baldomir and P. Hammond, Geometry of electromagnetic systems, Oxford,UK, Clarendon Press, 1996, xi+239 pp., Zbl 0919.76001.
[4] J.M.M. Barata, P.J.C.T. Santos N. Bernardo, and A.R.R. Silva, Experimental study of a ground vortex: the effect of the crossflow velocity, 49th AIAA Aerospace Sciences Meeting, AIAA, 2011, pp. 1–9.
[5] J.S. Birman and W.W. Menasco, Studying links via closed braids. V: the unlink, Trans. Amer. Math. Soc. 329 (1992), no. 2, 585–606.
[6] A. Di Concilio, C. Guadagni, J.F. Peters, and S. Ramanna, Descriptive proximities I: Properties and interplay between classical proximities and overlap, arXiv 1609 (2016), no. 06246v1, 1–12, Math. in Comp. Sci. 2017, https://doi.org/10.1007/s11786-017-0328-y, in press.
[7] , Descriptive proximities. properties and interplay between classical proximities and overlap, Math. Comput. Sci. 12 (2018), no. 1, 91–106, MR3767897.
[8] G.E. Cooke and R.L. Finney, Homology of cell complexes. based on lectures by norman e. steenrod, Princeton University Press; University of Tokyo Press, Tokyo, Princeton, New Jersey, 1967, xv+256 pp., MR0219059.
[9] E. Cui, Video vortex cat cycles part 1, Tech. report, University of Manitoba, Computational Intelligence Laboratory, Deparment of Electrical & Computer Engineering, U of MB, Winnipeg, MB R3T 5V6, Canada, 2018, https://youtu.be/rVGmkGTm4Oc.
[10] , Video vortex cat cycles part 2, Tech. report, University of Manitoba, Computational Intelligence Laboratory, Deparment of Electrical & Computer Engineering, U of MB, Winnipeg, MB R3T 5V6, Canada, 2018, https://youtu.be/yJBCdLhgcqk.
[11] A. Dareau, E. Levy, M.B. Aguilera, R. Bouganne, E. Akkermans, F. Gerbier, and J. Beugnon, Revealing the topology of quasicrystals with a diffraction experiment, arXiv, Physical Review Letters 1607 (2017), no. 00901v2, 1–7, doi.org/10.1103/PhysRevLett.119.215304.
[12] I.V. Dzedolik, Vortex properties of a photon flux in a dielectic waveguide, Technical Physics 75 (2005), no. 1, 137–140.
[13] , Vortex properties of a photon flux in a dielectric waveguide, Technical Physics [trans. from Zhurnal Tekhnicheskol Fiziki] 50 (2005), no. 1, 135–138.
[14] G. Dvali E. Adelberger and A. Gruzinov, Structured light meets structured matter, Phys. Rev. Letters 98 (2007), 010402–1–010402–4.
[15] H. Edelsbrunner and J.L. Harer, Computational topology. An introduction, Amer. Math. Soc., Providence, RI, 2010, xii+241 pp. ISBN: 978-0-8218-4925-5, MR2572029.
[16] M. Fermi, Persistent topology for natural data analysis - a survey, arXiv 1706 (2017), no. 00411v2, 1–18.
[17] , Why topology for machine learning and knowledge extraction, Machine Learning & Knowledge Extraction 1 (2018), no. 6, 1–6, https: //doi.org/10.3390/make1010006.
[18] A. Flammini and A. Stasiak, Natural classification of knots, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2017), no. 2078, 569–582, MR2288834.
[19] C. Guadagni, Bornological convergences on local proximity spaces and wm -metric spaces, Ph.D. thesis, Università degli Studi di Salerno, Salerno, Italy, 2015, Supervisor: A. Di Concilio, 79pp.
[20] M. Ostavari H. Boomari and A. Zarei, Recognizing visibility graphs of polygons with holes and internal-external visibility graphs of polygons, arXiv 1804 (2018), no. 05105v1, 1–16.
[21] M. Hance, Algebraic structures on nearness approximation spaces, Ph.D. thesis, University of Pennsylvania, Department of Physics and Astronomy, 2015, supervisor: H.H. Williams, vii+113pp.
[22] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, UK, 2002, xii+544 pp. ISBN: 0-521-79160-X, MR1867354.
[23] C. Jordan, Cours d’analyse de l’École polytechnique, tome i-iii, Éditions Jacques Gabay, Sceaux, 1991, reprint of 1915 edition, Tome I: MR1188186,Tome II: MR1188187, Tome III: MR1188188.
[24] W. Thomson (Lord Kelvin), On vortex atoms, Proc. Roy. Soc. Edin. 6 (1867), 94–105.
[25] S. Leader, On clusters in proximity spaces, Fundamenta Mathematicae 47 (1959), 205–213.
[26] N.M. Litchinitser, Structured light meets structured matter, Science, New Series 337 (2012), no. 6098, 1054–1055.
[27] , Structured light meets structured matter, Science, New Series 337 (2012), no. 6098, 1054–1055.
[28] R. Maehara, The Jordan curve theorem via the Brouwer fixed point theorem, Amer. Math. Monthly 91 (1984), no. 10, 641–643, MR0769530.
[29] J.R. Munkres, Topology, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 2000, xvi + 537 pp., 1st Ed. in 1975,MR0464128.
[30] J.P. Murphy and D.G. MacManus, Ground vortex aerodynamics under crosswind conditions, Experiments in Fluids 50 (2011), no. 1, 109–124.
[31] M. Pellikka, S. Suuriniemi, and L. Kettunen, Homology in electromagnetic boundary value problems, Boundary Value Problems 2010 (2010), no. 381953, 1–18, doi:10.1155/2010/381953.
[32] J. F. Peters and E. ˙Inan, Strongly proximal Edelsbrunner-Harer nerves, Proc. Jangjeon Math. Soc. 19 (2016), no. 3, 1–20, MR3618825, zbMATH Zbl 1360.54021.
[33] J.F. Peters, Local near sets: Pattern discovery in proximity spaces, Math. in Comp. Sci. 7 (2013), no. 1, 87–106, DOI 10.1007/s11786-013-0143-z, MR3043920, ZBL06156991.
[34] , Local near sets: pattern discovery in proximity spaces, Math. Comput. Sci. 7 (2013), no. 1, 87–106, MR3043920, ZBL06156991.
[35] , Computational proximity. Excursions in the topology of digital images., Intelligent Systems Reference Library 102 (2016), xxviii + 433pp, ISBN: 978-3-319-30260-7; 978-3-319-30262-1; DOI: 10.1007/978-3-319-30262-1; zbMATH Zbl 1382.68008;MR3727129.
[36] , Proximal relator spaces, Filomat 30 (2016), no. 2, 469–472, doi:10.2298/FIL1602469P, MR3497927.
[37] , Two forms of proximal, physical geometry. Axioms, sewing regions together, classes of regions, duality and parallel fibre bundles, Advan. in Math: Sci. J 5 (2016), no. 2, 241–268, Zbl 1384.54015, reviewed by D. Leseberg, Berlin.
[38] , Foundations of computer vision. Computational geometry, visual image structures and object shape recognition, Intelligent Systems Ref. Library 124, Springer International Pub. AG, Cham, Switzerland, 2017, xvii+431 pp., ISBN: 978-3-319-52483-2; 978-3-319-52481-8; DOI 10.1007/978-3-319-52483-2; MR3769444.
[39] , Proximal planar shape signatures. Homology nerves and descriptive proximity, Advan. in Math: Sci. J 6 (2017), no. 2, 71–85, Zbl 06855051.
[40] , Proximal planar shapes. Correspondence between shape and nerve complexes, arXiv 1708 (2017), no. 04147v1, 1–12, Bulletin of the Allahabad Math. Soc., Dharma Prokash Gupta Memorial Volume, 2018 in press.
[41] J.F. Peters, A. Tozzi, and S. Ramanna, Brain tissue tessellation shows absence of canonical microcircuits, Neuroscience Letters 626 (2016), 99–105, http://dx.doi.org/10.1016/j.neulet.2016.03.052.
[42] A.K. Travin P.R. Spalart, M. Kh. Strelets and M.L. Slur, Modeling the interaction of a vortex pair with the ground, Fluid Dynamics 36 (1999), no. 6, 899–908.
[43] A.R.R. Silva, D.F.G. Dur ao, J.M.M. Barata, P. Santos, and S. Ribeiro, Laser-doppler analysis of the separation zone of a ground vortex flow, 14th Symp on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, Universidade Beira Interior, 2008, pp. 7–10.
[44] Ju. M. Smirnov, On proximity spaces, American Math. Soc. Translations, Series 2, vol. 38 (AMS, ed.), Amer. Math. Soc., Providene, RI, 1964, pp. 3–36.
[45] Á Száz, Basic tools and mild continuities in relator spaces, Acta Math. Hungar. 50 (1987), no. 3-4, 177–201, MR0918156.
[46] S. De Toffoli and V. Giardino, Forms and roles of diagrams in knot theory, Erkenntnis 79 (2014), no. 4, 829–842, MR3260948.
[47] A. Tozzi, J.F. Peters, and E. Deli, Towards plasma-like collisionless trajectories in the brain, Neuroscience Letters 662 (2018), 105–109.
[48] E. Cech, Topological spaces, John Wiley & Sons Ltd., London, 1966, fr seminar, Brno, 1936-1939; rev. ed. Z. Frolik, M. Kate˘tov. Scientific editor, Vlastimil Ptak. Editor of the English translation, Charles O. Junge Publishing House of the Czechoslovak Academy of Sciences, Prague; Interscience Publishers John Wiley & Sons, London-New York-Sydney 1966 893 pp., MR0211373.
[49] H. van Leunen, The hilbert book model project, Tech. report, Deparment of Applied Physics, Technische Universiteit Eindhoven, 2018, https: //www.researchgate.net/project/The-Hilbert-Book-Model-Project.
[50] O. Veblen, Theory on plane curves in non-metrical analysis situs, Transactions of the American Mathematical Society 6 (1905), no. 1, 83–98, MR1500697.
[51] J.H.C. Whitehead, Combinatorial homotopy. I., Bulletin of the American Mathematical Society 55 (1949), no. 3, 213–245, Part 1, MR0030759.
[52] S. Willard, General topology, Dover Pub., Inc., Mineola, NY, 1970, xii + 369pp, ISBN: 0-486-43479-6 54-02, MR0264581.
[53] G.M. Ziegler, Lectures on polytopes, Springer, Berlin, 2007, x+370 pp. ISBN: 0-387-94365-X, MR1311028.

Proximal vortex cycles and vortex nerve structures. Non-concentric, nesting, possibly overlapping homology cell complexes

Yıl 2018, Cilt: 1 Sayı: 2, 56 - 72, 30.09.2018

James F. Peters

https://doi.org/10.33187/jmsm.425066

Cited By: 4

Öz

This article introduces proximal planar vortex 1-cycles, resembling the structure of vortex atoms introduced by William Thomson (Lord Kelvin) in 1867 and recent work on the proximity of sets that overlap either spatially or descriptively. Vortex cycles resemble Thomson's model of a vortex atom, inspired by P.G. Tait's smoke rings. A vortex cycle is a collection of non-concentric, nesting 1-cycles with nonempty interiors i.e., a collection of 1-cycles that share a nonempty set of interior points and which may or may not overlap). Overlapping 1-cycles in a vortex yield an Edelsbrunner-Harer nerve within the vortex. Overlapping vortex cycles constitute a vortex nerve complex. Several main results are given in this paper, namely, a Whitehead CW topology and a Leader uniform topology are outcomes of having a collection of vortex cycles (or nerves) equipped with a connectedness proximity and the case where each cluster of closed, convex vortex cycles and the union of the vortex cycles in the cluster have the same homotopy type.

Anahtar Kelimeler

Connectedness Proximity, CW Topology, Vortex Cycle, Vortex Nerve, Vortex Nerve

Kaynakça

[1] M.Z. Ahmad and J.F. Peters, Proximal C˘ech complexes in approximating digital image object shapes. Theory and application, Theory and Applications of Math. & Comp. Sci. 7 (2017), no. 2, 81–123, MR3769444.
[2] V.A. Baikov, R.R. Gilmanov, I.A. Taimanov, and A.A. Yakovlev, Topological characteristics of oil and gass reservoirs and their applications, Integrative Machine Learning, LNAI 10344 (A. Halzinger et. al., ed.), Springer, Berlin, 2017, pp. 182–193.
[3] D. Baldomir and P. Hammond, Geometry of electromagnetic systems, Oxford,UK, Clarendon Press, 1996, xi+239 pp., Zbl 0919.76001.
[4] J.M.M. Barata, P.J.C.T. Santos N. Bernardo, and A.R.R. Silva, Experimental study of a ground vortex: the effect of the crossflow velocity, 49th AIAA Aerospace Sciences Meeting, AIAA, 2011, pp. 1–9.
[5] J.S. Birman and W.W. Menasco, Studying links via closed braids. V: the unlink, Trans. Amer. Math. Soc. 329 (1992), no. 2, 585–606.
[6] A. Di Concilio, C. Guadagni, J.F. Peters, and S. Ramanna, Descriptive proximities I: Properties and interplay between classical proximities and overlap, arXiv 1609 (2016), no. 06246v1, 1–12, Math. in Comp. Sci. 2017, https://doi.org/10.1007/s11786-017-0328-y, in press.
[7] , Descriptive proximities. properties and interplay between classical proximities and overlap, Math. Comput. Sci. 12 (2018), no. 1, 91–106, MR3767897.
[8] G.E. Cooke and R.L. Finney, Homology of cell complexes. based on lectures by norman e. steenrod, Princeton University Press; University of Tokyo Press, Tokyo, Princeton, New Jersey, 1967, xv+256 pp., MR0219059.
[9] E. Cui, Video vortex cat cycles part 1, Tech. report, University of Manitoba, Computational Intelligence Laboratory, Deparment of Electrical & Computer Engineering, U of MB, Winnipeg, MB R3T 5V6, Canada, 2018, https://youtu.be/rVGmkGTm4Oc.
[10] , Video vortex cat cycles part 2, Tech. report, University of Manitoba, Computational Intelligence Laboratory, Deparment of Electrical & Computer Engineering, U of MB, Winnipeg, MB R3T 5V6, Canada, 2018, https://youtu.be/yJBCdLhgcqk.
[11] A. Dareau, E. Levy, M.B. Aguilera, R. Bouganne, E. Akkermans, F. Gerbier, and J. Beugnon, Revealing the topology of quasicrystals with a diffraction experiment, arXiv, Physical Review Letters 1607 (2017), no. 00901v2, 1–7, doi.org/10.1103/PhysRevLett.119.215304.
[12] I.V. Dzedolik, Vortex properties of a photon flux in a dielectic waveguide, Technical Physics 75 (2005), no. 1, 137–140.
[13] , Vortex properties of a photon flux in a dielectric waveguide, Technical Physics [trans. from Zhurnal Tekhnicheskol Fiziki] 50 (2005), no. 1, 135–138.
[14] G. Dvali E. Adelberger and A. Gruzinov, Structured light meets structured matter, Phys. Rev. Letters 98 (2007), 010402–1–010402–4.
[15] H. Edelsbrunner and J.L. Harer, Computational topology. An introduction, Amer. Math. Soc., Providence, RI, 2010, xii+241 pp. ISBN: 978-0-8218-4925-5, MR2572029.
[16] M. Fermi, Persistent topology for natural data analysis - a survey, arXiv 1706 (2017), no. 00411v2, 1–18.
[17] , Why topology for machine learning and knowledge extraction, Machine Learning & Knowledge Extraction 1 (2018), no. 6, 1–6, https: //doi.org/10.3390/make1010006.
[18] A. Flammini and A. Stasiak, Natural classification of knots, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2017), no. 2078, 569–582, MR2288834.
[19] C. Guadagni, Bornological convergences on local proximity spaces and wm -metric spaces, Ph.D. thesis, Università degli Studi di Salerno, Salerno, Italy, 2015, Supervisor: A. Di Concilio, 79pp.
[20] M. Ostavari H. Boomari and A. Zarei, Recognizing visibility graphs of polygons with holes and internal-external visibility graphs of polygons, arXiv 1804 (2018), no. 05105v1, 1–16.
[21] M. Hance, Algebraic structures on nearness approximation spaces, Ph.D. thesis, University of Pennsylvania, Department of Physics and Astronomy, 2015, supervisor: H.H. Williams, vii+113pp.
[22] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, UK, 2002, xii+544 pp. ISBN: 0-521-79160-X, MR1867354.
[23] C. Jordan, Cours d’analyse de l’École polytechnique, tome i-iii, Éditions Jacques Gabay, Sceaux, 1991, reprint of 1915 edition, Tome I: MR1188186,Tome II: MR1188187, Tome III: MR1188188.
[24] W. Thomson (Lord Kelvin), On vortex atoms, Proc. Roy. Soc. Edin. 6 (1867), 94–105.
[25] S. Leader, On clusters in proximity spaces, Fundamenta Mathematicae 47 (1959), 205–213.
[26] N.M. Litchinitser, Structured light meets structured matter, Science, New Series 337 (2012), no. 6098, 1054–1055.
[27] , Structured light meets structured matter, Science, New Series 337 (2012), no. 6098, 1054–1055.
[28] R. Maehara, The Jordan curve theorem via the Brouwer fixed point theorem, Amer. Math. Monthly 91 (1984), no. 10, 641–643, MR0769530.
[29] J.R. Munkres, Topology, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 2000, xvi + 537 pp., 1st Ed. in 1975,MR0464128.
[30] J.P. Murphy and D.G. MacManus, Ground vortex aerodynamics under crosswind conditions, Experiments in Fluids 50 (2011), no. 1, 109–124.
[31] M. Pellikka, S. Suuriniemi, and L. Kettunen, Homology in electromagnetic boundary value problems, Boundary Value Problems 2010 (2010), no. 381953, 1–18, doi:10.1155/2010/381953.
[32] J. F. Peters and E. ˙Inan, Strongly proximal Edelsbrunner-Harer nerves, Proc. Jangjeon Math. Soc. 19 (2016), no. 3, 1–20, MR3618825, zbMATH Zbl 1360.54021.
[33] J.F. Peters, Local near sets: Pattern discovery in proximity spaces, Math. in Comp. Sci. 7 (2013), no. 1, 87–106, DOI 10.1007/s11786-013-0143-z, MR3043920, ZBL06156991.
[34] , Local near sets: pattern discovery in proximity spaces, Math. Comput. Sci. 7 (2013), no. 1, 87–106, MR3043920, ZBL06156991.
[35] , Computational proximity. Excursions in the topology of digital images., Intelligent Systems Reference Library 102 (2016), xxviii + 433pp, ISBN: 978-3-319-30260-7; 978-3-319-30262-1; DOI: 10.1007/978-3-319-30262-1; zbMATH Zbl 1382.68008;MR3727129.
[36] , Proximal relator spaces, Filomat 30 (2016), no. 2, 469–472, doi:10.2298/FIL1602469P, MR3497927.
[37] , Two forms of proximal, physical geometry. Axioms, sewing regions together, classes of regions, duality and parallel fibre bundles, Advan. in Math: Sci. J 5 (2016), no. 2, 241–268, Zbl 1384.54015, reviewed by D. Leseberg, Berlin.
[38] , Foundations of computer vision. Computational geometry, visual image structures and object shape recognition, Intelligent Systems Ref. Library 124, Springer International Pub. AG, Cham, Switzerland, 2017, xvii+431 pp., ISBN: 978-3-319-52483-2; 978-3-319-52481-8; DOI 10.1007/978-3-319-52483-2; MR3769444.
[39] , Proximal planar shape signatures. Homology nerves and descriptive proximity, Advan. in Math: Sci. J 6 (2017), no. 2, 71–85, Zbl 06855051.
[40] , Proximal planar shapes. Correspondence between shape and nerve complexes, arXiv 1708 (2017), no. 04147v1, 1–12, Bulletin of the Allahabad Math. Soc., Dharma Prokash Gupta Memorial Volume, 2018 in press.
[41] J.F. Peters, A. Tozzi, and S. Ramanna, Brain tissue tessellation shows absence of canonical microcircuits, Neuroscience Letters 626 (2016), 99–105, http://dx.doi.org/10.1016/j.neulet.2016.03.052.
[42] A.K. Travin P.R. Spalart, M. Kh. Strelets and M.L. Slur, Modeling the interaction of a vortex pair with the ground, Fluid Dynamics 36 (1999), no. 6, 899–908.
[43] A.R.R. Silva, D.F.G. Dur ao, J.M.M. Barata, P. Santos, and S. Ribeiro, Laser-doppler analysis of the separation zone of a ground vortex flow, 14th Symp on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, Universidade Beira Interior, 2008, pp. 7–10.
[44] Ju. M. Smirnov, On proximity spaces, American Math. Soc. Translations, Series 2, vol. 38 (AMS, ed.), Amer. Math. Soc., Providene, RI, 1964, pp. 3–36.
[45] Á Száz, Basic tools and mild continuities in relator spaces, Acta Math. Hungar. 50 (1987), no. 3-4, 177–201, MR0918156.
[46] S. De Toffoli and V. Giardino, Forms and roles of diagrams in knot theory, Erkenntnis 79 (2014), no. 4, 829–842, MR3260948.
[47] A. Tozzi, J.F. Peters, and E. Deli, Towards plasma-like collisionless trajectories in the brain, Neuroscience Letters 662 (2018), 105–109.
[48] E. Cech, Topological spaces, John Wiley & Sons Ltd., London, 1966, fr seminar, Brno, 1936-1939; rev. ed. Z. Frolik, M. Kate˘tov. Scientific editor, Vlastimil Ptak. Editor of the English translation, Charles O. Junge Publishing House of the Czechoslovak Academy of Sciences, Prague; Interscience Publishers John Wiley & Sons, London-New York-Sydney 1966 893 pp., MR0211373.
[49] H. van Leunen, The hilbert book model project, Tech. report, Deparment of Applied Physics, Technische Universiteit Eindhoven, 2018, https: //www.researchgate.net/project/The-Hilbert-Book-Model-Project.
[50] O. Veblen, Theory on plane curves in non-metrical analysis situs, Transactions of the American Mathematical Society 6 (1905), no. 1, 83–98, MR1500697.
[51] J.H.C. Whitehead, Combinatorial homotopy. I., Bulletin of the American Mathematical Society 55 (1949), no. 3, 213–245, Part 1, MR0030759.
[52] S. Willard, General topology, Dover Pub., Inc., Mineola, NY, 1970, xii + 369pp, ISBN: 0-486-43479-6 54-02, MR0264581.
[53] G.M. Ziegler, Lectures on polytopes, Springer, Berlin, 2007, x+370 pp. ISBN: 0-387-94365-X, MR1311028.

Toplam 53 adet kaynakça vardır.

Ayrıntılar

Birincil Dil	İngilizce
Konular	Matematik
Bölüm	Makaleler
Yazarlar	James F. Peters
Yayımlanma Tarihi	30 Eylül 2018
Gönderilme Tarihi	18 Mayıs 2018
Kabul Tarihi	17 Temmuz 2018
Yayımlandığı Sayı	Yıl 2018 Cilt: 1 Sayı: 2

Kaynak Göster

APA	Peters, J. F. (2018). Proximal vortex cycles and vortex nerve structures. Non-concentric, nesting, possibly overlapping homology cell complexes. Journal of Mathematical Sciences and Modelling, 1(2), 56-72. https://doi.org/10.33187/jmsm.425066
AMA	Peters JF. Proximal vortex cycles and vortex nerve structures. Non-concentric, nesting, possibly overlapping homology cell complexes. Journal of Mathematical Sciences and Modelling. Eylül 2018;1(2):56-72. doi:10.33187/jmsm.425066
Chicago	Peters, James F. “Proximal Vortex Cycles and Vortex Nerve Structures. Non-Concentric, Nesting, Possibly Overlapping Homology Cell Complexes”. Journal of Mathematical Sciences and Modelling 1, sy. 2 (Eylül 2018): 56-72. https://doi.org/10.33187/jmsm.425066.
EndNote	Peters JF (01 Eylül 2018) Proximal vortex cycles and vortex nerve structures. Non-concentric, nesting, possibly overlapping homology cell complexes. Journal of Mathematical Sciences and Modelling 1 2 56–72.
IEEE	J. F. Peters, “Proximal vortex cycles and vortex nerve structures. Non-concentric, nesting, possibly overlapping homology cell complexes”, Journal of Mathematical Sciences and Modelling, c. 1, sy. 2, ss. 56–72, 2018, doi: 10.33187/jmsm.425066.
ISNAD	Peters, James F. “Proximal Vortex Cycles and Vortex Nerve Structures. Non-Concentric, Nesting, Possibly Overlapping Homology Cell Complexes”. Journal of Mathematical Sciences and Modelling 1/2 (Eylül 2018), 56-72. https://doi.org/10.33187/jmsm.425066.
JAMA	Peters JF. Proximal vortex cycles and vortex nerve structures. Non-concentric, nesting, possibly overlapping homology cell complexes. Journal of Mathematical Sciences and Modelling. 2018;1:56–72.
MLA	Peters, James F. “Proximal Vortex Cycles and Vortex Nerve Structures. Non-Concentric, Nesting, Possibly Overlapping Homology Cell Complexes”. Journal of Mathematical Sciences and Modelling, c. 1, sy. 2, 2018, ss. 56-72, doi:10.33187/jmsm.425066.
Vancouver	Peters JF. Proximal vortex cycles and vortex nerve structures. Non-concentric, nesting, possibly overlapping homology cell complexes. Journal of Mathematical Sciences and Modelling. 2018;1(2):56-72.