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On ultrametric-preserving functions. (English) Zbl 1505.54043


MSC:

54E35 Metric spaces, metrizability

References:

[1] Bernig, A.—Foertsch, T.—Schroeder, V.: Non standard metric products, Beiträge Algebra Geom. 44 (2003), 499-510. · Zbl 1049.54009
[2] Bestvina, M.: R-trees in topology, geometry and group theory. In: Handbook of Geometric Topology (R. J. Daverman, R. B. Sher, eds.), Nort-Holland, Amsterdam, 2002, pp. 55-91. · Zbl 0998.57003
[3] Beyrer, J.—Schroeder, V.: Trees and ultrametric Möbius structures, P-adic Numbers Ultrametr. Anal. Appl. 9 (2017), 247-256. · Zbl 1387.53022
[4] Borsík, J.—Doboš, J.: Functions whose composition with every metric is a metric, Math. Slovaca 31 (1981), 3-12. · Zbl 0482.54021
[5] Borsík, J.—Doboš, J.: On metric preserving functions, Real Anal. Exchange 13 (1988), 285-293. · Zbl 0684.54020
[6] Carlsson, G.—Mémoli, F.: Characterization, stability and convergence of hierarchical clustering methods, J. Machine Learn. Res. 11 (2010), 1425-1470. · Zbl 1242.62050
[7] Corazza, P.: Introduction to metric-preserving functions, Amer. Math. Monthly 104 (1999), 309-323. · Zbl 1077.54501
[8] Das, P. P.: Metricity preserving transforms, Pattern Recognition Letters 10 (1989), 73-76. · Zbl 0800.68775
[9] Delhommé, C.—Laflamme, C.—Pouzet, M.—Sauer, N.: Indivisible ultrametric spaces, Topology Appl. 155 (2008), 1462-1478. · Zbl 1158.54011
[10] Demaine, E. D.—Landau, G. M.—Weimann, O.: On Cartesian trees and range minimum queries. In: Proceedings of the 36th International Colloquium, ICALP 2009, Rhodes, Greece, 2009, Part I, Lecture Notes in Comput. Sci. 5555, pp. 341-353. Springer-Berlin-Heidelberg, 2009. · Zbl 1248.68165
[11] Deza, M. M.—Deza, E.: Encyclopedia of Distances, Springer-Verlag Berlin Heidelberg, 2009. · Zbl 1167.51001
[12] Doboš, J.: A survey of metric-preserving functions, Questions Answers Gen. Topology 13 (1995), 129-134. · Zbl 0917.54034
[13] Doboš, J.: On modification of the Euclidean metric on reals, Tatra Mt. Math. Publ. 8 (1996), 51-54. · Zbl 0928.26008
[14] Doboš, J.: Metric Preserving Functions, Štroffek, Košice, Slovakia, 1998.
[15] Doboš, J.—Piotrowski, Z.: Some remarks on metric-preserving functions, Real Anal. Exchange 19 (1994), 317-320. · Zbl 0795.26004
[16] Doboš, J.—Piotrowski, Z.: A note on metric preserving functions, Int. J. Math. Math. Sci. 19 (1996), 199-200. · Zbl 0840.26006
[17] Dordovskyi, D.—Dovgoshey, O.—Petrov, E.: Diameter and diametrical pairs of points in ultrametric spaces, P-adic Numbers Ultrametr. Anal. Appl. 3 (2011), 253-262. · Zbl 1304.54049
[18] Dovgoshey, O.—Dordovskyi, D.: Ultrametricity and metric betweenness in tangent spaces to metric spaces, P-adic Numbers Ultrametr. Anal. Appl. 2 (2010), 109-113. · Zbl 1276.53052
[19] Dovgoshey, O.—Martio, O.: Blow up of balls and coverings in metric spaces, Manuscripta Math. 127 (2008), 89-120. · Zbl 1155.54017
[20] Dovgoshey, O.—Martio, O.: Products of metric spaces, covering numbers, packing numbers and characterizations of ultrametric spaces, Rev. Roumaine Math. Pures. Appl. 54 (2009), 423-439. · Zbl 1212.54088
[21] Dovgoshey, O.—Martio, O.: Functions transferring metrics to metrics, Beiträge Algebra Geom. 54 (2013), 237-261. · Zbl 1276.54020
[22] Dovgoshey, O.—Petrov, E.: Subdominant pseudoultrametric on graphs, Sb. Math 204 (2013), 1131-1151. · Zbl 1276.05056
[23] Dovgoshey, O.—Petrov, E.: From isomorphic rooted trees to isometric ultrametric spaces, P-adic Numbers Ultrametr. Anal. Appl. 10 (2018), 287-298. · Zbl 1435.54011
[24] Dovgoshey, O.—Petrov, E.: Properties and morphisms of finite ultrametric spaces and their representing trees, P-adic Numbers Ultrametr. Anal. Appl. 11 (2019), 1-20. · Zbl 1433.54015
[25] Dovgoshey, O.—Petrov, E.—Kozub, G.: Metric products and continuation of isotone functions, Math. Slovaca 64 (2014), 187-208. · Zbl 1324.54048
[26] Dovgoshey, O.—Petrov, E.—Teichert, H.-M.: On spaces extremal for the Gomory-Hu inequality, P-adic Numbers Ultrametr. Anal. Appl. 7 (2015), 133-142. · Zbl 1343.05072
[27] Dovgoshey, O.—Petrov, E.—Teichert, H.-M.: How rigid the finite ultrametric spaces can be?, Fixed Point Theory Appl. 19 (2017), 1083-1102. · Zbl 1378.54028
[28] Fiedler, M.: Ultrametric sets in Euclidean point spaces, Electron. J. Linear Algebra 3 (1998), 23-30. · Zbl 0897.54020
[29] Foertsch, T.—Schroeder, V.: Minkowski versus Euclidean rank for products of metric spaces, Adv. Geom. 2 (2002), 123-131. · Zbl 0988.53033
[30] Gomory, R. E.—Hu, T. C.: Multi-terminal network flows, SIAM 9 (1961), 551-570. · Zbl 0112.12405
[31] Groot, J. D.: Non-Archimedean metrics in topology, Proc. A.M.S. 7 (1956), 948-956. · Zbl 0072.40201
[32] Gurvich, V.: Metric and ultrametric spaces of resistances, Discrete Appl. Math. 158 (2010), 1496-1505. · Zbl 1234.94091
[33] Gurvich, V.—Vyalyi, M.: Characterizing (quasi-)ultrametric finite spaces in terms of (directed) graphs, Discrete Appl. Math. 160 (2012), 1742-1756. · Zbl 1245.05058
[34] Gvishiani, A. D.—Gurvich, V. A.: Metric and ultrametric spaces of resistances, Commun. Moscow Math. Soc., Russ. Math. Surveys 42 (1987), 235-236. · Zbl 0708.90028
[35] Herburt, I.—Moszyńska, M.: On metric products, Colloq. Math. 62 (1991), 121-133. · Zbl 0782.54030
[36] Holly, J. E.: Pictures of ultrametric spaces, the p-adic numbers, and valued fields, Amer. Math. Monthly 108 (2001), 721-728. · Zbl 1039.12003
[37] Hughes, B.: Trees and ultrametric spaces: a categorical equivalence, Adv. Math. 189 (2004), 148-191. · Zbl 1061.57021
[38] Hughes, B.: Trees, ultrametrics, and noncommutative geometry, Pure Appl. Math. Q. 8 (2012), 221-312. · Zbl 1251.46038
[39] Ibragimov, Z.: Möbius maps between ultrametric spaces are local similarities, Ann. Acad. Sci. Fenn. Math. 37 (2012), 309-317. · Zbl 1294.30041
[40] Khemaratchatakumthorn, T.—Pongsriiam, P.: Remarks on b-metric and metric-preserving functions, Math. Slovaca 68 (2018), 1009-1016. · Zbl 1505.26014
[41] Khemaratchatakumthorn, T.—Pongsriiam, P.—Samphavat, S.: Further remarks on b-metrics, metric-preserving functions, and other related metrics, Int. J. Math. Comput. Sci. 14 (2019), 473-480. · Zbl 1414.26009
[42] Khemaratchatakumthorn, T.—Termwuttipong, I.: Metric-preserving functions, w-distances and Cauchy w-distances, Thai J. Math. 5 (2007), 51-56. · Zbl 1151.54330
[43] Kirk, W. A.—Shahzad, N.: Some fixed point results in ultrametric spaces, Topology Appl. 159 (2012), 3327-3334. · Zbl 1252.54038
[44] Lemin, A. J.: On isosceles metric spaces, Funct. Anal. Appl. (1984), 26-31.
[45] Lemin, A. J.: On the stability of the property of a space being isosceles, Russ. Math. Surveys 39 (1984), 283-284. · Zbl 0566.54016
[46] Lemin, A. J.: Proximity on isosceles spaces, Russ. Math. Surveys 39 (1984), 143-144. · Zbl 0556.54020
[47] Lemin, A. J.: Isometric embedding of isosceles (non-Archimedean) spaces in Euclidean spaces, Soviet Math. Dokl. 32 (1985), 740-744. · Zbl 0602.54031
[48] Lemin, A. J.: An application of the theory of isosceles (ultrametric) spaces to the Trnkova-Vinarek theorem, Comment. Math. Univ. Carolinae 29 (1988), 427-434. · Zbl 0677.54027
[49] Lemin, A. J.: The category of ultrametric spaces is isomorphic to the category of complete, atomic, tree-like, real graduated lattices (LAT^*, Algebra Universalis 50 (2003), 35-49. · Zbl 1094.54004
[50] Petrov, E.: Weak similarities of finite ultrametric and semimetric spaces, P-adic Numbers Ultrametr. Anal. Appl. 10 (2018), 108-117. · Zbl 1403.54019
[51] Petrov, E.—Dovgoshey, A.: On the Gomory-Hu inequality, J. Math. Sci. 198 (2014), 392-411. · Zbl 1348.54020
[52] Piotrowski, Z.—Vallin, R. W.: Functions which preserve Lebesgue spaces, Comment. Math. [Prace Mat.] 43 (2003), 249-255. · Zbl 1059.26005
[53] Pokorný, I.: Some remarks on metric-preserving functions, Tatra Mt. Math. Publ. 2 (1993), 65-68. · Zbl 0793.26010
[54] Pongsriiam, P.—Termwuttipong, I.: Remarks on ultrametrics and metric-preserving functions, Abstr. Appl. Anal. 2014 (2014), 9. · Zbl 1440.54010
[55] Qiu, D.: Geometry of non-Archimedian Gromov-Hausdorff distance, P-adic Numbers Ultrametr. Anal. Appl. 1 (2009), 317-337. · Zbl 1253.53041
[56] Qiu, D.: The structures of Hausdorff metric in non-Archimedian spaces, P-adic Numbers Ultrametr. Anal. Appl. 6 (2014), 33-53. · Zbl 1318.54015
[57] Rudin, W.: Principles of Mathematical Analysis, International Series in Pure & Applied Mathematics, McGraw-Hill, 1976. · Zbl 0148.02903
[58] Termwuttipong, I.—Oudkam, P.: Total boundedness, completeness and uniform limits of metric-preserving functions, Ital. J. Pure Appl. Math. 18 (2005), 187-196. · Zbl 1176.54021
[59] Tyson, J. T.—Wu, J.-M.: Characterizations of snowflake metric spaces, Ann. Acad. Sci. Fenn-M. 30 (2005), 313-336. · Zbl 1125.54014
[60] Vallin, R. W.: Continuity and differentiability aspects of metric preserving functions, Real Anal. Exchange 25 (1999/2000), 849-868. · Zbl 1016.26004
[61] Vaughan, J.: Universal ultrametric spaces of smallest weight, Topology Proc. 24 (1999), 611-619. · Zbl 1026.54030
[62] Vestfrid, I.: On the universal ultrametric space, Ukrainian Math. J. 46 (1994), 1890-1898. · Zbl 0871.54035
[63] Wilson, W. A.: On semi-metric spaces, Am. J. Math. 53 (1931), 361-373. · JFM 57.0735.01
[64] Wilson, W. A.: On certain types of continuous transformations of metric spaces, Amer. J. Math. 57 (1935), 62-68. · JFM 61.0634.06
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