[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 |