Classical and stochastic Löwner-Kufarev equations. (English) Zbl 1318.30015
Vasil’ev, Alexander (ed.), Harmonic and complex analysis and its applications. Cham: Birkhäuser/Springer (ISBN 978-3-319-01805-8/hbk; 978-3-319-01806-5/ebook). Trends in Mathematics, 39-134 (2014).
Summary: In this paper, we present a historical and scientific account of the development of the theory of the Löwner-Kufarev classical and stochastic equations spanning the 90-year period from the seminal paper by K. Löwner [Math. Ann. 89, 103–121 (1923; JFM 49.0714.01)] to recent generalizations and stochastic versions and their relations to conformal field theory.
For the entire collection see [Zbl 1278.30003].
For the entire collection see [Zbl 1278.30003].
MSC:
| 30C35 | General theory of conformal mappings |
| 01A70 | Biographies, obituaries, personalia, bibliographies |
Keywords:
conformal mapping; evolution family; Löwner-Kufarev equation; subordination chain; Brownian motionCitations:
JFM 49.0714.01References:
| [1] | Abate, M., Iteration Theory of Holomorphic Maps on Taut Manifolds (1989), Rende, Cosenza: Mediterranean Press, Rende, Cosenza · Zbl 0747.32002 |
| [2] | Abate, M.; Bracci, F.; Contreras, M. D.; Díaz-Madrigal, S., The evolution of Loewner’s differential equations, Newslett. Eur. Math. Soc., 78, 31-38 (2010) · Zbl 1230.30003 |
| [3] | Aharonov, D.; Reich, S.; Shoikhet, D., Flow invariance conditions for holomorphic mappings in Banach spaces, Math. Proc. R. Ir. Acad., 99A, 1, 93-104 (1999) · Zbl 0947.46033 |
| [4] | L.V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory (AMS Chelsea Publishing, Providence, RI, 2010) (Reprint of the 1973 original) · Zbl 1211.30002 |
| [5] | L.A. Aksent’ev, P.L. Shabalin, Sufficient conditions for univalence and quasiconformal extendibility of analytic functions, in Handbook of Complex Analysis: Geometric Function Theory, vol. 1 (North-Holland, Amsterdam, 2002), pp. 169-206 · Zbl 1067.30034 |
| [6] | Albeverio, S.; Høegh-Krohn, R., Uniqueness and the global markov property for euclidean fields, The case of trigonometric interactions. Commun. Math. Phys., 68, 95-128 (1979) · Zbl 0443.60099 |
| [7] | I.A. Aleksandrov, V.V. Chernikov, Pavel Parfen’evich Kufarev. Obituary (in Russian), Uspehi Mat. Nauk 24, 181-184 (1969) · Zbl 0175.25505 |
| [8] | Aleksandrov, I. A.; Kopanev, S. A., On mutual growth of the modulus of a univalent function and of the modulus of its derivative, Sibirsk. Mat. Zh., 7, 23-30 (1966) · Zbl 0152.27302 |
| [9] | Aleksandrov, I. A.; Kopanev, S. A., The range of values of the derivative on the class of holomorphic univalent functions, Ukrain. Mat. Zh., 22, 660-664 (1970) · Zbl 0203.38401 |
| [10] | I.A. Aleksandrov, V.I. Popov, Optimal controls and univalent functions. Ann. Univ. M. Curie-Sklodowska, Sec. A 22-24, 13-20 (1968-1970) · Zbl 0252.30025 |
| [11] | Aleksandrov, I. A.; Krjučkov, B. Ja.; Popov, V. I., The first coefficients of bounded holomorphic univalent functions, Dopovı̄dı̄ Akad. Nauk Ukraïn. RSR Ser. A, 1, 3-5 (1973) · Zbl 0262.30017 |
| [12] | Aleksandrov, I. A.; Popova, G. A., Extremal properties of univalent holomorphic functions with real coefficients, Sibirsk. Mat. Ž., 14, 915-926 (1973) · Zbl 0272.30018 |
| [13] | Aleksandrov, I. A.; Zavozin, G. G.; Kopanev, S. A., Optimal controls in coefficient problems for univalent functions, Differencial’nye Uravnenija, 12, 4, 599-611 (1976) · Zbl 0449.30010 |
| [14] | Aleksandrov, I. A., Parametric Continuations in the Theory of Univalent Functions (1976), Moscow: Nauka, Moscow · Zbl 0462.30008 |
| [15] | Aleksandrov, S. T.; Sobolev, V. V., Extremal problems in some classes of functions, univalent in the half plane, having a finite angular residue at infinity, Siberian Math. J., 27, 145-154 (1986) · Zbl 0601.30017 |
| [16] | Arosio, L., Resonances in Loewner equations. Adv. Math., 227, 1413-1435 (2011) · Zbl 1256.32022 |
| [17] | Arosio, L., Basins of attraction in Loewner equations, Ann. Acad. Sci. Fenn. Math., 37, 563-570 (2012) · Zbl 1281.32015 |
| [18] | Arosio, L.; Bracci, F., Infinitesimal generators and the Loewner equation on complete hyperbolic manifolds, Anal. Math. Phys., 1, 4, 337-350 (2011) · Zbl 1254.32037 |
| [19] | Arosio, L.; Bracci, F.; Fornæss Wold, E., Solving the Loewner PDE in complete hyperbolic starlike domains of \({\mathbb{C}}^N \), Adv. Math., 242, 209-216 (2013) · Zbl 1283.30054 |
| [20] | Arosio, L.; Bracci, F.; Hamada, H.; Kohr, G., An abstract approach to Löwner chains, J. Anal. Math., 119, 1, 89-114 (2013) · Zbl 1321.32005 |
| [21] | Astala, K.; Jones, P.; Kupiainen, A.; Saksman, E., Random curves by conformal welding, Compt. Rend. Math., 348, 5-6, 257-262 (2010) · Zbl 1190.30005 |
| [22] | Babelon, O.; Bernard, D.; Talon, M., Introduction to Classical Integrable Systems (2003), Cambridge: Cambridge University Press, Cambridge · Zbl 1045.37033 |
| [23] | Bauer, M.; Bernard, D., SLE_k growth processes and conformal field theories, Phys. Lett. B, 543, 1-2, 135-138 (2002) · Zbl 0997.60119 |
| [24] | Bauer, M.; Bernard, D., Conformal field theories of stochastic Loewner evolutions, Comm. Math. Phys., 239, 3, 493-521 (2003) · Zbl 1046.81091 |
| [25] | Bauer, R. O.; Friedrich, R. M., On radial stochastic Loewner evolution in multiply connected domains, J. Funct. Anal., 237, 565-588 (2006) · Zbl 1098.60086 |
| [26] | Bauer, R. O.; Friedrich, R. M., On chordal and bilateral SLE in multiply connected domains, Math. Z., 258, 241-265 (2008) · Zbl 1130.30006 |
| [27] | J. Basilewitsch, Sur les théorèmes de Koebe-Bieberbach. Rec. Math. [Mat. Sbornik] N.S. 1(43)(3), 283-291 (1936) · JFM 62.0373.03 |
| [28] | J. Basilewitsch, D’une propriété extrémale de la fonction \(F(z) = \frac{z} {{(1-z)}^2} \) . Rec. Math. [Mat. Sbornik] N.S. 12(54)(3), 315-319 (1943) · Zbl 0063.00235 |
| [29] | Bazilevič, I. E., On a case of integrability in quadratures of the Loewner-Kufarev equation, Mat. Sb. N.S., 37, 79, 471-476 (1955) · Zbl 0065.31002 |
| [30] | Bazilevič, I. E., Generalization of an integral formula for a subclass of univalent functions, Mat. Sb. N.S., 64, 106, 628-630 (1964) · Zbl 0133.32602 |
| [31] | Becker, J., Löwnersche, Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen. J. Reine Angew. Math., 255, 23-43 (1972) · Zbl 0239.30015 |
| [32] | Becker, J.; Pommerenke, Ch., Schlichtheitskriterien und Jordangebiete, J. Reine Angew. Math., 354, 74-94 (1984) · Zbl 0539.30009 |
| [33] | Beffara, V., The dimension of the SLE curves, Ann. Probab., 36, 4, 1421-1452 (2008) · Zbl 1165.60007 |
| [34] | Belavin, A. A.; Polyakov, A. M.; Zamolodchikov, A. B., Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys. B, 241, 2, 333-380 (1984) · Zbl 0661.17013 |
| [35] | E. Berkovich, B. Shein, Odyssey of Fritz Noether. Epilogue, Notes in Jewish History (2009), no. 11(114), July |
| [36] | Berkson, E.; Porta, H., Semigroups of holomorphic functions and composition operators, Mich. Math. J., 25, 101-115 (1978) · Zbl 0382.47017 |
| [37] | L. Bieberbach, Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Ein- heitskreises vermitteln. S.-B. Preuss. Akad. Wiss. 940-955 (1916) · JFM 46.0552.01 |
| [38] | Bill of indictment on L.A. Vishnevskiĭ. February 15, 1938. Novosibirsk |
| [39] | K.I. Bobenko, On the theory of extremal problems for univalent functions of class S. Trudy Mat. Inst. Steklov. vol. 101 (Izdat. “Nauka”, Moscow, 1972) · Zbl 0283.30020 |
| [40] | Bombieri, E., On the local maximum property of the Koebe function, Inv. Math., 4, 26-67 (1967) · Zbl 0174.12301 |
| [41] | Bracci, F.; Contreras, M. D.; Díaz-Madrigal, S., Pluripotential theory, semigroups and boundary behaviour of infinitesimal generators in strongly convex domains, J. Eur. Math. Soc., 12, 23-53 (2010) · Zbl 1185.32010 |
| [42] | Bracci, F.; Contreras, M. D.; Díaz-Madrigal, S., Evolution families and the Löwner equation I: the unit disk, J. Reine Angew. Math., 672, 1-37 (2012) · Zbl 1267.30025 |
| [43] | Bracci, F.; Contreras, M. D.; Díaz-Madrigal, S., Evolution Families and the Löwner Equation II: complex hyperbolic manifolds, Math. Ann., 344, 947-962 (2009) · Zbl 1198.32010 |
| [44] | Bracci, F.; Contreras, M. D.; Díaz-Madrigal, S., Semigroups versus evolution families in the Löwner theory, J. Anal. Math., 115, 1, 273-292 (2011) · Zbl 1310.30007 |
| [45] | L. de Branges, A proof of the Bieberbach conjecture. Preprint E-5-84, Leningrad Branch of the Steklov Institute of Mathematics, LOMI, Leningrad, 1984 · Zbl 0573.30014 |
| [46] | de Branges, L., A proof of the Bieberbach conjecture, Acta Math., 154, 1-2, 137-152 (1985) · Zbl 0573.30014 |
| [47] | Bshouty, D.; Hengartner, W., A variation of the Koebe mapping in a dense subset of S, Can. J. Math., 39, 7, 54-73 (1987) · Zbl 0624.30018 |
| [48] | Cardy, J.; Francoise, J.-P.; Naber, G.; Tsun Tsou, S., Boundary conformal field theory, Encyclopedia of Mathematical Physics, 333-340 (2006), Amsterdam: Elsevier, Amsterdam · Zbl 1170.00001 |
| [49] | Carleson, L.; Makarov, N., Aggregation in the plane and Loewner’s equation, Comm. Math. Phys., 216, 583-607 (2001) · Zbl 1042.82039 |
| [50] | Carverhill, A., Flows of stochastic dynamical systems: ergodic theory, Stochastics, 14, 4, 273-317 (1985) · Zbl 0536.58019 |
| [51] | Chan, D. Y.C.; Hughes, B. D.; Paterson, L.; Sirakoff, Ch., Simulating flow in porous media, Phys. Rev. A, 38, 8, 4106-4120 (1988) |
| [52] | Chelkak, D.; Smirnov, S., Universality in the 2D Ising model and conformal invariance of fermionic observables, Invent. Math., 189, 3, 515-580 (2012) · Zbl 1257.82020 |
| [53] | Contreras, M. D.; Díaz-Madrigal, S.; Gumenyuk, P., Löwner chains in the unit disk, Rev. Mat. Iberoam., 26, 3, 975-1012 (2010) · Zbl 1209.30011 |
| [54] | M.D. Contreras, S. Díaz-Madrigal, P. Gumenyuk, Local duality in Löwner equations. J. Nonlinear Convex Anal. (to appear). Available in http://arxiv.org/pdf/1202.2334.pdf · Zbl 1316.30025 |
| [55] | Contreras, M. D.; Díaz-Madrigal, S.; Gumenyuk, P., Loewner Theory in annulus I: evolution families and differential equations, Trans. Am. Math. Soc., 365, 2505-2543 (2013) · Zbl 1281.30007 |
| [56] | Contreras, M. D.; Díaz-Madrigal, S.; Gumenyuk, P., Loewner Theory in annulus II: Loewner chains, Anal. Math. Phys., 1, 351-385 (2011) · Zbl 1257.30004 |
| [57] | Conway, J., Functions of One Complex Variable II (1995), New York: Springer, New York · Zbl 0887.30003 |
| [58] | Date, E.; Kashiwara, M.; Miwa, T., Vertex operators and functions: transformation groups for soliton equations II, Proc. Jpn. Acad. Ser. A Math. Sci., 57, 387-392 (1981) · Zbl 0538.35066 |
| [59] | E. Date, M. Kashiwara, M. Jimbo, T. Miwa, Transformation groups for soliton equations. Nonlinear Integrable Systems—Classical Theory and Quantum Theory (Kyoto, 1981) (World Sci. Publishing, Singapore, 1983), pp. 39-119 · Zbl 0571.35098 |
| [60] | Davidovitch, B.; Hentschel, H. G.E.; Olami, Z.; Procaccia, I.; Sander, L. M.; Somfai, E., Diffusion limited aggregation and iterated conformal maps, Phys. Rev. E, 59, 2, 1368-1378 (1999) |
| [61] | Davidovitch, B.; Feigenbaum, M. J.; Hentschel, H. G.E.; Procaccia, I., Conformal dynamics of fractal growth patterns without randomness, Phys. Rev. E, 62, 2, 1706-1715 (2000) |
| [62] | A. Dick, Emmy Noether. 1882-1935 (Birkhäuser, Basel, 1970) |
| [63] | L. A. Dickey, Soliton Equations and Hamiltonian Systems, 2nd edn. Advanced Series in Mathematical Physics, vol. 26 (World Scientific, Singapore, 2003) · Zbl 1140.35012 |
| [64] | R.G. Douglas, Banach Algebra Techniques in Operator Theory, 2nd edn. Graduate Texts in Mathematics, vol. 179. (Springer, New York, 1998) · Zbl 0920.47001 |
| [65] | B. Doyon, Conformal loop ensembles and the stress-energy tensor: I. Fundamental notions of CLE. arXiv:0903.0372 (2009) |
| [66] | B. Doyon, Conformal loop ensembles and the stress-energy tensor: II. Construction of the stress-energy tensor. arXiv:0908.1511 (2009) · Zbl 1263.81253 |
| [67] | Doyon, B., Calculus on manifolds of conformal maps and CFT. J. Phys. A Math. Theor., 45, 31, 315202 (2012) · Zbl 1258.81076 |
| [68] | Duren, P. L.; Shapiro, H. S.; Shields, A. L., Singular measures and domains not of Smirnov type, Duke Math. J., 33, 247-254 (1966) · Zbl 0174.37501 |
| [69] | Dubédat, J., SLE and the free field: partition functions and couplings, J. Am. Math. Soc., 22, 4, 995-1054 (2009) · Zbl 1204.60079 |
| [70] | D.A. Dubovikov, An analog of the Loëwner equation for mappings of strip. Izv. Vyssh. Uchebn. Zaved. Mat. 51(8), 77-80 2007 (Russian); translation in Russian Math. (Iz. VUZ) 51(8), 74-77 (2007) · Zbl 1154.30008 |
| [71] | Duren, P. L.; Schiffer, M., The theory of the second variation in extremum problems for univalent functions, J. Anal. Math., 10, 193-252 (1962) · Zbl 0112.30302 |
| [72] | Duren, P. L., Univalent Functions (1983), New York: Springer, New York · Zbl 0514.30001 |
| [73] | Earle, C. J.; Epstein, A. L., Quasiconformal variation of slit domains, Proc. Am. Math. Soc., 129, 11, 3363-3372 (2001) · Zbl 0977.30008 |
| [74] | L.D. Faddeev, Discretized Virasoro Algebra. Contemporary Mathematics, vol. 391 (American Mathematical Society, Providence, RI, 2005), pp. 59-67 · Zbl 1268.81088 |
| [75] | Fekete, M.; Szegö, G., Eine Bemerkung über ungerade schlichte Funktionen, J. Lond. Math. Soc., 8, 2, 85-89 (1933) · JFM 59.0347.04 |
| [76] | FitzGerald, C. H.; Pommerenke, Ch., The de Branges theorem on univalent functions, Trans. Am. Math. Soc., 290, 683-690 (1985) · Zbl 0574.30018 |
| [77] | Fornæss, J. E.; Sibony, N., Increasing sequences of complex manifolds, Math. Ann., 255, 351-360 (1981) · Zbl 0438.32012 |
| [78] | Fornæss, J. E.; Stensønes, B., Stable manifolds of holomorphic hyperbolic maps, Int. J. Math., 15, 8, 749-758 (2004) · Zbl 1071.32016 |
| [79] | Friedland, S.; Schiffer, M., Global results in control theory with applications to univalent functions, Bull. Am. Math. Soc., 82, 913-915 (1976) · Zbl 0355.49014 |
| [80] | Friedland, S.; Schiffer, M., On coefficient regions of univalent functions, J. Anal. Math., 31, 125-168 (1977) · Zbl 0349.30008 |
| [81] | R. Friedrich, The global geometry of stochastic Loewner evolutions. Probabilistic Approach to Geometry. Advanced Studies in Pure Mathematics, vol. 57 (Mathematical Society of Japan, Tokyo, 2010), pp. 79-117 · Zbl 1198.82030 |
| [82] | Friedrich, R.; Werner, W., Conformal restriction, highest-weight representations and SLE, Comm. Math. Phys., 243, 1, 105-122 (2003) · Zbl 1030.60095 |
| [83] | Galin, L. A., Unsteady filtration with a free surface, Dokl. Akad. Nauk USSR, 47, 246-249 (1945) · Zbl 0061.46202 |
| [84] | Gamkrelidze, R. V., Foundations of Optimal Control Theory (1977), Tbilisi: Tbilisi University, Tbilisi |
| [85] | Garabedian, P. R.; Schiffer, M. A., A proof of the Bieberbach conjecture for the fourth coefficient, J. Rational Mech. Anal., 4, 427-465 (1955) · Zbl 0065.06902 |
| [86] | Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M., Korteweg-deVries equation and generalization, VI. Methods for exact solution. Comm. Pure Appl. Math., 27, 97-133 (1974) · Zbl 0291.35012 |
| [87] | Gehring, F. W.; Reich, E., Area distortion under quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I, 388, 1-14 (1966) · Zbl 0141.27105 |
| [88] | Gervais, J.-L., Infinite family of polynomial functions of the Virasoro generators with vanishing Poisson brackets, Phys. Lett. B, 160, 4-5, 277-278 (1985) |
| [89] | l. Gibbons, S.P. Tsarev, Conformal maps and reductions of the Benney equations. Phys. Lett. A 258(4-6), 263-271 (1999) · Zbl 0936.35184 |
| [90] | G.M. Goluzin, Über die Verzerrungssätze der schlichten konformen Abbildungen. Rec. Math. [Mat. Sbornik] N.S. 1(34)(1), 127-135 (1936) · Zbl 0014.22101 |
| [91] | Goluzin, G. M., Sur les théorèmes de rotation dans la théorie des fonctions univalentes, Rec. Math. [Mat. Sbornik] N.S., 1, 43-1, 293-296 (1936) · JFM 62.0373.04 |
| [92] | Goluzin, G. M., On the parametric representation of functions univalent in a ring, Mat. Sbornik N.S., 29, 71, 469-476 (1951) · Zbl 0044.30702 |
| [93] | Goluzin, G. M., Geometrical Theory of Functions of a Complex Variable (1966), Moscow: Nauka, Moscow · Zbl 0148.30603 |
| [94] | G. S. Goodman, Pontryagins Maximal Principle as a variational method in conformal mapping. Abstracts of Brief Scientific Communications, sec. 13 (ICM Moscow, 1966), pp. 7-8 |
| [95] | G.S. Goodman, Univalent functions and optimal control, Ph.D. Thesis, Stanford University, 1968 |
| [96] | V.V. Goryainov, V.Ya. Gutlyanskiĭ, Extremal Problems in the Class S_M, Mathematics collection (Russian) (Izdat. “Naukova Dumka”, Kiev, 1976), pp. 242-246 |
| [97] | Gorjainov, V. V., On a parametric method of the theory of univalent functions, Math. Notes, 27, 3-4, 275-279 (1980) · Zbl 0442.30015 |
| [98] | Gorjainov, V. V., Boundary functions of a system of functionals composed of values of a univalent function and its derivative, Soviet Math. (Iz. VUZ), 26, 7, 91-94 (1982) · Zbl 0507.30015 |
| [99] | V.V. Gorjainov, Extremals in Estimates of Functionals Depending on Values of a Univalent Function and Its Derivative. Theory of Mappings and Approximation of Functions (“Naukova Dumka”, Kiev, 1983), pp. 38-49 · Zbl 0576.30006 |
| [100] | Gorjainov, V. V., The Parametric Method and Extremal Conformal Mappings, Soviet Math. Dokl., 28, 1, 205-208 (1983) · Zbl 0567.30018 |
| [101] | V.V. Goryainov, Semigroups of conformal mappings. Mat. Sb. (N.S.) 129(171)(4), 451-472 (1986) (Russian); translation in Math. USSR Sbornik 57, 463-483 (1987) · Zbl 0618.30009 |
| [102] | V.V. Goryajnov, Evolution families of analytic functions and time-inhomogeneous Markov branching processes. Dokl. Akad. Nauk 347(6) 729-731 (1996); translation in Dokl. Math. 53(2), 256-258 (1996) · Zbl 0895.60090 |
| [103] | V.V. Goryainov, The Löwner - Kufarev equation and extremal problems for conformal mappings with fixed points on the boundary (in Russian). Talk at the International Workshop on Complex Analysis and its Applications, Steklov Mathematical Institute, Moscow, 26 December 2011 |
| [104] | Goryainov, V. V.; Ba, I., Semigroups of conformal mappings of the upper half-plane into itself with hydrodynamic normalization at infinity, Ukrainian Math. J., 44, 1209-1217 (1992) · Zbl 0873.30006 |
| [105] | V.V. Goryainov, O.S. Kudryavtseva, One-parameter semigroups of analytic functions, fixed points and the Koenigs function. Mat. Sb. 202(7), 43-74 (2011) (Russian); translation in Sbornik: Mathematics 202(7), 971-1000 (2011) · Zbl 1258.30008 |
| [106] | Goryainov, V. V., Semigroups of analytic functions in analysis and applications, Uspekhi Mat. Nauk, 67, 6, 5-52 (2012) · Zbl 1269.30002 |
| [107] | Graham, I.; Hamada, H.; Kohr, G., Parametric representation of univalent mappings in several complex variables, Can. J. Math., 54, 324-351 (2002) · Zbl 1004.32007 |
| [108] | Graham, I.; Kohr, G., Geometric Function Theory in One and Higher Dimensions (2003), New York: Marcel Dekker, New York · Zbl 1042.30001 |
| [109] | Graham, I.; Hamada, H.; Kohr, G.; Kohr, M., Asymptotically spirallike mappings in several complex variables, J. Anal. Math., 105, 267-302 (2008) · Zbl 1148.32009 |
| [110] | Graham, I.; Kohr, G.; Pfaltzgraff, J. A., The general solution of the Loewner differential equation on the unit ball in \({\mathbb{C}}^n \), Contemp. Math. (AMS), 382, 191-203 (2005) · Zbl 1089.32008 |
| [111] | Graham, I.; Kohr, G., Geometric Function Theory in One and Higher Dimensions (2003), New York: Marcel Dekker, New York · Zbl 1042.30001 |
| [112] | Greiner, R.; Roth, O., On support points of univalent functions and a disproof of a conjecture of Bombieri, Proc. Am. Math. Soc., 129, 12, 3657-3664 (2001) · Zbl 0981.30018 |
| [113] | Grier, D.; Ben-Jacob, E.; Clarke, R.; Sander, L. M., Morphology and Microstructure in Electrochemical Deposition of Zinc, Phys. Rev. Lett., 56, 1264-1267 (1986) |
| [114] | B. Gustafsson, A. Vasil’ev, Conformal and Potential Analysis in Hele-Shaw Cells (Birkhäuser, Basel, 2006) · Zbl 1122.76002 |
| [115] | Ya, V., Gutlyanskiĭ, Parametric representation of univalent functions. Dokl. Akad. Nauk SSSR, 194, 750-753 (1970) |
| [116] | V.Ya. Gutljanskĭ, Parametric Representations and Extremal Problems in the Theory of Univalent Functions (in Russian), Dissertation Dokt. Fiz.-Mat. Nauk, Mat. Inst. Akad. Nauk Ukrain. SSR, Kiev, 1972 |
| [117] | Hamada, H.; Kohr, G.; Muir, J. R., Extension of L^d-Loewner chains to higher dimensions, J. Anal. Math., 120, 1, 357-392 (2013) · Zbl 1284.32009 |
| [118] | T.E. Harris, The Theory of Branching Processes. Corrected reprint of the 1963 original, Dover Phoenix Editions (Dover Publications, Mineola, NY, 2002) · Zbl 1037.60001 |
| [119] | Halsey, T. C., Diffusion-Limited Aggregation: A Model for Pattern Formation. Phys. Today, 53, 11, 36 (2000) |
| [120] | Hastings, M. B., Renormalization theory of stochastic growth, Phys. Rev. E, 55, 1, 135-152 (1987) |
| [121] | Hastings, M. B.; Levitov, L. S., Laplacian growth as one-dimensional turbulence, Phys. D, 116, 1-2, 244-252 (1998) · Zbl 0962.76542 |
| [122] | Hayman, W. K., Multivalent Functions (1958), Cambridge: Cambridge University Press, Cambridge · Zbl 0082.06102 |
| [123] | M.H. Heins, Semigroups of holomorphic maps of a Riemann surface into itself which are homomorphs of the set of positive reals considered additively, in E.B. Christoffel—The Influence of His Work on Mathematics and the Physical Sciences (Bikhäuser, Basel, 1981), pp. 314-331 · Zbl 0474.30022 |
| [124] | Hele-Shaw, H. S., The flow of water, Nature, 58, 33-36 (1898) · JFM 30.0680.02 |
| [125] | Hohlov, Yu. E.; Howison, S. D.; Huntingford, C.; Ockendon, J. R.; Lacey, A. A., A model for non-smooth free boundaries in Hele-Shaw flows, Q. J. Mech. Appl. Math., 47, 107-128 (1994) · Zbl 0795.76032 |
| [126] | Ivanov, G.; Vasil’ev, A., Lwner evolution driven by a stochastic boundary point, Anal. Math. Phys., 1, 4, 387-412 (2011) · Zbl 1255.60169 |
| [127] | Ivanov, G.; Prokhorov, D.; Vasil’ev, A., Non-slit and singular solutions to the Loewner equation, Bull. Sci. Math., 136, 3, 328-341 (2012) · Zbl 1259.30007 |
| [128] | V.G. Kac, Simple irreducible graded Lie algebras of finite growth. Math. USSR Izv. 2, 1271-1311 (1968); Izv. Akad. Nauk USSR Ser. Mat. 32, 1923-1967 (1968) · Zbl 0222.17007 |
| [129] | Kager, W.; Nienhuis, B.; Kadanoff, L. P., Exact solutions for Löwner evolutions, J. Stat. Phys., 115, 3-4, 805-822 (2004) · Zbl 1056.30005 |
| [130] | N.-G. Kang, N.G. Makarov, Gaussian free field and conformal field theory. arXiv: 1101.1024 [math.PR], 2011 · Zbl 1280.81004 |
| [131] | N.-G. Kang, N.G. Makarov, Radial SLE martingale-observables. arXiv:1209.1615 [math.PR], 2012 |
| [132] | Kesten, H., Hitting probabilities of random walks on \({\mathbb{Z}}^d \), Stoch. Proc. Appl., 25, 165-184 (1987) · Zbl 0626.60067 |
| [133] | Kirillov, A. A.; Yuriev, D. V., Representations of the Virasoro algebra by the orbit method, J. Geom. Phys., 5, 3, 351-363 (1988) · Zbl 0698.17015 |
| [134] | Kobayashi, S., Hyperbolic Complex Spaces (1998), Berlin Heidelberg: Springer, Berlin Heidelberg · Zbl 0917.32019 |
| [135] | Komatu, Y., Über eine Verschärfung des Löwnerschen Hilfssatzes, Proc. Imp. Acad. Tokyo, 18, 354-359 (1942) · Zbl 0060.23407 |
| [136] | Komatu, Y., On conformal slit mapping of multiply-connected domains, Proc. Jpn. Acad., 26, 7, 26-31 (1950) · Zbl 0041.20405 |
| [137] | E.L. Korotyaev, P. Kargaev, Inverse problems generated by conformal mappings on complex plane with parallel slits. 2009, unpublished preprint. |
| [138] | Korotyaev, E. L.; Kargaev, P., Estimates for periodic Zakharov-Shabat operators, J. Differ. Equat., 249, 76-93 (2010) · Zbl 1204.34119 |
| [139] | E.L. Korotyaev, S. Kuksin, KdV Hamiltonian as function of actions. arXiv 1110.4475, 2011 |
| [140] | I.K. Kostov, I. Krichever, M. Mineev-Weinstein, P.B. Wiegmann, A. Zabrodin, The τ-function for analytic curves. Random Matrix Models and Their Applications. Mathematical Sciences Research Institute Publications, vol. 40 (Cambridge University Press, Cambridge, 2001), pp. 285-299 · Zbl 0991.37050 |
| [141] | Krichever, I.; Mineev-Weinstein, M.; Wiegmann, P.; Zabrodin, A., Laplacian growth and Whitham equations of soliton theory, Phys. D, 198, 1-2, 1-28 (2004) · Zbl 1078.76025 |
| [142] | Krichever, I.; Marshakov, A.; Zabrodin, A., Integrable structure of the Dirichlet boundary problem in multiply-connected domains, Comm. Math. Phys., 259, 1, 1-44 (2005) · Zbl 1091.37019 |
| [143] | Krzyż, J., The radius of close-to-convexity within the family of univalent functions, Bull. Acad. Polon. Sci., 10, 201-204 (1962) · Zbl 0116.05803 |
| [144] | Kufarev, P. P., On one-parameter families of analytic functions, Rec. Math. [Mat. Sbornik] N.S., 13, 55, 87-118 (1943) · Zbl 0061.14905 |
| [145] | Kufarev, P. P., On integrals of a simplest differential equation with movable polar singularity in the right-hand side, Tomsk. Gos. Univ. Uchen. Zap., 1, 35-48 (1946) |
| [146] | Kufarev, P. P., Theorem on solutions to some differential equation, Tomsk. Gos. Univ. Učenye Zapiski, 5, 20-21 (1947) |
| [147] | Kufarev, P. P., A remark on integrals of Löwner’s equation, Dokl. Akad. Nauk SSSR (N.S.), 57, 655-656 (1947) · Zbl 0029.03702 |
| [148] | Kufarev, P. P., On conformal mapping of complementary regions, Dokl. Akad. Nauk SSSR (N.S.), 73, 5, 881-884 (1950) · Zbl 0037.33602 |
| [149] | Kufarev, P. P.; Fales, A. E., On an extremal problem for complementary regions, Dokl. Akad. Nauk SSSR (N.S.), 81, 6, 995-998 (1950) · Zbl 0044.30401 |
| [150] | Kufarev, P. P.; Fales, A. E., On an extremal problem for complementary regions, Tomsk. Gos. Univ. Ucen. Zap., 17, 25-35 (1947) |
| [151] | Kufarev, P. P.; Sobolev, V. V.; Sporysheva, L. V., A certain method of investigation of extremal problems for functions that are univalent in the half-plane, Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat., 200, 142-164 (1968) · Zbl 0209.10401 |
| [152] | Kuvaev, M. P.; Kufarev, P. P., An equation of Löwner’s type for multiply connected regions, Tomsk. Gos. Univ. Uv. c. Zap. Mat. Meh., 25, 19-34 (1955) |
| [153] | Kuz’mina, G. V., Moduli of families of curves and quadratic differentials, Trudy Mat. Inst. Steklov., 139, 241 (1980) · Zbl 0482.30015 |
| [154] | Lavrent’ev, M. A., To the theory of conformal maps, Trav. Inst. Math. Stekloff, 5, 159-245 (1934) · Zbl 0010.26402 |
| [155] | Lawler, G. F.; Kaimonovich, V., An introduction to the stochastic Loewner evolution, Random Walks and Geometry, 261-293 (2004), Berlin: de Gruyter, Berlin · Zbl 1061.60107 |
| [156] | Lawler, G. F., Conformally Invariant Processes in the Plane, Mathematical Surveys and Monographs (2005), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1074.60002 |
| [157] | Lawler, G.; Schramm, O.; Werner, W., Values of Brownian Intersection Exponents. I. Half-plane Exponents, Acta Math., 187, 2, 237-273 (2001) · Zbl 1005.60097 |
| [158] | Lebedev, N. A., On parametric representation of functions regular and univalent in a ring (in Russian), Dokl. Akad. Nauk SSSR (N.S.), 103, 767-768 (1955) · Zbl 0065.31001 |
| [159] | N.A. Lebedev, The Area Principle in the Theory of Univalent Functions (Izdat. “Nauka”, Moscow, 1975) · Zbl 0747.30015 |
| [160] | Lévy, P., Processus Stochastiques et Mouvement Brownian (1947), Paris: Gauthier-Villars, Paris |
| [161] | Li, E. P., On the theory of univalent functions on a circular ring, Dokl. Akad. Nauk SSSR (N.S.), 92, 475-477 (1953) |
| [162] | Lind, J., A sharp condition for the Löwner equation to generate slits, Ann. Acad. Sci. Fenn. Math., 30, 1, 143-158 (2005) · Zbl 1069.30012 |
| [163] | Lind, J.; Marshall, D.; Rohde, S., Collisions and spirals of Loewner traces, Duke Math. J., 154, 3, 527-573 (2010) · Zbl 1206.30024 |
| [164] | Littlewood, J. E.; Paley, R. E.A. C., A proof that an odd schlicht function has bounded coefficients, J. Lond. Math. Soc., 7, 167-169 (1932) · Zbl 0005.01803 |
| [165] | Löwner, K., Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, Math. Ann., 89, 103-121 (1923) · JFM 49.0714.01 |
| [166] | Löwner, K., Über monotone Matrixfunktionen, Math. Z., 38, 177-216 (1934) · JFM 60.0055.01 |
| [167] | Loewner, Ch., Conservation laws in compressible fluid flow and associated mapping, J. Ration. Mech. Anal., 2, 537-561 (1953) · Zbl 0051.17403 |
| [168] | B. Mandelbrot, Fractal aspects of the iteration of z → λ z(1 − z) for complex λ and z, in Non-Linear Dynamics, ed. by Robert H.G. Helleman. Annals of the New York Academy of Sciences, vol. 357 (New York Academy of Sciences, New York, 1979), pp. 249-259 · Zbl 0478.58017 |
| [169] | Markina, I.; Prokhorov, D.; Vasil’ev, A., Sub-Riemannian geometry of the coefficients of univalent functions, J. Funct. Anal., 245, 2, 475-492 (2007) · Zbl 1147.30015 |
| [170] | Markina, I.; Vasil’ev, A., Virasoro algebra and dynamics in the space of univalent functions, Contemp. Math., 525, 85-116 (2010) · Zbl 1234.81127 |
| [171] | I. Markina, A. Vasil’ev, Evolution of smooth shapes and integrable systems. ArXiv: 1108.1007 [math-ph], 2011, pp. 25 |
| [172] | Marshakov, A.; Wiegmann, P.; Zabrodin, A., Integrable structure of the Dirichlet boundary problem in two dimensions, Comm. Math. Phys., 227, 1, 131-153 (2002) · Zbl 1065.37048 |
| [173] | Marshall, D. E.; Rohde, S., The Löwner differential equation and slit mappings, J. Am. Math. Soc., 18, 4, 763-778 (2005) · Zbl 1078.30005 |
| [174] | Mineev-Weinstein, M.; Wiegmann, P. B.; Zabrodin, A., Integrable structure of interface dynamics, Phys. Rev. Lett., 84, 22, 5106-5109 (2000) |
| [175] | Mineev-Weinstein, M.; Zabrodin, A., Whitham-Toda hierarchy in the Laplacian growth problem, J. Nonlinear Math. Phys., 8, 212-218 (2001) · Zbl 1008.76013 |
| [176] | Moody, R. V., A new class of Lie algebras, J. Algebra, 10, 211-230 (1968) · Zbl 0191.03005 |
| [177] | Muñoz Porras, J.; Pablos Romo, F., Generalized reciprocity laws, Trans. Am. Math. Soc., 360, 7, 3473-3492 (2008) · Zbl 1194.14037 |
| [178] | Nelson, E., The free Markov field, J. Funct. Anal., 12, 221-227 (1973) · Zbl 0273.60079 |
| [179] | Niemeyer, L.; Pietronero, L.; Wiessmann, H. J., Fractal dimension of dielectric breakdown, Phys. Rev. Lett., 52, 12, 1033-1036 (1984) |
| [180] | J. O’Connor, E.F. Robertson, Charles Loewner, MacTutor History of Mathematics archive, University of St Andrews |
| [181] | K. Osterwalder, R. Schrader, Axioms for Euclidean Green’s functions. Comm. Math. Phys. 31, 83-112 (1973); 42, 281-305 (1975) · Zbl 0303.46034 |
| [182] | Peschl, E., Zur Theorie der schlichten Funktionen, J. Reine Angew. Math., 176, 61-94 (1936) · JFM 62.0371.03 |
| [183] | Peskin, M. E.; Schroeder, D. V., An Introduction to Quantum Field Theory (1995), Cambridge: Perseus Books, Cambridge |
| [184] | Pfaltzgraff, J. A., Subordination chains and univalence of holomorphic mappings in \({\mathbb{C}}^n \), Math. Ann., 210, 55-68 (1974) · Zbl 0275.32012 |
| [185] | Pfaltzgraff, J. A., Subordination chains and quasiconformal extension of holomorphic maps in \({\mathbb{C}}^n \), Ann. Acad. Sci. Fenn. Ser. A I Math., 1, 13-25 (1975) · Zbl 0314.32001 |
| [186] | G. Pick, Über die konforme Abbildung eines Kreises auf ein schlichtes und zugleich beschränktes Gebiet. S.-B. Kaiserl. Akad. Wiss. Wien, Math.-Natur. Kl. Abt. II a 126, 247-263 (1917) · JFM 46.0553.01 |
| [187] | Ya, P., Polubarinova-Kochina, On a problem of the motion of the contour of a petroleum shell. Dokl. Akad. Nauk USSR, 47, 4, 254-257 (1945) |
| [188] | Ya, P., Polubarinova-Kochina, Concerning unsteady motions in the theory of filtration. Prikl. Matem. Mech., 9, 1, 79-90 (1945) · Zbl 0061.46109 |
| [189] | Pommerenke, Ch., Über die Subordination analytischer Funktionen, J. Reine Angew. Math., 218, 159-173 (1965) · Zbl 0184.30601 |
| [190] | Ch. Pommerenke, Univalent Functions, with a Chapter on Quadratic Differentials by G. Jensen (Vandenhoeck & Ruprecht, Göttingen, 1975) · Zbl 0283.30034 |
| [191] | Pommerenke, Ch., Boundary Behaviour of Conformal Maps (1992), Berlin: Springer, Berlin · Zbl 0762.30001 |
| [192] | Pontryagin, L. S.; Boltyanskii, V. G.; Gamkrelidze, R. V.; Mishchenko, E. F., The Mathematical Theory of Optimal Processes (1964), New York: A Pergamon Press Book. The Macmillan Co., New York · Zbl 0117.31702 |
| [193] | Poreda, T., On generalized differential equations in Banach spaces, Diss. Math., 310, 1-50 (1991) · Zbl 0745.35048 |
| [194] | V.I. Popov, The range of a system of functionals on the class S. Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 182(3), 106-132 (1965) · Zbl 0222.30017 |
| [195] | V.I. Popov, L.S. Pontrjagin’s maximum principle in the theory of univalent functions. Dokl. Akad. Nauk SSSR 188, 532-534 (1969) |
| [196] | Popova, N. V., Connection between the Löwner equation and the equation \(\frac{\mathit{dw}} {\mathit{dt}} = \frac{1} {w-\lambda (t)} \), Izv. Acad. Sci. Belorussian SSR, 6, 97-98 (1954) |
| [197] | Pressley, A.; Segal, G., Loop Groups (1986), New York: Oxford Mathematical Monographs (Oxford Science Publications, The Clarendon Press, Oxford University Press, New York · Zbl 0618.22011 |
| [198] | Prokhorov, D. V., The geometric characterization of functions from Bazilevič subclasses, Sov. Math. (Iz. VUZ), 19, 2, 106-107 (1975) |
| [199] | Prokhorov, D. V., Optimal control method in an extremal problem on the class of univalent functions, Dokl. Akad. Nauk SSSR, 275, 4, 798-801 (1984) |
| [200] | Prokhorov, D. V., Value sets for systems of functionals in the classes of univalent functions, Mat. Sb., 181, 12, 1659-1677 (1990) · Zbl 0717.30013 |
| [201] | Prokhorov, D. V., The value set of initial coefficients for bounded univalent typically real functions, Sib. Mat. Zh., 32, 5, 132-141 (1990) · Zbl 0749.30013 |
| [202] | Prokhorov, D. V., Reachable Set Methods in Extremal Problems for Univalent Functions (1993), Saratov: Saratov University, Saratov · Zbl 0814.30016 |
| [203] | Prokhorov, D. V., Even coefficient estimates for bounded univalent functions, Ann. Polon. Math., 58, 3, 267-273 (1993) · Zbl 0784.30010 |
| [204] | Prokhorov, D. V., Coefficients of functions close to the identity function, Complex Variables Theor. Appl., 33, 1-4, 255-263 (1997) · Zbl 0902.30013 |
| [205] | D.V. Prokhorov, Coefficients of holomorphic functions. Complex analysis and representation theory, 2. J. Math. Sci. (New York) 106(6), 3518-3544 (2001) · Zbl 1080.30023 |
| [206] | Prokhorov, D. V.; Nikulin, A. A., Asymptotic estimates for the coefficients of bounded univalent functions, Sibirsk. Mat. Zh., 47, 5, 1128-1138 (2006) · Zbl 1150.30320 |
| [207] | Prokhorov, D.; Vasil’ev, A., Univalent functions and integrable systems, Comm. Math. Phys., 262, 2, 393-410 (2006) · Zbl 1103.37045 |
| [208] | Prokhorov, D.; Vasil’ev, A., Optimal control in Bombieri’s and Tammi’s conjectures, Georgian Math. J., 12, 4, 743-761 (2005) · Zbl 1095.30019 |
| [209] | D. Prokhorov, A. Vasil’ev, Singular and tangent slit solutions to the Löwner equation, in Analysis and Mathematical Physics, Trends in Mathematics (Birkhäuser, Basel, 2009), pp. 455-463 · Zbl 1297.30012 |
| [210] | Reed, M.; Simon, B., Methods of Modern Mathematical Physics. I. Functional Analysis, 400 (1980), New York: Academic Press, New York · Zbl 0459.46001 |
| [211] | Reich, S., D Shoikhet, Metric domains, holomorphic mappings and nonlinear semigroups, Abstr. Appl. Anal., 3, 1-2, 203-228 (1998) · Zbl 0993.47040 |
| [212] | Reich, S.; Shoikhet, D., Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces (2005), London: Imperial College Press, London · Zbl 1089.46002 |
| [213] | Robertson, M. S., A remark in the odd schlicht functions, Bull. Am. Math. Soc., 42, 6, 366-370 (1936) · Zbl 0014.40702 |
| [214] | Robertson, M. S., Applications of the subordination principle to univalent functions, Pacific J. Math., 11, 315-324 (1961) · Zbl 0109.04902 |
| [215] | Robinson, R. M., Bounded univalent functions, Trans. Am. Math. Soc., 52, 426-449 (1942) · Zbl 0060.21802 |
| [216] | Rohde, S.; Schramm, O., Basic properties of SLE, Ann. Math., 161, 2, 883-924 (2005) · Zbl 1081.60069 |
| [217] | Rohde, S., Oded Schramm: from circle packing to SLE, Ann. Probab., 39, 5, 1621-1667 (2011) · Zbl 1236.60004 |
| [218] | Rosenblum, M.; Rovnyak, J., Topics in Hardy Classes and Univalent Functions (1994), Basel: Birkhäuser Verlag, Basel · Zbl 0816.30001 |
| [219] | Röckner, M., Markov property of generalized fields and axiomatic potential theory, Math. Ann., 264, 153-177 (1983) · Zbl 0519.60047 |
| [220] | Röckner, M., Generalized Markov fields and Dirichlet forms, Acta Applicandae Math., 3, 285-311 (1985) · Zbl 0588.60044 |
| [221] | Rushkin, I.; Bettelheim, E.; Gruzberg, I. A.; Wiegmann, P., Critical curves in conformally invariant statistical systems, J. Phys. A Math. Theor., 40, 2165-2195 (2007) · Zbl 1160.82324 |
| [222] | M. Sato, Y. Sato, Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold. Nonlinear Partial Differential Equations in Applied Science Tokyo, 1982. North-Holland Mathematics Studies, vol. 81 (North-Holland, Amsterdam, 1983), pp. 259-271 · Zbl 0528.58020 |
| [223] | Savakis, A.; Maggelakis, S., Models of shrinking clusters with applications to epidermal wound healing, Math. Comput. Model., 25, 6, 1-6 (1997) · Zbl 0882.92021 |
| [224] | Segal, G.; Wilson, G., Loop groups and equations of KdV type, Publ. Math. IHES, 61, 5-65 (1985) · Zbl 0592.35112 |
| [225] | Schaeffer, A. C.; Spencer, D. C., The coefficients of schlicht functions, II, 12, 107-125 (1945) · Zbl 0063.06763 |
| [226] | A.C. Schaeffer, D.C. Spencer, Coefficient Regions for Schlicht Functions. American Mathematical Society Colloquium Publications, vol. 35 (American Mathematical Society, New York, NY, 1950) · Zbl 0049.06003 |
| [227] | Schiffer, M.; Spencer, D. C., Functionals of Finite Riemann Surfaces (1954), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 0059.06901 |
| [228] | Schiffer, M.; Tammi, O., On the fourth coefficient of bounded univalent functions, Trans. Am. Math. Soc., 119, 67-78 (1965) · Zbl 0173.08203 |
| [229] | M. Schottenloher, A Mathematical Introduction to Conformal Field Theory, 2nd edn. Lecture Notes in Physics, vol. 759 (Springer, Berlin, 2008) · Zbl 1161.17014 |
| [230] | Schramm, O., Scaling limits of loop-erased random walks and uniform spanning trees, Isr. J. Math., 118, 221-288 (2000) · Zbl 0968.60093 |
| [231] | Schramm, O.; Sheffield, S., Contour lines of the two-dimensional discrete Gaussian free field, Acta Math., 202, 1, 21-137 (2009) · Zbl 1210.60051 |
| [232] | O. Schramm, S. Sheffield, A contour line of the continuum Gaussian free field. arXiv: 1008.2447, 2010 · Zbl 1331.60090 |
| [233] | Shah, D.-S., Le-le Fan, On the parametric representation of quasi-conformal mappings. Sci. Sin., 11, 149-162 (1962) · Zbl 0144.33603 |
| [234] | Sheffield, S., Exploration trees and conformal loop ensembles, Duke Math. J., 147, 1, 79-129 (2009) · Zbl 1170.60008 |
| [235] | Sheil-Small, T., Some remarks on Bazilevič functions, J. Anal. Math., 43, 1-11 (1983) · Zbl 0566.30009 |
| [236] | Shein, B. M., Private communication (2012) |
| [237] | Siewierski, L., Sharp estimation of the coefficients of bounded univalent functions close to identity, Diss. Math., 86, 1-153 (1971) · Zbl 0229.30019 |
| [238] | A.G. Siskakis, Semigroups of composition operators and the Cesàro operator on H^p(D), PhD Thesis, University of Illinois, 1985 |
| [239] | Siskakis, A. G., Semigroups of composition operators on spaces of analytic functions, a review, Contemp. Math., 213, 229-252 (1998) · Zbl 0904.47030 |
| [240] | Smirnov, S., Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits, C. R. Acad. Sci. Paris Sér. I Math., 333, 239-244 (2001) · Zbl 0985.60090 |
| [241] | T.T. Soong, Random Differential Equations in Science and Engineering. Mathematics in Science and Engineering, vol. 103 (Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1973) · Zbl 0348.60081 |
| [242] | Lad. Špaček, Contribution à la théorie des fonctions univalentes. Čas. Mat. Fys. 62(2), 12-19 (1932) · Zbl 0006.06403 |
| [243] | Takebe, T.; Teo, L.-P.; Zabrodin, A., Löwner equations and dispersionless hierarchies, J. Phys. A, 39, 37, 11479-11501 (2006) · Zbl 1104.37046 |
| [244] | Tammi, O., On the maximalization of the coefficient a_3 of bounded schlicht functions, Ann. Acad. Sci. Fennicae. Ser. A. I. Math. Phys., 149, 1-14 (1953) · Zbl 0050.30201 |
| [245] | Tammi, O., On optimizing parameters of the power inequality for a_4 in the class of bounded univalent functions, Ann. Acad. Sci. Fennicae. Ser. A. I. Math. Phys., 560, 1-24 (1973) · Zbl 0274.30012 |
| [246] | Tammi, O., Extremal Problems for Bounded Univalent Functions, Lecture Notes in Mathematics (1978), New York: Springer, New York · Zbl 0375.30006 |
| [247] | Tammi, O., Extremal Problems for Bounded Univalent Functions, II, Lecture Notes in Mathematics (1982), New York: Springer, New York · Zbl 0481.30020 |
| [248] | Tricomi, F., Sull’iterazione delle funzioni di linee, Giorn. Math. Battaglini, 55, 1-8 (1917) |
| [249] | Tsai, J., The Loewner driving function of trajectory arcs of quadratic differentials, J. Math. Anal. Appl., 360, 561-576 (2009) · Zbl 1177.30059 |
| [250] | Vasil’ev, A., Mutual change of initial coefficients of univalent functions, Matemat. Zametki, 38, 1, 56-65 (1985) · Zbl 0589.30012 |
| [251] | A. Vasil’ev, Value set \(\{f(r_1),f(r_2)\}\) for the class of univalent functions with real coefficients. Sib. Mat. Zh. 27(6), 28-32 (1986) · Zbl 0616.30006 |
| [252] | Vasil’ev, A., Optimal control methods in an extremal problem on a class of solutions of the Loewner-Kufarev equation, Diff. Uravneniya, 26, 3, 386-392 (1990) · Zbl 0711.34078 |
| [253] | Vasil’ev, A., From the Hele-Shaw experiment to integrable systems: a historical overview, Complex Anal. Oper. Theor., 3, 2, 551-585 (2009) · Zbl 1279.01050 |
| [254] | Vicsek, T., Fractal Growth Phenomena (1989), Singapore: World Scientific Publishing, Singapore · Zbl 0744.28005 |
| [255] | Voda, M., Solution of a Loewner chain equation in several variables, J. Math. Anal. Appl., 375, 1, 58-74 (2011) · Zbl 1209.32007 |
| [256] | Wick, G. C., The evaluation of the collision matrix, Phys. Rev., 80, 2, 268-272 (1950) · Zbl 0040.13006 |
| [257] | Wiegmann, P. B.; Zabrodin, A., Conformal maps and integrable hierarchies, Comm. Math. Phys., 213, 3, 523-538 (2000) · Zbl 0973.37042 |
| [258] | A.S. Wightman, Hilberts Sixth Problem: Mathematical Treatment of the Axioms of Physics, ed. by F.E. Browder. Proceedings of Symposia in Pure Mathematics (part 1), vol. 28 (American Mathematical Society, Providence, RI, 1976), pp. 241-268 · Zbl 0339.46051 |
| [259] | Witten, E., Quantum field theory, Grassmannians, and algebraic curves, Comm. Math. Phys., 113, 4, 529-600 (1988) · Zbl 0636.22012 |
| [260] | Witten, T. A. Jr; Sander, L. M., Diffusion-limited aggregation, and kinetic critical phenomenon, Phys. Rev. Lett., 47, 2, 1400-1403 (1981) |
| [261] | Witten, T. A.; Sander, L. M., Diffusion-limited aggregation. Phys. Rev. B, 27, 9, 5686-5697 (1983) |
| [262] | Zabusky, N. J.; Krushkal, M. D., Interaction of solitons in a collisionless plasma and initial states, Phys. Rev. Lett., 15, 240-243 (1965) · Zbl 1201.35174 |
| [263] | Zaharov, V. E.; Faddeev, L. D., The Korteweg-de Vries equation is a fully integrable Hamiltonian system, Funkcional. Anal. i Prilozh., 5, 5, 18-27 (1971) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
