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Journal articles on the topic 'Loewner equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 27 April 2022

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1

Schleißinger, Sebastian. "The chordal Loewner equation and monotone probability theory." Infinite Dimensional Analysis, Quantum Probability and Related Topics 20, no. 03 (September 2017): 1750016. http://dx.doi.org/10.1142/s0219025717500163.

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In Ref. 5, O. Bauer interpreted the chordal Loewner equation in terms of noncommutative probability theory. We follow this perspective and identify the chordal Loewner equations as the non-autonomous versions of evolution equations for semigroups in monotone and anti-monotone probability theory. We also look at the corresponding equation for free probability theory.

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2

Starnes, Andrew. "The Loewner equation for multiple hulls." Annales Academiae Scientiarum Fennicae Mathematica 44, no. 1 (February 2019): 581–99. http://dx.doi.org/10.5186/aasfm.2019.4435.

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3

Gruzberg, Ilya A., and Leo P. Kadanoff. "The Loewner Equation: Maps and Shapes." Journal of Statistical Physics 114, no. 5/6 (March 2004): 1183–98. http://dx.doi.org/10.1023/b:joss.0000013973.40984.3b.

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4

Rohde, Steffen, Huy Tran, and Michel Zinsmeister. "The Loewner equation and Lipschitz graphs." Revista Matemática Iberoamericana 34, no. 2 (May 28, 2018): 937–48. http://dx.doi.org/10.4171/rmi/1010.

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5

McDonald, Robb. "Geodesic Loewner paths with varying boundary conditions." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2242 (October 2020): 20200466. http://dx.doi.org/10.1098/rspa.2020.0466.

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Equations of the Loewner class subject to non-constant boundary conditions along the real axis are formulated and solved giving the geodesic paths of slits growing in the upper half complex plane. The problem is motivated by Laplacian growth in which the slits represent thin fingers growing in a diffusion field. A single finger follows a curved path determined by the forcing function appearing in Loewner’s equation. This function is found by solving an ordinary differential equation whose terms depend on curvature properties of the streamlines of the diffusive field in the conformally mapped ‘mathematical’ plane. The effect of boundary conditions specifying either piecewise constant values of the field variable along the real axis, or a dipole placed on the real axis, reveal a range of behaviours for the growing slit. These include regions along the real axis from which no slit growth is possible, regions where paths grow to infinity, or regions where paths curve back toward the real axis terminating in finite time. Symmetric pairs of paths subject to the piecewise constant boundary condition along the real axis are also computed, demonstrating that paths which grow to infinity evolve asymptotically toward an angle of bifurcation of π /5.

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6

Graham, Ian, Hidetaka Hamada, and Gabriela Kohr. "Parametric Representation of Univalent Mappings in Several Complex Variables." Canadian Journal of Mathematics 54, no. 2 (April 1, 2002): 324–51. http://dx.doi.org/10.4153/cjm-2002-011-2.

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AbstractLet B be the unit ball of with respect to an arbitrary norm. We prove that the analog of the Carathéodory set, i.e. the set of normalized holomorphic mappings from B into of “positive real part”, is compact. This leads to improvements in the existence theorems for the Loewner differential equation in several complex variables. We investigate a subset of the normalized biholomorphic mappings of B which arises in the study of the Loewner equation, namely the set S0(B) of mappings which have parametric representation. For the case of the unit polydisc these mappings were studied by Poreda, and on the Euclidean unit ball they were studied by Kohr. As in Kohr’s work, we consider subsets of S0(B) obtained by placing restrictions on the mapping from the Carathéodory set which occurs in the Loewner equation. We obtain growth and covering theorems for these subsets of S0(B) as well as coefficient estimates, and consider various examples. Also we shall see that in higher dimensions there exist mappings in S(B) which can be imbedded in Loewner chains, but which do not have parametric representation.

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7

Roth, Oliver. "Pontryagin’s Maximum Principle for the Loewner Equation in Higher Dimensions." Canadian Journal of Mathematics 67, no. 4 (August 1, 2015): 942–60. http://dx.doi.org/10.4153/cjm-2014-027-6.

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AbstractIn this paper we develop a variational method for the Loewner equation in higher dimensions. As a result we obtain a version of Pontryagin’s maximum principle from optimal control theory for the Loewner equation in several complex variables. Based on recent work of Arosio, Bracci, and Wold, we then apply our version of the Pontryagin maximum principle to obtain first-order necessary conditions for the extremal mappings for a wide class of extremal problems over the set of normalized biholomorphic mappings on the unit ball in ℂn.

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8

Marshall, Donald E., and Steffen Rohde. "The Loewner differential equation and slit mappings." Journal of the American Mathematical Society 18, no. 4 (June 10, 2005): 763–78. http://dx.doi.org/10.1090/s0894-0347-05-00492-3.

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9

Prokhorov, D. "Exact Solutions of the Multiple Loewner Equation." Lobachevskii Journal of Mathematics 41, no. 11 (November 2020): 2248–56. http://dx.doi.org/10.1134/s1995080220110189.

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10

Roth, Oliver, and Sebastian Schleissinger. "The Schramm-Loewner equation for multiple slits." Journal d'Analyse Mathématique 131, no. 1 (March 2017): 73–99. http://dx.doi.org/10.1007/s11854-017-0002-y.

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11

Böhm, Christoph, and Wolfgang Lauf. "A Komatu–Loewner Equation for Multiple Slits." Computational Methods and Function Theory 14, no. 4 (April 2, 2014): 639–63. http://dx.doi.org/10.1007/s40315-014-0064-0.

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12

Technau, Marc, and Niclas Technau. "A Loewner Equation for Infinitely Many Slits." Computational Methods and Function Theory 17, no. 2 (October 3, 2016): 255–72. http://dx.doi.org/10.1007/s40315-016-0179-6.

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13

ZADOROZHNAYA, Olga V., and Vladimir K. KOCHETKOV. "INTEGRAL REPRESENTATION OF SOLUTIONS OF AN ORDINARY DIFFERENTIAL EQUATION AND THE LOEWNER– KUFAREV EQUATION." Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, no. 67 (2020): 28–39. http://dx.doi.org/10.17223/19988621/67/3.

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The article presents a method of integral representation of solutions of ordinary differential equations and partial differential equations with a polynomial right-hand side part, which is an alternative to the construction of solutions of differential equations in the form of different series. The method is based on the introduction of additional analytical functions establishing the equation of connection between the introduced functions and the constituent components of the original differential equation. The implementation of the coupling equations contributes to the representation of solutions of the differential equation in the integral form, which allows solving some problems of mathematics and mathematical physics. The first part of the article describes the coupling equation for an ordinary differential equation of the first order with a special polynomial part of a higher order. Here, the integral representation of the solution of a differential equation with a second-order polynomial part is indicated in detail. In the second part of the paper, we consider the integral representation of the solution of a partial differential equation with the polynomial second-order part of the Loewner–Kufarev equation, which is an equation for univalent functions.

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14

Lau, Ka-Sing, and Hai-Hua Wu. "On tangential slit solution of the Loewner equation." Annales Academiae Scientiarum Fennicae Mathematica 41 (2016): 681–91. http://dx.doi.org/10.5186/aasfm.2016.4142.

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15

Kadanoff, Leo P., and Marko Kleine Berkenbusch. "Trace for the Loewner equation with singular forcing." Nonlinearity 17, no. 4 (May 8, 2004): R41—R54. http://dx.doi.org/10.1088/0951-7715/17/4/r01.

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16

Kadanoff, L. P., and M. Kleine Berkenbusch. "Trace for the Loewner equation with singular forcing." Nonlinearity 18, no. 2 (February 2, 2005): 937. http://dx.doi.org/10.1088/0951-7715/18/2/c01.

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17

Lind, Joan, and Steffen Rohde. "Spacefilling curves and phases of the Loewner equation." Indiana University Mathematics Journal 61, no. 6 (2012): 2231–49. http://dx.doi.org/10.1512/iumj.2012.61.4794.

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18

Wu, HaiHua, and XinHan Dong. "Driving functions and traces of the Loewner equation." Science China Mathematics 57, no. 8 (August 19, 2013): 1615–24. http://dx.doi.org/10.1007/s11425-013-4698-6.

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19

OEVEL, W., and W. SCHIEF. "SQUARED EIGENFUNCTIONS OF THE (MODIFIED) KP HIERARCHY AND SCATTERING PROBLEMS OF LOEWNER TYPE." Reviews in Mathematical Physics 06, no. 06 (December 1994): 1301–38. http://dx.doi.org/10.1142/s0129055x94000468.

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It is shown that products of eigenfunctions and (integrated) adjoint eigenfunctions associated with the (modified) Kadomtsev-Petviashvili (KP) hierarchy form generators of a symmetry transformation. Linear integro-differential representations for these symmetries are found. For special cases the corresponding nonlinear equations are the compatibility conditions of linear scattering problems of Loewner type. The examples include the 2+1-dimensional sine-Gordon equation with space variables occuring on an equal footing introduced recently by Konopelchenko and Rogers. This equation represents a special squared eigenfunction symmetry of the Ishimori hierarchy.

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20

Prokhorov, D. V., and K. A. Samsonova. "Integrals of the Loewner Equation with Exponential Driving Function." Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics 13, no. 4 (2013): 98–108. http://dx.doi.org/10.18500/1816-9791-2013-13-4-98-108.

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21

Prokhorov, Dmitri, and Kristina Samsonova. "Value range of solutions to the chordal Loewner equation." Journal of Mathematical Analysis and Applications 428, no. 2 (August 2015): 910–19. http://dx.doi.org/10.1016/j.jmaa.2015.03.065.

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22

Prokhorov, D. V., A. M. Zakharov, and A. V. Zherdev. "Solutions of the Loewner equation with combined driving functions." Izvestia of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics 21, no. 3 (August 25, 2021): 317–25. http://dx.doi.org/10.18500/1816-9791-2021-21-3-317-325.

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23

Shibasaki, Yusuke, and Minoru Saito. "Loewner Equation with Chaotic Driving Function Describes Neurite Outgrowth Mechanism." Journal of the Physical Society of Japan 88, no. 6 (June 15, 2019): 063801. http://dx.doi.org/10.7566/jpsj.88.063801.

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24

Voda, Mircea. "Solution of a Loewner chain equation in several complex variables." Journal of Mathematical Analysis and Applications 375, no. 1 (March 2011): 58–74. http://dx.doi.org/10.1016/j.jmaa.2010.08.057.

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25

Filippo, Bracci, Manuel D. Contreras, and Santiago Díaz-Madrigal. "Evolution families and the Loewner equation I: the unit disc." Journal für die reine und angewandte Mathematik (Crelles Journal) 2012, no. 672 (January 2012): 1–37. http://dx.doi.org/10.1515/crelle.2011.167.

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26

Durán, M., and G. L. Vasconcelos. "Loewner equation for Laplacian growth: A Schwarz-Christoffel-transformation approach." Journal of Physics: Conference Series 246 (September 1, 2010): 012023. http://dx.doi.org/10.1088/1742-6596/246/1/012023.

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27

Arosio, Leandro, and Filippo Bracci. "Infinitesimal generators and the Loewner equation on complete hyperbolic manifolds." Analysis and Mathematical Physics 1, no. 4 (December 2011): 337–50. http://dx.doi.org/10.1007/s13324-011-0020-3.

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28

Bracci, Filippo, Manuel D. Contreras, and Santiago Díaz-Madrigal. "Evolution families and the Loewner equation II: complex hyperbolic manifolds." Mathematische Annalen 344, no. 4 (March 10, 2009): 947–62. http://dx.doi.org/10.1007/s00208-009-0340-x.

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29

Zherdev, A. V. "Value range of solutions to the chordal Loewner equation with restriction on the driving function." Issues of Analysis 26, no. 2 (June 2019): 92–104. http://dx.doi.org/10.15393/j3.art.2019.6270.

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30

Murayama, Takuya. "Chordal Komatu–Loewner equation for a family of continuously growing hulls." Stochastic Processes and their Applications 129, no. 8 (August 2019): 2968–90. http://dx.doi.org/10.1016/j.spa.2018.08.012.

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31

Schleißinger, Sebastian. "On driving functions generating quasislits in the chordal Loewner–Kufarev equation." Complex Variables and Elliptic Equations 60, no. 1 (April 10, 2014): 134–44. http://dx.doi.org/10.1080/17476933.2014.904296.

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32

Böhm, Christoph, and Sebastian Schleißinger. "Constant coefficients in the radial Komatu–Loewner equation for multiple slits." Mathematische Zeitschrift 279, no. 1-2 (September 14, 2014): 321–32. http://dx.doi.org/10.1007/s00209-014-1370-y.

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33

del Monaco, Andrea, Ikkei Hotta, and Sebastian Schleißinger. "Tightness Results for Infinite-Slit Limits of the Chordal Loewner Equation." Computational Methods and Function Theory 18, no. 1 (May 25, 2017): 9–33. http://dx.doi.org/10.1007/s40315-017-0205-3.

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34

Duren, Peter, Ian Graham, Hidetaka Hamada, and Gabriela Kohr. "Solutions for the generalized Loewner differential equation in several complex variables." Mathematische Annalen 347, no. 2 (October 29, 2009): 411–35. http://dx.doi.org/10.1007/s00208-009-0429-2.

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35

de Carvalho Neto, Edgar Marcelino, Thiago A. de Assis, Caio M. C. de Castilho, and Roberto F. S. Andrade. "Field enhancement optimization of growing curved structures using the Loewner equation." Journal of Applied Physics 130, no. 2 (July 14, 2021): 025108. http://dx.doi.org/10.1063/5.0050282.

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36

Lawler, Gregory F., and Fredrik Viklund. "The Loewner difference equation and convergence of loop-erased random walk." Latin American Journal of Probability and Mathematical Statistics 19, no. 1 (2022): 565. http://dx.doi.org/10.30757/alea.v19-22.

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37

Hamada, H. "Polynomially bounded solutions to the Loewner differential equation in several complex variables." Journal of Mathematical Analysis and Applications 381, no. 1 (September 2011): 179–86. http://dx.doi.org/10.1016/j.jmaa.2011.02.080.

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38

Roth, Oliver, and Sebastian Schleißinger. "Rogosinski's lemma for univalent functions, hyperbolic Archimedean spirals and the Loewner equation." Bulletin of the London Mathematical Society 46, no. 5 (July 3, 2014): 1099–109. http://dx.doi.org/10.1112/blms/bdu054.

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39

Goryainov, V. V. "Loewner–Kufarev Equation for a Strip with an Analogue of Hydrodynamic Normalization." Lobachevskii Journal of Mathematics 39, no. 6 (August 2018): 759–66. http://dx.doi.org/10.1134/s1995080218060070.

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40

Iancu, Mihai. "On Reachable Families of the Loewner Differential Equation in Several Complex Variables." Complex Analysis and Operator Theory 10, no. 2 (April 29, 2015): 353–68. http://dx.doi.org/10.1007/s11785-015-0461-z.

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41

Böhm, Christoph, and Sebastian Schleißinger. "The Loewner equation for multiple slits, multiply connected domains and branch points." Arkiv för Matematik 54, no. 2 (October 2016): 339–70. http://dx.doi.org/10.1007/s11512-016-0231-9.

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42

Yuan, Jiangtao, and Caihong Wang. "Some Properties of Furuta Type Inequalities and Applications." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/457367.

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This work is to consider Furuta type inequalities and their applications. Firstly, some Furuta type inequalities underA≥B≥0are obtained via Loewner-Heinz inequality; as an application, a proof of Furuta inequality is given without using the invertibility of operators. Secondly, we show a unified satellite theorem of grand Furuta inequality which is an extension of the results by Fujii et al. At the end, a kind of Riccati type operator equation is discussed via Furuta type inequalities.

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43

Chen, Zhen-Qing, Masatoshi Fukushima, and Steffen Rohde. "Chordal Komatu-Loewner equation and Brownian motion with darning in multiply connected domains." Transactions of the American Mathematical Society 368, no. 6 (October 2, 2015): 4065–114. http://dx.doi.org/10.1090/tran/6441.

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44

Murayama, Takuya. "On the slit motion obeying chordal Komatu–Loewner equation with finite explosion time." Journal of Evolution Equations 20, no. 1 (June 4, 2019): 233–55. http://dx.doi.org/10.1007/s00028-019-00519-3.

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45

ZADOROZHNAYA, Olga Vladimirovna, and Vladimir Konstantinovich KOCHETKOV. "THE STRUCTURE OF INTEGRALS OF THE SECOND LOEWNER–KUFAREV DIFFERENTIAL EQUATION IN A PARTICULAR CASE." Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, no. 55 (August 1, 2018): 12–21. http://dx.doi.org/10.17223/19988621/55/2.

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46

Hottovy, Scott, and Samuel N. Stechmann. "A Spatiotemporal Stochastic Model for Tropical Precipitation and Water Vapor Dynamics." Journal of the Atmospheric Sciences 72, no. 12 (November 24, 2015): 4721–38. http://dx.doi.org/10.1175/jas-d-15-0119.1.

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Abstract A linear stochastic model is presented for the dynamics of water vapor and tropical convection. Despite its linear formulation, the model reproduces a wide variety of observational statistics from disparate perspectives, including (i) a cloud cluster area distribution with an approximate power law; (ii) a power spectrum of spatiotemporal red noise, as in the “background spectrum” of tropical convection; and (iii) a suite of statistics that resemble the statistical physics concepts of critical phenomena and phase transitions. The physical processes of the model are precipitation, evaporation, and turbulent advection–diffusion of water vapor, and they are represented in idealized form as eddy diffusion, damping, and stochastic forcing. Consequently, the form of the model is a damped version of the two-dimensional stochastic heat equation. Exact analytical solutions are available for many statistics, and numerical realizations can be generated for minimal computational cost and for any desired time step. Given the simple form of the model, the results suggest that tropical convection may behave in a relatively simple, random way. Finally, relationships are also drawn with the Ising model, the Edwards–Wilkinson model, the Gaussian free field, and the Schramm–Loewner evolution and its possible connection with cloud cluster statistics. Potential applications of the model include several situations where realistic cloud fields must be generated for minimal cost, such as cloud parameterizations for climate models or radiative transfer models.

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47

Fukushima, Masatoshi. "Komatu-Loewner differential equations." Sugaku Expositions 33, no. 2 (October 13, 2020): 239–60. http://dx.doi.org/10.1090/suga/455.

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48

Arosio, Leandro. "Resonances in Loewner equations." Advances in Mathematics 227, no. 4 (July 2011): 1413–35. http://dx.doi.org/10.1016/j.aim.2011.03.010.

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49

Arosio, Leandro. "Basins of attraction in Loewner equations." Annales Academiae Scientiarum Fennicae Mathematica 37 (August 2012): 563–70. http://dx.doi.org/10.5186/aasfm.2012.3742.

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50

Arosio, Leandro. "Loewner equations on complete hyperbolic domains." Journal of Mathematical Analysis and Applications 398, no. 2 (February 2013): 609–21. http://dx.doi.org/10.1016/j.jmaa.2012.09.018.

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