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Journal articles on the topic 'Fully nonlinear operators'

Author: Grafiati

Published: 4 June 2021

Last updated: 17 February 2022

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1

Esteban, Maria J., Patricio Felmer, and Alexander Quaas. "Eigenvalues for Radially Symmetric Fully Nonlinear Operators." Communications in Partial Differential Equations 35, no. 9 (August 10, 2010): 1716–37. http://dx.doi.org/10.1080/03605301003674848.

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2

Väth, Martin. "Degeneracy results for fully nonlinear integral operators." Indagationes Mathematicae 31, no. 5 (September 2020): 893–916. http://dx.doi.org/10.1016/j.indag.2020.06.009.

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3

Niu, Pengcheng, Leyun Wu, and Xiaoxue Ji. "Positive solutions to nonlinear systems involving fully nonlinear fractional operators." Fractional Calculus and Applied Analysis 21, no. 2 (April 25, 2018): 552–74. http://dx.doi.org/10.1515/fca-2018-0030.

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Abstract In this paper we consider the following fractional system $$\begin{array}{} \displaystyle \left\{ \begin{gathered} F(x,u(x),v(x),{\mathcal{F}_\alpha }(u(x))) = 0,\\ G(x,v(x),u(x),{\mathcal{G}_\beta }(v(x))) = 0, \\ \end{gathered} \right. \end{array}$$ where 0 < α, β < 2, 𝓕α and 𝓖β are the fully nonlinear fractional operators: $$\begin{array}{} \displaystyle {\mathcal{F}_\alpha }(u(x)) = {C_{n,\alpha }}PV\int_{{\mathbb{R}^n}} {\frac{{f(u(x) - u(y))}} {{{{\left| {x - y} \right|}^{n + \alpha }}}}dy} ,\\ \displaystyle{\mathcal{G}_\beta }(v(x)) = {C_{n,\beta }}PV\int_{{\mathbb{R}^n}} {\frac{{g(v(x) - v(y))}} {{{{\left| {x - y} \right|}^{n + \beta }}}}dy} . \end{array}$$ A decay at infinity principle and a narrow region principle for solutions to the system are established. Based on these principles, we prove the radial symmetry and monotonicity of positive solutions to the system in the whole space and a unit ball respectively, and the nonexistence in a half space by generalizing the direct method of moving planes to the nonlinear system.

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4

Kuo, Hung-Ju, and Neil S. Trudinger. "Schauder estimates for fully nonlinear elliptic difference operators." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132, no. 6 (December 2002): 1395–406. http://dx.doi.org/10.1017/s030821050000216x.

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In this paper, we are concerned with discrete Schauder estimates for solutions of fully nonlinear elliptic difference equations. Our estimates are discrete versions of second derivative Hölder estimates of Evans, Krylov and Safonov for fully nonlinear elliptic partial differential equations. They extend previous results of Holtby for the special case of functions of pure second-order differences on cubic meshes. As with Holtby's work, the fundamental ingredients are the pointwise estimates of Kuo-Trudinger for linear difference schemes on general meshes.

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5

Birindelli, Isabeau, and Stefania Patrizi. "A Neumann eigenvalue problem for fully nonlinear operators." Discrete & Continuous Dynamical Systems - A 28, no. 2 (2010): 845–63. http://dx.doi.org/10.3934/dcds.2010.28.845.

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6

Caffarelli, Luis, Luis Duque, and Hernán Vivas. "The two membranes problem for fully nonlinear operators." Discrete & Continuous Dynamical Systems - A 38, no. 12 (2018): 6015–27. http://dx.doi.org/10.3934/dcds.2018152.

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7

Patrizi, Stefania. "The Neumann problem for singular fully nonlinear operators." Journal de Mathématiques Pures et Appliquées 90, no. 3 (September 2008): 286–311. http://dx.doi.org/10.1016/j.matpur.2008.04.007.

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8

Tarsia, Antonio. "Near operators theory and fully nonlinear elliptic equations." Journal of Global Optimization 40, no. 1-3 (January 22, 2008): 443–53. http://dx.doi.org/10.1007/s10898-007-9227-0.

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9

Birindelli, Isabeau, Françoise Demengel, and Fabiana Leoni. "Ergodic pairs for singular or degenerate fully nonlinear operators." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 75. http://dx.doi.org/10.1051/cocv/2018070.

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We study the ergodic problem for fully nonlinear operators which may be singular or degenerate when the gradient of solutions vanishes. We prove the convergence of both explosive solutions and solutions of Dirichlet problems for approximating equations. We further characterize the ergodic constant as the infimum of constants for which there exist bounded sub-solutions. As intermediate results of independent interest, we prove a priori Lipschitz estimates depending only on the norm of the zeroth order term, and a comparison principle for equations having no zero order terms.

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10

Montanari, Annamaria, and Ermanno Lanconelli. "Pseudoconvex fully nonlinear partial differential operators: strong comparison theorems." Journal of Differential Equations 202, no. 2 (August 2004): 306–31. http://dx.doi.org/10.1016/j.jde.2004.03.017.

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11

Kim, Yong-Cheol, and Ki-Ahm Lee. "Regularity results for fully nonlinear parabolic integro-differential operators." Mathematische Annalen 357, no. 4 (June 27, 2013): 1541–76. http://dx.doi.org/10.1007/s00208-013-0948-8.

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12

Demengel, Françoise. "Ergodic pairs for degenerate pseudo Pucci's fully nonlinear operators." Discrete & Continuous Dynamical Systems 41, no. 7 (2021): 3465. http://dx.doi.org/10.3934/dcds.2021004.

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13

Goffi, Alessandro, and Francesco Pediconi. "A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds." Journal of Geometric Analysis 31, no. 8 (February 16, 2021): 8641–65. http://dx.doi.org/10.1007/s12220-021-00607-2.

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AbstractWe investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and $$\infty $$ ∞ -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

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14

JUNGES MIOTTO, T. "THE ALEKSANDROV–BAKELMAN–PUCCI ESTIMATES FOR SINGULAR FULLY NONLINEAR OPERATORS." Communications in Contemporary Mathematics 12, no. 04 (August 2010): 607–27. http://dx.doi.org/10.1142/s0219199710003956.

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The main scope of this paper is to obtain Aleksandrov–Bakelman–Pucci estimates (ABP estimates) for viscosity solutions of singular fully nonlinear operator, which includes the p-Laplacian operator, p > 1.

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15

Felmer, Patricio, and Darío Valdebenito. "Eigenvalues for radially symmetric fully nonlinear singular or degenerate operators." Nonlinear Analysis: Theory, Methods & Applications 75, no. 18 (December 2012): 6524–40. http://dx.doi.org/10.1016/j.na.2012.07.029.

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16

Bardi, Martino, and Annalisa Cesaroni. "Liouville properties and critical value of fully nonlinear elliptic operators." Journal of Differential Equations 261, no. 7 (October 2016): 3775–99. http://dx.doi.org/10.1016/j.jde.2016.06.006.

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17

Trudinger, Neil S. "Hölder gradient estimates for fully nonlinear elliptic equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 108, no. 1-2 (1988): 57–65. http://dx.doi.org/10.1017/s0308210500026512.

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SynopsisIn this paper we prove interior and global Hölder estimates for Lipschitz viscosity solutions of second order, nonlinear, uniformly elliptic equations. The smoothness hypotheses on the operators are more general than previously considered for classical solutions, so that our estimates are also new in this case and readily extend to embrace obstacle problems. In particular Isaac's equations of stochastic differential game theory constitute a special case of our results, and moreover our techniques, in combination with recent existence theorems of Ishii, lead to existence theorems for continuously differentiable viscosity solutions of the uniformly elliptic Isaac's equation.

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18

Birindelli, Isabeau, and Françoise Demengel. "Comparison principle and Liouville type results for singular fully nonlinear operators." Annales de la faculté des sciences de Toulouse Mathématiques 13, no. 2 (2004): 261–87. http://dx.doi.org/10.5802/afst.1070.

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19

Bae, Jongchun. "Regularity for Fully Nonlinear Equations Driven by Spatial-Inhomogeneous Nonlocal Operators." Potential Analysis 43, no. 4 (July 4, 2015): 611–24. http://dx.doi.org/10.1007/s11118-015-9488-z.

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20

Kim, Soojung, Yong-Cheol Kim, and Ki-Ahm Lee. "Regularity for Fully Nonlinear Integro-differential Operators with Regularly Varying Kernels." Potential Analysis 44, no. 4 (January 15, 2016): 673–705. http://dx.doi.org/10.1007/s11118-015-9525-y.

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21

Quaas, Alexander, and Boyan Sirakov. "Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators." Advances in Mathematics 218, no. 1 (May 2008): 105–35. http://dx.doi.org/10.1016/j.aim.2007.12.002.

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22

Dávila, Gonzalo, Patricio Felmer, and Alexander Quaas. "Harnack inequality for singular fully nonlinear operators and some existence results." Calculus of Variations and Partial Differential Equations 39, no. 3-4 (July 22, 2010): 557–78. http://dx.doi.org/10.1007/s00526-010-0325-3.

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23

da Silva, João Vítor, and Hernán Vivas. "The obstacle problem for a class of degenerate fully nonlinear operators." Revista Matemática Iberoamericana 37, no. 5 (January 21, 2021): 1991–2020. http://dx.doi.org/10.4171/rmi/1256.

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24

Mukhamedov, Farrukh, Mansoor Saburov, and Izzat Qaralleh. "Onξ(s)-Quadratic Stochastic Operators on Two-Dimensional Simplex and Their Behavior." Abstract and Applied Analysis 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/942038.

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A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for quadratic stochastic operators which are the simplest nonlinear operators. To study this problem, several classes of QSO were investigated. We studyξ(s)-QSO defined on 2D simplex. We first classifyξ(s)-QSO into 20 nonconjugate classes. Further, we investigate the dynamics of three classes of such operators.

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25

Birindelli, Isabeau, and Françoise Demengel. "Existence and Regularity Results for Fully Nonlinear Operators on the Model of the Pseudo Pucci’s Operators." Journal of Elliptic and Parabolic Equations 2, no. 1-2 (April 2016): 171–87. http://dx.doi.org/10.1007/bf03377400.

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26

Blanc, Pablo. "A lower bound for the principal eigenvalue of fully nonlinear elliptic operators." Communications on Pure & Applied Analysis 19, no. 7 (2020): 3613–23. http://dx.doi.org/10.3934/cpaa.2020158.

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27

Kim, Minhyun, and Ki-Ahm Lee. "Regularity for fully nonlinear integro-differential operators with kernels of variable orders." Nonlinear Analysis 193 (April 2020): 111312. http://dx.doi.org/10.1016/j.na.2018.07.009.

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28

Kim, Yong-Cheol, and Ki-Ahm Lee. "Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels." Manuscripta Mathematica 139, no. 3-4 (December 7, 2011): 291–319. http://dx.doi.org/10.1007/s00229-011-0516-z.

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29

Birindelli, I., and F. Demengel. "Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators." Journal of Differential Equations 249, no. 5 (September 2010): 1089–110. http://dx.doi.org/10.1016/j.jde.2010.03.015.

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30

Quaas, Alexander, and Boyan Sirakov. "On the principal eigenvalues and the Dirichlet problem for fully nonlinear operators." Comptes Rendus Mathematique 342, no. 2 (January 2006): 115–18. http://dx.doi.org/10.1016/j.crma.2005.11.003.

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31

Birindelli, I., and F. Demengel. "Eigenvalue and Dirichlet problem for fully-nonlinear operators in non-smooth domains." Journal of Mathematical Analysis and Applications 352, no. 2 (April 2009): 822–35. http://dx.doi.org/10.1016/j.jmaa.2008.11.012.

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32

Esteban, Maria J., Patricio L. Felmer, and Alexander Quaas. "Superlinear elliptic equation for fully nonlinear operators without growth restrictions for the data." Proceedings of the Edinburgh Mathematical Society 53, no. 1 (January 12, 2010): 125–41. http://dx.doi.org/10.1017/s0013091507001393.

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AbstractWe deal with existence and uniqueness of the solution to the fully nonlinear equation−F(D2u) + |u|s−1u = f(x) in ℝn,where s > 1 and f satisfies only local integrability conditions. This result is well known when, instead of the fully nonlinear elliptic operator F, the Laplacian or a divergence-form operator is considered. Our existence results use the Alexandroff-Bakelman-Pucci inequality since we cannot use any variational formulation. For radially symmetric f, and in the particular case where F is a maximal Pucci operator, we can prove our results under fewer integrability assumptions, taking advantage of an appropriate variational formulation. We also obtain an existence result with boundary blow-up in smooth domains.

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33

CHARRO, FERNANDO, and IRENEO PERAL. "ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS: COMPARISON, EXISTENCE AND UNIQUENESS." Communications in Contemporary Mathematics 11, no. 01 (February 2009): 131–64. http://dx.doi.org/10.1142/s0219199709003296.

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We study existence of solutions to [Formula: see text] where F is elliptic and homogeneous of degree m, and either f(λ,u) = λ uqor f(λ,u) = λ uq+ ur, for 0 < q < m < r, and λ > 0. Furthermore, in the first case, we obtain that the solution is unique as a consequence of a comparison principle up to the boundary. Several examples, including uniformly elliptic operators and the infinity-laplacian, are considered.

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34

Ferrari, Fausto. "An application of the theorem on Sums to viscosity solutions of degenerate fully nonlinear equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 2 (January 26, 2019): 975–92. http://dx.doi.org/10.1017/prm.2018.147.

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AbstractWe prove Hölder continuous regularity of bounded, uniformly continuous, viscosity solutions of degenerate fully nonlinear equations defined in all of ℝn space. In particular, the result applies also to some operators in Carnot groups.

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35

Holtby, Derek W. "HIGHER-ORDER ESTIMATES FOR FULLY NONLINEAR DIFFERENCE EQUATIONS. II." Proceedings of the Edinburgh Mathematical Society 44, no. 1 (February 2001): 87–102. http://dx.doi.org/10.1017/s0013091598000200.

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AbstractThe purpose of this work is to establish a priori $C^{2,\alpha}$ estimates for mesh function solutions of nonlinear difference equations of positive type in fully nonlinear form on a uniform mesh, where the fully nonlinear finite difference operator $\F$ is concave in the second-order variables. The estimate is an analogue of the corresponding estimate for solutions of concave fully nonlinear elliptic partial differential equations. We use the results for the special case that the operator does not depend explicitly upon the independent variables (the so-called frozen case) established in part I to approach the general case of explicit dependence upon the independent variables. We make our approach for the diagonal case via a discretization of the approach of Safonov for fully nonlinear elliptic partial differential equations using the discrete linear theory of Kuo and Trudinger and an especially agreeable mesh function interpolant provided by Kunkle. We generalize to non-diagonal operators using an idea which, to the author’s knowledge, is novel. In this paper we establish the desired Hölder estimate in the large, that is, on the entire mesh $n$-plane. In a subsequent paper a truly interior estimate will be established in a mesh $n$-box.AMS 2000 Mathematics subject classification: Primary 35J60; 35J15; 39A12. Secondary 39A70; 39A10; 65N06; 65N22; 65N12

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36

Argiolas, Roberto. "Single phase problem with curvature for a class of fully nonlinear elliptic operators." Ricerche di Matematica 57, no. 1 (May 15, 2008): 1–12. http://dx.doi.org/10.1007/s11587-008-0022-0.

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37

MADSEN, P. A., H. B. BINGHAM, and HUA LIU. "A new Boussinesq method for fully nonlinear waves from shallow to deep water." Journal of Fluid Mechanics 462 (July 10, 2002): 1–30. http://dx.doi.org/10.1017/s0022112002008467.

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A new method valid for highly dispersive and highly nonlinear water waves is presented. It combines a time-stepping of the exact surface boundary conditions with an approximate series expansion solution to the Laplace equation in the interior domain. The starting point is an exact solution to the Laplace equation given in terms of infinite series expansions from an arbitrary z-level. We replace the infinite series operators by finite series (Boussinesq-type) approximations involving up to fifth-derivative operators. The finite series are manipulated to incorporate Padé approximants providing the highest possible accuracy for a given number of terms. As a result, linear and nonlinear wave characteristics become very accurate up to wavenumbers as high as kh = 40, while the vertical variation of the velocity field becomes applicable for kh up to 12. These results represent a major improvement over existing Boussinesq-type formulations in the literature. A numerical model is developed in a single horizontal dimension and it is used to study phenomena such as solitary waves and their impact on vertical walls, modulational instability in deep water involving recurrence or frequency downshift, and shoaling of regular waves up to breaking in shallow water.

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38

Ivochkina, N. M., and N. V. Filimonenkova. "G˚arding Cones and Bellman Equations in the Theory of Hessian Operators and Equations." Contemporary Mathematics. Fundamental Directions 63, no. 4 (December 15, 2017): 615–26. http://dx.doi.org/10.22363/2413-3639-2017-63-4-615-626.

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In this work, we continue investigation of algebraic properties of G˚arding cones in the space of symmetric matrices. Based on this theory, we propose a new approach to study of fully nonlinear diﬀerential operators and second-order partial diﬀerential equations. We prove new-type comparison theorems for evolution Hessian operators and establish a relation between Hessian and Bellman equations.

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39

FAN, HONG-YI, and LI-YUN HU. "OPERATOR-SUM REPRESENTATION OF DENSITY OPERATORS AS SOLUTIONS TO MASTER EQUATIONS OBTAINED VIA THE ENTANGLED STATE APPROACH." Modern Physics Letters B 22, no. 25 (October 10, 2008): 2435–68. http://dx.doi.org/10.1142/s0217984908017072.

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We solve various master equations to obtain density operators' infinite operator-sum representation via a new approach, i.e., by virtue of the thermo-entangled state representation that has a fictitious mode as a counterpart mode of the system mode. The corresponding Kraus operators from the point of view of quantum channel are derived, whose normalization conditions are proved. Miscellaneous characters possessed by different quantum channels, such as decoherence, phase diffusion, damping, and amplification, can be shown explicitly in the entangled state representation of the density operators. Squeezing transformation is applied to the density operator for decoherence to generate a master equation for describing the phase sensitive process. Partial trace method for deriving new density operators is also introduced. Throughout our discussion, the technique of integration within an ordered product (IWOP) of operators is fully used.

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40

Kim, Yong-Cheol, and Ki-Ahm Lee. "Regularity Results for Fully Nonlinear Integro-Differential Operators with Nonsymmetric Positive Kernels: Subcritical Case." Potential Analysis 38, no. 2 (April 14, 2012): 433–55. http://dx.doi.org/10.1007/s11118-012-9280-2.

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41

Crasta, Graziano, and Ilaria Fragalà. "The Brunn–Minkowski inequality for the principal eigenvalue of fully nonlinear homogeneous elliptic operators." Advances in Mathematics 359 (January 2020): 106855. http://dx.doi.org/10.1016/j.aim.2019.106855.

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42

Cutrì, Alessandra, and Nicoletta Tchou. "Fully nonlinear degenerate operators associated with the Heisenberg group: barrier functions and qualitative properties." Comptes Rendus Mathematique 344, no. 9 (May 2007): 559–63. http://dx.doi.org/10.1016/j.crma.2007.03.003.

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43

Chen, Tao, Tong Kang, and Jun Li. "An A-ϕ Scheme for Type-II Superconductors." East Asian Journal on Applied Mathematics 7, no. 4 (November 2017): 658–78. http://dx.doi.org/10.4208/eajam.141016.300517a.

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AbstractA fully discrete A-ϕ finite element scheme for a nonlinear model of type-II superconductors is proposed and analyzed. The nonlinearity is due to a field dependent conductivity with the regularized power-law form. The challenge of this model is the error estimate for the nonlinear term under the time derivative. Applying the backward Euler method in time discretisation, the well-posedness of the approximation problem is given based on the theory of monotone operators. The fully discrete system is derived by standard finite element method. The error estimate is suboptimal in time and space.

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44

Adzhemyan, L. Ts, N. V. Antonov, and T. L. Kim. "Composite operators, operator expansion, and Galilean invariance in the theory of fully developed turbulence. Infrared corrections to Kolmogorov scaling." Theoretical and Mathematical Physics 100, no. 3 (September 1994): 1086–99. http://dx.doi.org/10.1007/bf01018574.

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45

Amendola, M. E., L. Rossi, and A. Vitolo. "Harnack Inequalities and ABP Estimates for Nonlinear Second-Order Elliptic Equations in Unbounded Domains." Abstract and Applied Analysis 2008 (2008): 1–19. http://dx.doi.org/10.1155/2008/178534.

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We are concerned with fully nonlinear uniformly elliptic operators with a superlinear gradient term. We look for local estimates, such as weak Harnack inequality and local maximum principle, and their extension up to the boundary. As applications, we deduce ABP-type estimates and weak maximum principles in general unbounded domains, a strong maximum principle, and a Liouville-type theorem.

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46

Indrei, Emanuel, and Andreas Minne. "Nontransversal intersection of free and fixed boundaries for fully nonlinear elliptic operators in two dimensions." Analysis & PDE 9, no. 2 (March 24, 2016): 487–502. http://dx.doi.org/10.2140/apde.2016.9.487.

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47

Rasetti, M. "A fully consistent Lie algebraic representation of quantum phase and number operators." Journal of Physics A: Mathematical and General 37, no. 38 (September 9, 2004): L479—L487. http://dx.doi.org/10.1088/0305-4470/37/38/l01.

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48

Wang, Zhan. "Modelling nonlinear electrohydrodynamic surface waves over three-dimensional conducting fluids." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2200 (April 2017): 20160817. http://dx.doi.org/10.1098/rspa.2016.0817.

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The evolution of the free surface of a three-dimensional conducting fluid in the presence of gravity, surface tension and vertical electric field due to parallel electrodes, is considered. Based on the analysis of the Dirichlet–Neumann operators, a series of fully nonlinear models is derived systematically from the Euler equations in the Hamiltonian framework without assumptions on competing length scales can therefore be applied to systems of arbitrary fluid depth and to disturbances with arbitrary wavelength. For special cases, well-known weakly nonlinear models in shallow and deep fluids can be generalized via introducing extra electric terms. It is shown that the electric field has a great impact on the physical system and can change the qualitative nature of the free surface: (i) when the separation distance between two electrodes is small compared with typical wavelength, the Boussinesq, Benney–Luke (BL) and Kadomtsev–Petviashvili (KP) equations with modified coefficients are obtained, and electric forces can turn KP-I to KP-II and vice versa; (ii) as the parallel electrodes are of large separation distance but the thickness of the liquid is much smaller than typical wavelength, we generalize the BL and KP equations by adding pseudo-differential operators resulting from the electric field; (iii) for a quasi-monochromatic plane wave in deep fluid, we derive the cubic nonlinear Schrödinger (NLS) equation, but its type (focusing or defocusing) is strongly influenced by the value of the electric parameter. For sufficient surface tension, numerical studies reveal that lump-type solutions exist in the aforementioned three regimes. Particularly, even when the associated NLS equation is defocusing for a wave train, lumps can exist in fully nonlinear models.

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49

Antonov, N. V., S. V. Borisenok, and V. I. Girina. "Renormalization group in the theory of fully developed turbulence. Composite operators of canonical dimension 8." Theoretical and Mathematical Physics 106, no. 1 (January 1996): 75–83. http://dx.doi.org/10.1007/bf02070765.

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50

Ferrari, Fausto. "Two-phase problems for a class of fully nonlinear elliptic operators: Lipschitz free boundaries are C 1,γ." American Journal of Mathematics 128, no. 3 (2006): 541–71. http://dx.doi.org/10.1353/ajm.2006.0023.

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