Nonlinearity: Problems, Solutions and Applications. Volume 1

Ludmila A. Uvarova, Ph.D., Alexey B. Nadykto, Ph.D. and Anatoly V. Latyshev, Ph.D. (Editors)
Moscow State University of Technology, Moscow, Russian Federation

Series: Theoretical and Applied Mathematics
BISAC: MAT017000

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The book has two volumes and consists of forty-four chapters, which are divided into five sections: (i) Mathematical treatment of non-linear problems, including the differential equations, numerical methods, algorithms and solutions; (ii) theoretical and computational studies dedicated to the physics and chemistry of advanced materials, nanostructured systems and fractal systems; (iii) articles dedicated to non-linear processes in complex biological processes, systems and objects; (iv) theoretical and modeling studies of kinetics, dynamics and thermochemistry of micro-, meso- and macro-scale systems; and (v) multidisciplinary research focused on forecasting, control and management problems. (Imprint: Nova)

Section I. Differential Equations, Numerical Methods, Algorithms and Solutions

Chapter 1. Computational Methods for Two-Dimensional Neural Fields
P.M. Lima, E. Buckwar (CEMAT, Instituto Superior Tecnico, Universidade de Lisboa, Portugal)

Chapter 2. Mathematical Simulation of the Heat and Mass Transfer in a Plane Channel with Infinite Parallel Walls under the Action of a Temperature Gradient
O.V. Germider and V.N. Popov (Department of Mathematics, Northern (Arctic) Federal University named after M.V. Lomonosov, Arhangelsk, Russia)

Chapter 3. The Hopfions in the Skyrme – Faddeev Spinor Model
Yu. P. Rybakov and V.I. Molotkov (Department of Theoretical Physics and Mechanics, RUDN University, Moscow, Russia)

Chapter 4. Mathematical Simulation of the Mass Transfer in a Long Rectangular Channel under the Action of a Temperature Gradient
O.V. Germider and V.N. Popov (Department of Mathematics, Northern (Arctic) Federal University named after M.V. Lomonosov, Arhangelsk, Russia)

Chapter 5. The Nonexistence of Solutions for Some Elliptic Inequalities and Systems with Variable Exponents and Singular Coefficients on the Boundary
O.A. Salieva (A lied Mathematics Department, Moscow State Technological University “STANKIN”, Moscow, Russia)

Chapter 6. The Analysis of a Motion Model of Orbital Tether Systems Based on Properties of Limit Cycles
V. Vorontsova and O. Druzhinina (Kazan Federal University, Kazan, Russia)

Chapter 7. The Nonexistence of Solutions for Some Nonlinear Inequalities with a Transformed Argument in Bounded Domains
O.A. Salieva (A lied Mathematics Department, Moscow State Technological University “STANKIN”, Moscow, Russia)

Chapter 8. TS Models and Semidefinite Lyapunov Functions in Stability Analysis of Nonlinear Delay Systems
N. Sedova (Department of Mathematics, Information and Aviation Technologies, Ulyanovsk State University, Ulyanovsk, Russia)

Chapter 9. Matrix Fourier Integral Transforms for Coupled Mathematical Models
O.E. Iaremko (Department of computer science, Penza State University, Penza, Russia)

Chapter 10. Some Approaches to the Design of Partial Solutions for Nonlinear Equations of Mathematical Physics (pp.171-200)
V.A. Kadymov, E.A. Yanovskaya (Department of A lied Mathematics, Moscow State University of Humanities and Economics, Moscow, Russia)

Section II. Kinetics, Dynamics and Thermochemistry of Micro-, Meso- and Macro-Scale Systems

Chapter 11. The Evolution of Polymer Systems during Electrospinning: From a Semi-Dilute Polymer Solution to a Non-Equilibrium State
A. Arinstein (Department of Mechanical Engineering, Technion–Israel Institute of Technology, Haifa, Israel

Chapter 12. Structural Methods of Design Identification Systems
N. Karabutov (Department of ControlProblems, Moscow Technological University (MIREA), Moscow, Russia)

Chapter 13. Non-Linear Longitudinal Current in Quantum Plasma Generated by N Transverse Electromagnetic Waves
A.V. Latyshev and V.I. Askerova (Department of Mathematical Analysis and Geometry, Moscow State Regional University, Moscow, Russia)

Chapter 14. Mathematical Modeling of Railway Track Structure under Changing Rigidity Parameters
A. Loktev, V. Sychev, E. Gridasova and R. Stepanov (Departament of Transport Construction, Moscow State University of Railway Engineering (MIIT), Moscow, Russia)

Chapter 15. An Experimental Study of the Effects of the Wind on a Metal Bridge Crossing with Two Independent Parallel Spans
A. Loktev O. Poddaeva, A. Fedosova and P. Churin (Departament of Transport Construction, Moscow State University of Railway Engineering (MIIT), Moscow, Russia)

Chapter 16. The Photophoretic Motion of Moderately Large Spherical Aerosol Particles with Arbitrary Temperature Differences
N.V. Malay and E.R. Shchukin (Belgorod National Research University, Belgorod, Russia)

Chapter 17. The Evolution of Turbulence Structure over Inhomogeneously Heated Surfaces
V.V. Nosov, V.P. Lukin, E.V. Nosov, A.V. Torgaev (V.E. Zuev Institute of Atmospheric Optics of SB RAS, Tomsk, Russia)

Chapter 18. The Computer Simulation of Nonlinear Processes in Gas-Metal Microsystems
V. Podryga and S. Polyakov (Keldysh Institute of A lied Mathematics of Russian Academy of Sciences,Moscow, Russia)

Chapter 19. Magnetic Excitations in a Chiral Graphene Model
Yu.P. Rybakov and M. Iskandar (Department of Theoretical Physics and Mechanics, RUDN University, Moscow, Russia)

Chapter 20. Thermophoresis of Non-Uniform Moderately Large Spherical Aerosol Particles
E.R. Shchukin, N.V. Malay and Z.L. Shulimanova (Joint Institute for High Temperatures of the Russian Academy of Science, Moscow, Russia)

Chapter 21. The Thermophoresis of a Cylindrical Aerosol Particle of Non-Uniform Thermal Conductivity
E.R. Shchukin, L.A. Uvarova, Z.L. Shulimanova and N.V. Malay (Joint Institute for High Temperatures of the Russian Academy of Science, Moscow, Russia)

Chapter 22. On Some Methods of Construction of Nonlinear Structure-Property Relationships for Organic Compounds
M. Skvortsova (Institute of Fine Chemical Technologies, Moscow Technological University, Moscow, Russia)

Chapter 23. Modeling the Structures of Organic Compounds: From Molecular Graphs to Molecular Hypergraphs
I. Faskhutdinova, N. Mikhailova and M. Skvortsova (Institute of Fine Chemical Technologies, Moscow Technological University, Moscow, Russia)

Chapter 24. The Mathematical and Physical Modeling of Distribution Operations in Crimp Conical Shells
E. N. Sosenushkin, V. A. Kadymov, E. A. Yanovskaya, A. A. Tatarintsev and A. E. Sosenushkin (Department of A lied Mathematics, Moscow State Technological University “STANKIN”, Moscow, Russia)

Chapter 25. Modeling of Heat Transfer in the System of Small Spherical and Cylinrical Particles under the Action of Elecromagnetic Radiation
L.A. Uvarova, I.V. Krivenko, M.A. Smirnova and A.F. Ivannikov (Department of A lied Mathematics, Moscow State Technological University “STANKIN”, Moscow, Russia)

Chapter 26. HPC Simulation of Gasdynamic Flows on Macroscopic and Molecular Levels
T. Kudryashova, V. Podryga and S. Polyakov (Keldysh Institute of A lied Mathematics of Russian Academy of Sciences, Moscow, Russia)

About the Editors

Index

Chapter 1

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Fredholm integral equations, J. Comput. Applied Math. 189 (2006) 568­579.
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field equations, Physica D 239 (2010) 561--578.
Computational Methods for Two­Dimensional Neural Fields 31
[6] R. FitzHugh, Impulses and physiological states in theoretical models of nerve mem­
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[7] S.E. Folias and P.C.Bressloff, Breathers in Two­dimensional neural media, Phys. Rev.
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[8] Han Guoqiang, Asymtotic error expansion for a nonlinear Volterra­Fredholm integral
equation, J. Comput. Applied Math. 59 (1995) 49­59.
[9] A. Hodgkin and A. Huxley, A qualitative description of nerve current and its applica­
tion to conduction and excitation in nerve, J. Physiology, 117 (1952), 500--544 .
[10] A. Hutt and N. Rougier, Activity spread and breathers induced by finite transmission
speeds in two­dimensional neuronal fields, Physical Review, E 82 (2010) 055701.
[11] A. Hutt and N. Rougier, Numerical Simulations of One­ and Two­dimensional Neural
Fields Involving Space­Dependent Delays, in S. Coombes et al., Eds., Neural Fields
Theory and Applications, Springer, 2014.
[12] Z. Jackiewicz, M. Rahman, B.D. Welfert, Numerical Solution of a Fredholm integro­
differential equation modelling neural networks, Applied Numerical Mathematics , 56
(2006) 423­432.
[13] Z. Jackiewicz, M. Rahman, B.D. Welfert, Numerical Solution of a Fredholm integro­
differential equation modelling #­neural networks, Appl. Math. Comput., 195 (2008)
523­536.
[14] J.­P. Kauthen, Continuous time collocation methods for Volterra­Fredholm integral
equations, Num. Math., 56 (1989) 409--424.
[15] P. M. Lima and E. Buckwar, Numerical solution of the neural field equation in the
two­dimensional case, SIAM Journal on Scientific Computing , 37 (6) (2015) B962­
B979
[16] P.M. Lima and E. Buckwar, Numerical investigation of the two­dimensional neural
field equation with delay, Proceedings of the 2nd International Conference on Mathe­
matics and Computers in Sciences and in Industry, MCSI 2015 7423954 (2016) 131­
137.
[17] J. Lund, A. Angelucci, P. Bressloff, Anatomical substracts for functional columns in
macaque monkey primary visual cortex, Cerebral Cortex, 13 (2003) 15­24.
[18] J. Nagumo, S.Arimoto and S. Yoshizawa, An active pulse transmission line simulating
nerve axon, Proceedings of the IRE, 50 (1962) 2061--2070.
[19] R. Pothast and P. beim Graben, Existence and properties of solutions for neural field
equations, Math. Meth. Appl. Sci., 33 (2010) 935­949.
[20] J. Rankin, D. Avitabile, J. Baladron, G. Faye, and D.J.B. Lloyd, Continuation of Lo­
calised coherent stuctures in nonlocal neural field equations, SIAM Journal on Scien­
tific Computing, 37 (1)(2014) B70­B93.
32 Pedro M. Lima and Evelyn Buckwar
[21] R. Veltz and O. Faugeras, Stability of the stationary solutions of neural field equations
with propagation delays, Journal of Mathematical Neuroscience (2011) 1:28.
[22] H.R. Wilson and J.D. Cowan, Excitatory and inhibitory interactions in localized pop­
ulations of model neurons, Bipophys. J., 12 (1972) 1­24.
[23] Weng­Jing Xie, Fu­Rong Lin, A fast numerical solution method for two dimensional
Fredholm integral equations of the second kind, Applied Numerical Mathematics , 59
(2009) 1709­1719.

Chapter 2

[1] Sharipov, F. M., & Seleznev, V. D. (2008). Motion of Rarefied Gases in Channels and Microchannels. The Ural Branch of RAS, Ekaterinburg.
[2] Kogan, M. N. (1969). Rarefied gas dynamics. New York: Plenum Press.
[3] Germider, O. V., Popov,V. N., & Yushkanov, A. A. (2015).Computation of heat flow in a long rectangular channel of constant cross section in the framework of the kinetic Approach. Bulletin of Moscow State Regional University. Series: Physics-Mathematics, 2, 96-106.
[4] Latyshev, A. V., & Yushkanov, A. A. (2004). Analytic solution of boundary problems for kinetic Equations. Moscow: MGOU.
[5] Courant, P. (1962). Partial differential equations. New York–London: Interscience Publishers.
[6] Graur, I., & Sharipov, F. (2008). Gas flow through an elliptical tube over the whole range of the gas rarefaction. European Journal of Mechanics В Fluids, 27, 335-345.
[7] Germider, O. V., Popov, V. N., & Yushkanov, A. A. (2016). Computation of the Heat Flux in a Cylindrical Duct within the Framework of the Kinetic Approach. Journal of Engineering Physics and Thermophysics, 89(5), 1338-1343.
[8] Kamphorst, C. H., Rodrigues, P., & Barichello, L. B. (2014). A closed-form solution of a kinetic integral equation for rarefied gas flow in a cylindrical duct. Applied Mathematics, 5, 1516-1527.
[9] Siewert, C. E., & Valougeorgis, D. (2002). An analytical discrete-ordinates solution of the S-model kinetic equations for flow in a cylindrical tube. Quantitative Spectroscopy and Radiative Transfer, 72, 531-550.
[10] Breyiannis G., Varoutis, S., & Valougeorgis, D. (2008) Rarefied gas flow in concentric annular tube: Estimation of the Poiseuille number and the exact hydraulic diameter.European Journal of Mechanics B/Fluids, 27, 609–622.
[11] Germider, O. V., & Popov,V. N. (2016). Mathematical modelling of the mass transfer process between two coaxial cylinders in the problem of thermal creep. IOP Conference Series: Materials Science and Engineering, 58, 1-7.
[12] Shakhov, E. M. (2003). Rarefied gas flow between coaxial cylinders under the action of a pressure gradient. Zhurnal Vychislitelnoi Matematiki i Matematicheskoi Fiziki, 43, 1007–1016.
[13] Naris, S., Valougeorgis, D. (2008). Rarefied gas flow in a triangular duct based on a boundary fitted lattice. Eur. J. Mech. B/ Fluids, 27, 810-822.
[14] Rykov, V. A., Titarev,V. A.,Shakhov, E. M. (2011). Rarefied Poiseuille Flow in Elliptical and Rectangular Tubes. Fluid Dynamics, 46, 456–466.
[15] Titarev, V. A., & Shakhov, E. M. (2010). Nonisothermal gas flow in a long channel analyzed on the basis of the kinetic S-model. Computational Mathematics and Mathematical Physics, 50(12), 2131-2144.


Chapter 3

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[4] Kundu, A., & Rybakov, Y. P. (1982). Closed-vortex-type solitons with Hopf index. Journal of Physics A: Mathematical and General, 15(1), 269.
[5] Nicole, D. A. (1978). Solitons with non-vanishing Hopf index. Journal of Physics G: Nuclear Physics, 4(9), 1363.
[6] de Vega, H. J. (1978). Closed vortices and the Hopf index in classical field theory. Physical Review D, 18(8), 2945.
[7] Zahed, I., & Brown, G. E. (1986). The Skyrme model. Physics Reports, 142(1-2), 1-102.
[8] Makhan’kov, V. G., Rybakov, Y. P., & Sanyuk, V. I. (2013). The Skyrme model: fundamentals, methods, applications. Berlin, New York: Springer-Verlag (third edition) (Springer series in nuclear and particle physics).
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[11] Rybakov, Y. P. (2015). Axially-symmetric topological configurations in the Skyrme and Faddeev chiral models. Eurasian Mathematical Journal, 6(2), 82-89.
[12] Rybakov, Y. P. (2011). Soliton configurations in generalized Mie electrodynamics. Physics of Atomic Nuclei, 74(7), 1073-1076.
[13] Rybakov, Yu. P. (2013). 8-Spinors and structure of solitons in generalized Mie electrodynamics. Physics of Atomic Nuclei, 76, 219-223.
[14] Rybakov, Y. P. (2015). Structure of topological solitons in nonlinear spinor model. Physics of Particles and Nuclei Letters, 12(3), 420-422.
[15] Rybakov, Y. P. (2013). Topological solitons in 8-spinor Mie electrodynamics. Physics of Atomic Nuclei, 76(10), 1284-1288.
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[19] Rybakov, Y. P. (1972). The Bohm - Vigier subquantum fluctuations and nonlinear field theory. International Journal of Theoretical Physics, 5(2), 131-138.
[20] Rybakov, Yu. P. (1974). On the causal interpretation of quantum mechanics. Foundations of Physics, 4, 149-161.
[21] Rybakov, Yu. P. (1977). La théorie statistique des champs et la mécanique quantique. Ann. Fond. L. de Broglie, 2(3), 181-203.
[22] Rybakov, Yu. P., & Saha, B. (1995). Soliton model of atom. Foundations of Physics, 25, 1723- 1731.
[23] Rybakov, Yu. P., & Saha, B. (1996). Interaction of a charged 3D soliton with a Coulomb center. Physics Letters, 222 A, 5-13.
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[25] Rybakov, Yu. P., & Kamalov, T. F. (2007). Entangled solitons and stochastic
Q-bits. Physics of Particles and Nuclei Letters, 4, 208–213.
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Chapter 4

[1] Sharipov, F. M., & Seleznev, V. D. (2008). Motion of Rarefied Gases in Channels and Microchannels. The Ural Branch of RAS, Ekaterinburg.
[2] Graur, I.; Ho, M.T. (2014). Rarefied gas flow through a long rectangular channel of variable cross section.Vacuum, 101, 328-332.
[3] Graur, I., Sharipov, F. (2008). Gas Flow through an Elliptical Tube over the whole Range of the Gas Rarefaction. European Journal of Mechanics В Fluids, 27, 335-345.
[4] Kamphorst, C. H., Rodrigues P., Barichello, L. B. (2014). A closed-form solution of a kinetic integral equation for rarefied gas flow in a cylindrical duct. Applied Mathematics, 5, 1516-1527.
[5] Naris, S., & Valougeorgis, D. (2008). Rarefied gas flow in a triangular duct based on a boundary fitted lattice. European Journal of Mechanics-B/Fluids, 27(6), 810-822.
[6] Sharipov, F. (1999). Rarefied gas flow through a long rectangular channel. Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films, 17(5), 3062-3066.
[7] Siewert, C. E., Valougeorgis, D. (2002). An analytical discrete-ordinates solution of the S-model kinetic equations for flow in a cylindrical tube. Quantitative Spectroscopy and Radiative Transfer,72, 531-550.
[8] Titarev, V. A., & Shakhov, E. M. (2010). Kinetic analysis of an isothermal flow in a long microchannel with rectangular cross section. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 50(7), 1285-1302.
[9] Rykov, V. A., Titarev, V. A., Shakhov, E. M. (2011). Rarefied Poiseuille flow in elliptical and rectangular tubes. Fluid Dynamics, 46, 456–466.
[10] Germider, O. V., Popov, V. N., & Yushkanov, A. A. (2016). Computation of the Heat Flux in a Cylindrical Duct Within the Framework of the Kinetic Approach. Journal of Engineering Physics and Thermophysics, 89(5), 1338-1343.
[11] Latyshev, A. V., Yushkanov, A. A. (2004). Analytic solution of boundary problems for kinetic Equations. Moscow: MGOU.
[12] Germider, O. V., Popov, V. N.(2016). Mathematical modeling of the heat transfer process in a rectangular channel depending on Knudsen number. Zh. SVMO 18, 85-93.
[13] Kogan, M. N. (1969). Rarefied gas dynamics. New York: Plenum Press.
[14] Courant, P. (1962). Partial Differential Equations. New York–London: Interscience Publishers.
[15] Germider, O. V., Popov,V. N.,Yushkanov, A. A. (2016). Computation of the gas mass and heat fluxes in a rectangular channel in the free molecular regime. Technical Physics 61, 835–840.

Chapter 5

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[9] Galakhov, E., & Salieva, O. (2015). Blow-up of solutions of some nonlinear inequalities with singularities on unbounded sets. Mathematical Notes, 98, 222-229.
[10] Drábek, P., & Pohozaev, S. (1997). Positive solutions for the p-Laplacian: application of the fibering method. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 127(04), 703-726.
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[16] Farina, A., & Serrin, J. (2011). Entire solutions of completely coercive quasilinear elliptic equations II. Journal of Differential Equations, 250, 4409-4436.
[17] Filippucci, R., Pucci, P., & Rigoli, M. (2010). Nonlinear weighted p-Laplacian elliptic inequalities with gradient terms. Communications in Contemporary Mathematics, 12, 501-535.
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Chapter 6

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Chapter 7

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Chapter 8

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Chapter 9

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Chapter 10

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Chapter 11

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Chapter 13

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Chapter 14

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Chapter 15

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[13] Loktev, A. A. (2010). Dynamic contact of a spherical indenter and a prestressed orthotropic Uflyand-Mindlin Plate. Acta Mechanica, 222(1-2), 17-25.
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Chapter 16

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Chapter 17

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Chapter 20

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Chapter 21

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Chapter 22

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Chapter 23

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[8] Stankevitch, I. V. (1988). Graphs in structural chemistry. In: N. S. Zefirov and S. I. Kutchanov (Eds.). Application of Graph Theory in Chemistry. (pp. 7-69). Novosibirsk: Nauka. (in Russian).
[9] Todeschini, R., and Consonni, V. (2000). Handbook of Molecular Descriptors, Weinheim: Wiley-VCH.
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Chapter 24

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Chapter 25

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Chapter 26

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[15] Polyakov, S. V., Karamzin, Y. N., Kudryashova, T. A., Kosolapov, O. A., & Sukov, S. A. (2012). Hybrid supercomputer platform and development of applications for solving problems of continuum mechanics by grid methods. News of Russian Southern Federal University. Technical Sciences, 6(131), 105-115.

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