# American Institute of Mathematical Sciences

July  2013, 12(4): 1587-1633. doi: 10.3934/cpaa.2013.12.1587

## On Dirichlet, Poncelet and Abel problems

 1 Institute of Applied Mathematics, Donetsk, 83114, Ukraine 2 Donetsk Institute for Physics and Technology, Donetsk, 83114, Ukraine

Received  June 2011 Revised  June 2011 Published  November 2012

We propose interconnections between some problems of PDE, geometry, algebra, calculus and physics. Uniqueness of a solution of the Dirichlet problem and of some other boundary value problems for the string equation inside an arbitrary biquadratic algebraic curve is considered. It is shown that a solution is non-unique if and only if a corresponding Poncelet problem for two conics has a periodic trajectory. A set of problems is proven to be equivalent to the above problem. Among them are the solvability problem of the algebraic Pell-Abel equation and the indeterminacy problem of a new moment problem that generalizes the well-known trigonometrical moment problem. Solvability criteria of the above-mentioned problems can be represented in form $\theta\in Q$ where number $\theta=m/n$ is built by means of data for a problem to solve. We also demonstrate close relations of the above-mentioned problems to such problems of modern mathematical physics as elliptic solutions of the Toda chain, static solutions of the classical Heisenberg $XY$-chain and biorthogonal rational functions on elliptic grids in the theory of the Padé interpolation.
Citation: Vladimir P. Burskii, Alexei S. Zhedanov. On Dirichlet, Poncelet and Abel problems. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1587-1633. doi: 10.3934/cpaa.2013.12.1587
##### References:
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I, Izvestija AN SSSR, serija matematicheskaja, 25 (1961), 21-86.  Google Scholar [7] R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc., 54 (1985), 1-55.  Google Scholar [8] G. A. Baker and P. Graves-Morris, Padé approximants. Parts I and II, in "Encyclopedia of Mathematics and its Applications," 13, 14, Addison-Wesley Publishing Co., Reading, Mass., 1981.  Google Scholar [9] H. Bateman and A. Erdélyi, "Higher Transcendental Functions," 3, McGraw-Hill, New York, 1955, Bateman manuscript project. Google Scholar [10] R. Baxter, "Exactly Solvable Models in Statistical Mechanics," London, Academic Press, 1982.  Google Scholar [11] M. V. Beloglyadov, On the Dirichlet problem for the vibrating string equation in domain with a bi-quadratic boundary, Trudy IAMM NASU, 14 (2007), 14-29 (In Russian).  Google Scholar [12] E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its and V. B. Matveev, "Algebro-geometrical Approach to Non-linear Integrable Equations," Springer Series in Nonlinear Dynamics, XII, Berlin: Springer-Verlag, 1994. Google Scholar [13] E. D. Belokolos and V. Z. Enolskii, Reduction of Abelian functions and algebraically integrable systems, Journal of Mathematical Sciences, Part I: 106 (2001), 3395-3486; Part II: 108 (2002), 295-374. Google Scholar [14] Yu. M. Berezanskii, "Expansion by Eigenfunctions of Selfadjoint Operators," Naukova Dumka, Kiev, 1965 (In Russian). Google Scholar [15] M. Berger, "Géométrie," CEDIC, Paris, 1978. Google Scholar [16] M. Berger, "Geometry Revealed, A Jacob's Ladder to Modern Higher Geometry," Springer, 2010.  Google Scholar [17] D. Bourgin and R. Duffin, The Dirlchlet problem for the vibrating string equations, Bull. Am. Math. Soc., 45 (1939), 851-858.  Google Scholar [18] A. B. Bogatyrev, Chebyshev representation for rational function, Sbornik Mathematics, 201 (2010), 1579-1598.  Google Scholar [19] V. P. Burskii, On solution uniqueness of some boundary value problems for differential equations in domains with algebraic boundary, Ukr. Math. Journal, 45 (1993), 993-1003.  Google Scholar [20] V. P. Burskii, On boundary value problems for differential equations with constant coefficients in a plane domain and a moment problem, Ukr. Math. Journal, 48 (1993), 1659-1668.  Google Scholar [21] V. P. Burskii, "Investigation Methods of Boundary Value Problems for General Differential Equations," Kiev, Naukova dumka, 2002 (In Russian). Google Scholar [22] V. P. Burskii and A. S. Zhedanov, On Dirichlet problem for string equation, Poncelet problem, Pell-Abel equation, and some other related problems, Ukr. Math. Journal, 58 (2006), 487-504.  Google Scholar [23] V. P. Burskii and A. S. Zhedanov, Dirichlet and Neumann problems for string equation, Poncelet problem and Pell-Abel equation, Symmetry, Integrability and Geometry: Methods and Applications, 2006, V. 2, rec.No: 041.  Google Scholar [24] V. P. Burskii and A. S. Zhedanov, Boundary value problems for string equation, Poncelet problem, and Pell-Abel equation: links and relations, Contemporary Mathematics. Fundamental Directions, 16 (2006), pp. 59. Google Scholar [25] A. A. Chernikov, R. Z. Sagdeev and G. M. Zaslavsky, Stochastic webs. Progress in chaotic dynamics, Phys. D, 33 (1988), 6576.  Google Scholar [26] O. Egecioglu and C. K. Koc, A fast algorithm for rational interpolation via orthogonal polynomials, Math. Comp., 53 (1989), 249-264.  Google Scholar [27] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, "Higher Transcendental Functions. I," McGraw-Hill, New York, 1953 Bateman manuscript project.  Google Scholar [28] M. V. Fokin, Solvability of the Dirichlet problem for the string equation, Doklady AN SSSR, 272 (1983), 801-805 (in Russian).  Google Scholar [29] J. P. Francoise and O. Ragnisco, An iterative process on quartics and integrable symplectic maps, in "Symmetries and Integrability of Difference Equations," P. A. Clarkson and F. W. Nijhoff eds., Cambridge University Press, 1998.  Google Scholar [30] Ya. I. Granovskii and A. S. Zhedanov, Integrability of the classical $XY$-chain, Pis'ma to Zh. Exp. Theor. Phys., 44 (1986), 237-239 (Russian).  Google Scholar [31] P. Griffiths and J. Harris, Poncelet theorem in space, Comment. Math. Helvetici, 52 (1977), 145-160.  Google Scholar [32] P. Griffiths and J. Harris, On a Cayley's explicit solution to Poncelet's porism, Enseign. Math., 24 (1978), 31-40.  Google Scholar [33] P. Griffiths and J. Harris, "Principles of Algebraic Geometry," v. I, II, John Wiley and Sons, Inc., 1978.  Google Scholar [34] J. Hadamard, Equations aux derivees partielles, L 'Enseignment Mathematique, 36 (1936), 25-42. Google Scholar [35] G. H. Halphen, "Traité des Fonctions Elliptiques et de Leures Applications," II, Gauthier朧illar, Paris, 1886. Google Scholar [36] A. Huber, Erste Randwertaufgabe fur geschlossene Bereiche bei der Gleichung $U_{xy}=f(x,y)$, Monatshefte für Mathematik und Physik, 39 (1932), 79-100.  Google Scholar [37] E. L. Ince, Ordinary differential equations, ., ().   Google Scholar [38] F. John, The Dirichlet problem for a hyperbolic equation, Am. J. Math., 63 (1941), 141-154.  Google Scholar [39] A. Iatrou and J. A. G. Roberts, Integrable mappings of the plane preseving biquadratic invariants curves II, Nonlinearity, 15 (2002), 459-489.  Google Scholar [40] A. Iatrou, Real Jacobian elliptic function parameterization for a genuinely asymmetric biquadratic curve, arXiv: nlin. SI/0306051 v1 25, Jun 2003. Google Scholar [41] S. M. Kerawala, Poncelet Porism in Two Circles, Bull. Calcutta Math. Soc., 39 (1947), 85-105.  Google Scholar [42] J. L. King, Three problems in search of a measure, Amer. Math. Monthly, 101 (1994), 609-628.  Google Scholar [43] M. M. Lavrent'ev, Mathematical problems of tomography and hyperbolic mappings, Sib. Math. J., 42 (2001), 916-925.  Google Scholar [44] V. F. Lazutkin, "KAM Theory and Semiclassical Approximation to Eigenfunctions," Springer Verlag, Berlin, Hei-delberg, New York (1993), Ergebnisse der Mathematik und ihrer Grenzgebiete: 3. Folge, Band 24.  Google Scholar [45] A. Magnus, Rational interpolation to solutions of Riccati difference equations on elliptic lattices,, Preprint http://www.math.ucl.ac.be/membres/magnus/., ().   Google Scholar [46] V. A. Malyshev, Abel equation, Algebra and Analysis, 13 (2001), 1-55 (In Russian).  Google Scholar [47] J. Meinguet, On the solubility of the Cauchy interpolation problem, Approximation Theory (Proc. Sympos., Lancaster, 1969), 137-163. Academic Press, London.  Google Scholar [48] L. J. Mordell, "Diophantine Equations," Academic Press, 1969.  Google Scholar [49] Z. Nitecki, "Differentiable Dinamics," MIT Press, Cambridge Mass - London, 1971.  Google Scholar [50] S. G. Ovsepjan, On ergodisity of continuous automorphizms and solution uniqueness of the Dirichlet problem for the string equation. II, Izv. AN Arm. SSR., 2 (1967), 195-209.  Google Scholar [51] B. Yo. Ptashnik, Incorrect boundary value problems for differential equations with partual derivatives, Kiev, Naukova dumka, 1984 (In Russian).  Google Scholar [52] J. F. Ritt, Periodic functions with a multiplication theorem, Trans. Amer. Math. Soc., 23 (1922), 16-25.  Google Scholar [53] I. J. Schoenberg, On Jacobi-Bertrand's proof of a theorem of Poncelet, in "Studies in Pure Mathematics, To the Memory of Paul Turan," 623-627, Birkhuser, Basel, 1983.  Google Scholar [54] L. M. Sodin and P. M. Yuditskii, Functions least deviating from zero on closed sets of real axis,, Algebra and Analysis, 4 (): 1.   Google Scholar [55] V. Spiridonov and A. Zhedanov, Spectral transformation chains and some new biorthogonal rational functions, Commun. Math. Phys., 210 (2000), 49-83.  Google Scholar [56] V. P. Spiridonov and A. S. Zhedanov, To the theory of biorthogonal rational functions, RIMS Kokyuroku, 1302 (2003), 172-192.  Google Scholar [57] V. Spiridonov and A. Zhedanov, Elliptic grids, rational functions, and Padé interpolation, Ramanujan J., 13 (2007), 285-310.  Google Scholar [58] T. Stieltjes, Sur l'équation d'Euler, Bul.Sci.Math., Paris, sér. 2, 12 (1888), 222-227. Google Scholar [59] A. A. Telitsyna, The Dirichlet problem for wave equation in plane domain with biquadratic boundary, Trudy IAMM NASU, 13 (2007), 198-210 (In Russian).  Google Scholar [60] M. Toda, "Theory of Nonlinear Lattices," Springer Series in Solid-State Sciences, vol. 20, Springer-Verlag, Berlin, 1989.  Google Scholar [61] A. P. Veselov, Integrable systems with discrete time and difference operators, Functional Analysis and its Applications, 22 (1988), 1-13 (Russian).  Google Scholar [62] A. P. Veselov, Integrable maps, Russian Math. Surveys, 46 (1991), 1-51.  Google Scholar [63] L. Vinet and A. Zhedanov, Generalized Bochner theorem: characterization of the Askey-Wilson polynomials, J. Comput. Appl. Math., 211 (2008), 45-56.  Google Scholar [64] T. I. Zelenjak, Selected topics of quality theory of equations with partial derivatives, Novosibirsk: NGU, 1970 (In Russian). Google Scholar [65] A. Zhedanov, Biorthogonal rational functions and the generalized eigenvalue problem, J. Approx. Theory, 101 (1999), 303-329.  Google Scholar [66] A. Zhedanov, Padé interpolation table and biorthogonal rational functions, Proceedings of the Workshop on Elliptic Integrable Systems November 8-11, 2004, Kyoto, Rokko Lectures in Mathematics, No. 18, 323-363. http://www.math.kobe-u.ac.jp/publications/rlm18/20.pdf. Google Scholar [67] http:, //en.wikipedia.org/wiki/Archimedes, ., ().   Google Scholar

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##### References:
 [1] N. I. Akhiezer, "Elements of the Theory of Elliptic Functions," 2nd edition, Nauka, Moscow, 1970. Translations Math. Monographs, 79, AMS, Providence, 1990.  Google Scholar [2] N. I. Akhiezer, "Lectures on Approximation Theory," Nauka, M., 1965 (In Russian). Google Scholar [3] R. A. Alexandrjan, On the Dirichlet problem for the string equation and on completeness of a system of function in a disk, Doklady AN USSR., 73 (1950) (In Russian). Google Scholar [4] R. A. Alexandrjan, Spectral properties of operators generated by systems differential equations of Sobolev type, Trudy Mosc. Math. Obshchestva, 9 (1960), 455-505 (In Russian).  Google Scholar [5] G. S. Akopyan and R. A. Aleksandryan, On the completeness of a system of eigen- and vector-polynomials of a linear differential operator pencil in ellipsoidal domains, Dokl. Akad. Nauk Arm. SSR, 86 (1988), 147-152 (In Russian).  Google Scholar [6] V. I. Arnold, Small demominators. I, Izvestija AN SSSR, serija matematicheskaja, 25 (1961), 21-86.  Google Scholar [7] R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc., 54 (1985), 1-55.  Google Scholar [8] G. A. Baker and P. Graves-Morris, Padé approximants. Parts I and II, in "Encyclopedia of Mathematics and its Applications," 13, 14, Addison-Wesley Publishing Co., Reading, Mass., 1981.  Google Scholar [9] H. Bateman and A. Erdélyi, "Higher Transcendental Functions," 3, McGraw-Hill, New York, 1955, Bateman manuscript project. Google Scholar [10] R. Baxter, "Exactly Solvable Models in Statistical Mechanics," London, Academic Press, 1982.  Google Scholar [11] M. V. Beloglyadov, On the Dirichlet problem for the vibrating string equation in domain with a bi-quadratic boundary, Trudy IAMM NASU, 14 (2007), 14-29 (In Russian).  Google Scholar [12] E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its and V. B. Matveev, "Algebro-geometrical Approach to Non-linear Integrable Equations," Springer Series in Nonlinear Dynamics, XII, Berlin: Springer-Verlag, 1994. Google Scholar [13] E. D. Belokolos and V. Z. Enolskii, Reduction of Abelian functions and algebraically integrable systems, Journal of Mathematical Sciences, Part I: 106 (2001), 3395-3486; Part II: 108 (2002), 295-374. Google Scholar [14] Yu. M. Berezanskii, "Expansion by Eigenfunctions of Selfadjoint Operators," Naukova Dumka, Kiev, 1965 (In Russian). Google Scholar [15] M. Berger, "Géométrie," CEDIC, Paris, 1978. Google Scholar [16] M. Berger, "Geometry Revealed, A Jacob's Ladder to Modern Higher Geometry," Springer, 2010.  Google Scholar [17] D. Bourgin and R. Duffin, The Dirlchlet problem for the vibrating string equations, Bull. Am. Math. Soc., 45 (1939), 851-858.  Google Scholar [18] A. B. Bogatyrev, Chebyshev representation for rational function, Sbornik Mathematics, 201 (2010), 1579-1598.  Google Scholar [19] V. P. Burskii, On solution uniqueness of some boundary value problems for differential equations in domains with algebraic boundary, Ukr. Math. Journal, 45 (1993), 993-1003.  Google Scholar [20] V. P. Burskii, On boundary value problems for differential equations with constant coefficients in a plane domain and a moment problem, Ukr. Math. Journal, 48 (1993), 1659-1668.  Google Scholar [21] V. P. Burskii, "Investigation Methods of Boundary Value Problems for General Differential Equations," Kiev, Naukova dumka, 2002 (In Russian). Google Scholar [22] V. P. Burskii and A. S. Zhedanov, On Dirichlet problem for string equation, Poncelet problem, Pell-Abel equation, and some other related problems, Ukr. Math. Journal, 58 (2006), 487-504.  Google Scholar [23] V. P. Burskii and A. S. Zhedanov, Dirichlet and Neumann problems for string equation, Poncelet problem and Pell-Abel equation, Symmetry, Integrability and Geometry: Methods and Applications, 2006, V. 2, rec.No: 041.  Google Scholar [24] V. P. Burskii and A. S. Zhedanov, Boundary value problems for string equation, Poncelet problem, and Pell-Abel equation: links and relations, Contemporary Mathematics. Fundamental Directions, 16 (2006), pp. 59. Google Scholar [25] A. A. Chernikov, R. Z. Sagdeev and G. M. Zaslavsky, Stochastic webs. Progress in chaotic dynamics, Phys. D, 33 (1988), 6576.  Google Scholar [26] O. Egecioglu and C. K. Koc, A fast algorithm for rational interpolation via orthogonal polynomials, Math. Comp., 53 (1989), 249-264.  Google Scholar [27] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, "Higher Transcendental Functions. I," McGraw-Hill, New York, 1953 Bateman manuscript project.  Google Scholar [28] M. V. Fokin, Solvability of the Dirichlet problem for the string equation, Doklady AN SSSR, 272 (1983), 801-805 (in Russian).  Google Scholar [29] J. P. Francoise and O. Ragnisco, An iterative process on quartics and integrable symplectic maps, in "Symmetries and Integrability of Difference Equations," P. A. Clarkson and F. W. Nijhoff eds., Cambridge University Press, 1998.  Google Scholar [30] Ya. I. Granovskii and A. S. Zhedanov, Integrability of the classical $XY$-chain, Pis'ma to Zh. Exp. Theor. Phys., 44 (1986), 237-239 (Russian).  Google Scholar [31] P. Griffiths and J. Harris, Poncelet theorem in space, Comment. Math. Helvetici, 52 (1977), 145-160.  Google Scholar [32] P. Griffiths and J. Harris, On a Cayley's explicit solution to Poncelet's porism, Enseign. Math., 24 (1978), 31-40.  Google Scholar [33] P. Griffiths and J. Harris, "Principles of Algebraic Geometry," v. I, II, John Wiley and Sons, Inc., 1978.  Google Scholar [34] J. Hadamard, Equations aux derivees partielles, L 'Enseignment Mathematique, 36 (1936), 25-42. Google Scholar [35] G. H. Halphen, "Traité des Fonctions Elliptiques et de Leures Applications," II, Gauthier朧illar, Paris, 1886. Google Scholar [36] A. Huber, Erste Randwertaufgabe fur geschlossene Bereiche bei der Gleichung $U_{xy}=f(x,y)$, Monatshefte für Mathematik und Physik, 39 (1932), 79-100.  Google Scholar [37] E. L. Ince, Ordinary differential equations, ., ().   Google Scholar [38] F. John, The Dirichlet problem for a hyperbolic equation, Am. J. Math., 63 (1941), 141-154.  Google Scholar [39] A. Iatrou and J. A. G. Roberts, Integrable mappings of the plane preseving biquadratic invariants curves II, Nonlinearity, 15 (2002), 459-489.  Google Scholar [40] A. Iatrou, Real Jacobian elliptic function parameterization for a genuinely asymmetric biquadratic curve, arXiv: nlin. SI/0306051 v1 25, Jun 2003. Google Scholar [41] S. M. Kerawala, Poncelet Porism in Two Circles, Bull. Calcutta Math. Soc., 39 (1947), 85-105.  Google Scholar [42] J. L. King, Three problems in search of a measure, Amer. Math. Monthly, 101 (1994), 609-628.  Google Scholar [43] M. M. Lavrent'ev, Mathematical problems of tomography and hyperbolic mappings, Sib. Math. J., 42 (2001), 916-925.  Google Scholar [44] V. F. Lazutkin, "KAM Theory and Semiclassical Approximation to Eigenfunctions," Springer Verlag, Berlin, Hei-delberg, New York (1993), Ergebnisse der Mathematik und ihrer Grenzgebiete: 3. Folge, Band 24.  Google Scholar [45] A. Magnus, Rational interpolation to solutions of Riccati difference equations on elliptic lattices,, Preprint http://www.math.ucl.ac.be/membres/magnus/., ().   Google Scholar [46] V. A. Malyshev, Abel equation, Algebra and Analysis, 13 (2001), 1-55 (In Russian).  Google Scholar [47] J. Meinguet, On the solubility of the Cauchy interpolation problem, Approximation Theory (Proc. Sympos., Lancaster, 1969), 137-163. Academic Press, London.  Google Scholar [48] L. J. Mordell, "Diophantine Equations," Academic Press, 1969.  Google Scholar [49] Z. Nitecki, "Differentiable Dinamics," MIT Press, Cambridge Mass - London, 1971.  Google Scholar [50] S. G. Ovsepjan, On ergodisity of continuous automorphizms and solution uniqueness of the Dirichlet problem for the string equation. II, Izv. AN Arm. SSR., 2 (1967), 195-209.  Google Scholar [51] B. Yo. Ptashnik, Incorrect boundary value problems for differential equations with partual derivatives, Kiev, Naukova dumka, 1984 (In Russian).  Google Scholar [52] J. F. Ritt, Periodic functions with a multiplication theorem, Trans. Amer. Math. Soc., 23 (1922), 16-25.  Google Scholar [53] I. J. Schoenberg, On Jacobi-Bertrand's proof of a theorem of Poncelet, in "Studies in Pure Mathematics, To the Memory of Paul Turan," 623-627, Birkhuser, Basel, 1983.  Google Scholar [54] L. M. Sodin and P. M. Yuditskii, Functions least deviating from zero on closed sets of real axis,, Algebra and Analysis, 4 (): 1.   Google Scholar [55] V. Spiridonov and A. Zhedanov, Spectral transformation chains and some new biorthogonal rational functions, Commun. Math. Phys., 210 (2000), 49-83.  Google Scholar [56] V. P. Spiridonov and A. S. Zhedanov, To the theory of biorthogonal rational functions, RIMS Kokyuroku, 1302 (2003), 172-192.  Google Scholar [57] V. Spiridonov and A. Zhedanov, Elliptic grids, rational functions, and Padé interpolation, Ramanujan J., 13 (2007), 285-310.  Google Scholar [58] T. Stieltjes, Sur l'équation d'Euler, Bul.Sci.Math., Paris, sér. 2, 12 (1888), 222-227. Google Scholar [59] A. A. Telitsyna, The Dirichlet problem for wave equation in plane domain with biquadratic boundary, Trudy IAMM NASU, 13 (2007), 198-210 (In Russian).  Google Scholar [60] M. Toda, "Theory of Nonlinear Lattices," Springer Series in Solid-State Sciences, vol. 20, Springer-Verlag, Berlin, 1989.  Google Scholar [61] A. P. Veselov, Integrable systems with discrete time and difference operators, Functional Analysis and its Applications, 22 (1988), 1-13 (Russian).  Google Scholar [62] A. P. Veselov, Integrable maps, Russian Math. Surveys, 46 (1991), 1-51.  Google Scholar [63] L. Vinet and A. Zhedanov, Generalized Bochner theorem: characterization of the Askey-Wilson polynomials, J. Comput. Appl. Math., 211 (2008), 45-56.  Google Scholar [64] T. I. Zelenjak, Selected topics of quality theory of equations with partial derivatives, Novosibirsk: NGU, 1970 (In Russian). Google Scholar [65] A. Zhedanov, Biorthogonal rational functions and the generalized eigenvalue problem, J. Approx. Theory, 101 (1999), 303-329.  Google Scholar [66] A. Zhedanov, Padé interpolation table and biorthogonal rational functions, Proceedings of the Workshop on Elliptic Integrable Systems November 8-11, 2004, Kyoto, Rokko Lectures in Mathematics, No. 18, 323-363. http://www.math.kobe-u.ac.jp/publications/rlm18/20.pdf. Google Scholar [67] http:, //en.wikipedia.org/wiki/Archimedes, ., ().   Google Scholar
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