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Higher pentagram maps, weighted directed networks, and cluster dynamics
1. | Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States |
2. | Department of Mathematics, Michigan State University, East Lansing, MI 48823, United States |
3. | Department of Mathematics, Penn State University, University Park, PA 16802 |
4. | Department of Mathematics and Department of Computer Science, University of Haifa, Haifa, Mount Carmel 31905, Israel |
References:
[1] |
A. Bobenko and T. Hoffmann, Hexagonal circle patterns and integrable systems: Patterns with constant angles,, Duke Math. J., 116 (2003), 525.
doi: 10.1215/S0012-7094-03-11635-X. |
[2] |
A. Bobenko and Yu. Suris, "Discrete Differential Geometry. Integrable Structure,", Amer. Math. Soc., (2008).
|
[3] |
H. Busemann and P. Kelly, "Projective geometry and projective metrics,", Academic Press, (1953).
|
[4] |
B. Dubrovin, I. Krichever and S. Novikov, Integrable systems. I,, in, 4 (2001), 177.
|
[5] |
L. Faddeev and L. Takhtajan, "Hamiltonian Methods in the Theory of Solitons,", Springer-Verlag, (1987).
|
[6] |
S. Fomin and A. Zelevinsky, Cluster algebras. IV. Coefficients,, Compos. Math., 143 (2007), 112.
doi: 10.1112/S0010437X06002521. |
[7] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson geometry of directed networks in a disk,, Selecta Math., 15 (2009), 61.
doi: 10.1007/s00029-009-0523-z. |
[8] |
M. Gekhtman, M. Shapiro and A. Vainshtein, "Cluster Algebras and Poisson Geometry,", Amer. Math. Soc., (2010).
|
[9] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson geometry of directed networks in an annulus,, J. Europ. Math. Soc., ().
|
[10] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Generalized Bäcklund-Darboux transformations for Coxeter-Toda flows from a cluster algebra perspective,, Acta Math., 206 (2011), 245.
doi: 10.1007/s11511-011-0063-1. |
[11] |
M. Glick, The pentagram map and $Y$-patterns,, Adv. Math., 227 (2011), 1019.
doi: 10.1016/j.aim.2011.02.018. |
[12] |
, M. Glick,, private communication., (). Google Scholar |
[13] |
A. Goncharov and R. Kenyon, Dimers and cluster integrable systems,, preprint, (). Google Scholar |
[14] |
B. Khesin and F. Soloviev, Integrability of a space pentagram map,, in preparation., (). Google Scholar |
[15] |
B. Konopelchenko and W. Schief, Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy,, J. Phys. A, 35 (2002), 6125.
doi: 10.1088/0305-4470/35/29/313. |
[16] |
G. Mari Beffa, On generalizations of the pentagram map: Discretizations of AGD flows,, preprint, (). Google Scholar |
[17] |
S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons,, Ann. Inst. Fourier, (). Google Scholar |
[18] |
F. Nijhoff and H. Capel, The discrete Korteweg-de Vries equation,, Acta Appl. Math., 39 (1995), 133.
doi: 10.1007/BF00994631. |
[19] |
M. Olshanetsky, A. Perelomov, A. Reyman and M. Semenov-Tian-Shansky, Integrable systems. II,, in, 16 (1994), 83.
|
[20] |
V. Ovsienko, R. Schwartz and S. Tabachnikov, Quasiperiodic motion for the Pentagram map,, Electron. Res. Announc. Math. Sci., 16 (2009), 1.
|
[21] |
V. Ovsienko, R. Schwartz and S. Tabachnikov, The Pentagram map: A discrete integrable system,, Commun. Math. Phys., 299 (2010), 409.
doi: 10.1007/s00220-010-1075-y. |
[22] |
V. Ovsienko, R. Schwartz and S. Tabachnikov, Liouville-Arnold integrability of the pentagram map on closed polygons,, preprint, (). Google Scholar |
[23] |
A. Postnikov, Total positivity, Grassmannians, and networks,, preprint, (). Google Scholar |
[24] |
O. Schramm, Circle patterns with the combinatorics of the square grid,, Duke Math. J., 86 (1997), 347.
doi: 10.1215/S0012-7094-97-08611-7. |
[25] |
R. Schwartz, The pentagram map,, Experiment. Math., 1 (1992), 71.
|
[26] |
R. Schwartz, The pentagram map is recurrent,, Experiment. Math., 10 (2001), 519.
|
[27] |
R. Schwartz, Discrete monodromy, pentagrams, and the method of condensation,, J. Fixed Point Theory Appl., 3 (2008), 379.
doi: 10.1007/s11784-008-0079-0. |
[28] |
R. Schwartz and S. Tabachnikov, Elementary surprises in projective geometry,, Math. Intelligencer, 32 (2010), 31.
doi: 10.1007/s00283-010-9137-8. |
[29] |
R. Schwartz and S. Tabachnikov, The pentagram integrals on inscribed polygons,, Electron. J. Comb., 18 (2011). Google Scholar |
[30] |
F. Soloviev, Integrability of the pentagram map,, preprint, (). Google Scholar |
[31] |
Yu. Suris, On some integrable systems related to the Toda lattice,, J. Phys. A, 30 (1997), 2235.
doi: 10.1088/0305-4470/30/6/041. |
show all references
References:
[1] |
A. Bobenko and T. Hoffmann, Hexagonal circle patterns and integrable systems: Patterns with constant angles,, Duke Math. J., 116 (2003), 525.
doi: 10.1215/S0012-7094-03-11635-X. |
[2] |
A. Bobenko and Yu. Suris, "Discrete Differential Geometry. Integrable Structure,", Amer. Math. Soc., (2008).
|
[3] |
H. Busemann and P. Kelly, "Projective geometry and projective metrics,", Academic Press, (1953).
|
[4] |
B. Dubrovin, I. Krichever and S. Novikov, Integrable systems. I,, in, 4 (2001), 177.
|
[5] |
L. Faddeev and L. Takhtajan, "Hamiltonian Methods in the Theory of Solitons,", Springer-Verlag, (1987).
|
[6] |
S. Fomin and A. Zelevinsky, Cluster algebras. IV. Coefficients,, Compos. Math., 143 (2007), 112.
doi: 10.1112/S0010437X06002521. |
[7] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson geometry of directed networks in a disk,, Selecta Math., 15 (2009), 61.
doi: 10.1007/s00029-009-0523-z. |
[8] |
M. Gekhtman, M. Shapiro and A. Vainshtein, "Cluster Algebras and Poisson Geometry,", Amer. Math. Soc., (2010).
|
[9] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson geometry of directed networks in an annulus,, J. Europ. Math. Soc., ().
|
[10] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Generalized Bäcklund-Darboux transformations for Coxeter-Toda flows from a cluster algebra perspective,, Acta Math., 206 (2011), 245.
doi: 10.1007/s11511-011-0063-1. |
[11] |
M. Glick, The pentagram map and $Y$-patterns,, Adv. Math., 227 (2011), 1019.
doi: 10.1016/j.aim.2011.02.018. |
[12] |
, M. Glick,, private communication., (). Google Scholar |
[13] |
A. Goncharov and R. Kenyon, Dimers and cluster integrable systems,, preprint, (). Google Scholar |
[14] |
B. Khesin and F. Soloviev, Integrability of a space pentagram map,, in preparation., (). Google Scholar |
[15] |
B. Konopelchenko and W. Schief, Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy,, J. Phys. A, 35 (2002), 6125.
doi: 10.1088/0305-4470/35/29/313. |
[16] |
G. Mari Beffa, On generalizations of the pentagram map: Discretizations of AGD flows,, preprint, (). Google Scholar |
[17] |
S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons,, Ann. Inst. Fourier, (). Google Scholar |
[18] |
F. Nijhoff and H. Capel, The discrete Korteweg-de Vries equation,, Acta Appl. Math., 39 (1995), 133.
doi: 10.1007/BF00994631. |
[19] |
M. Olshanetsky, A. Perelomov, A. Reyman and M. Semenov-Tian-Shansky, Integrable systems. II,, in, 16 (1994), 83.
|
[20] |
V. Ovsienko, R. Schwartz and S. Tabachnikov, Quasiperiodic motion for the Pentagram map,, Electron. Res. Announc. Math. Sci., 16 (2009), 1.
|
[21] |
V. Ovsienko, R. Schwartz and S. Tabachnikov, The Pentagram map: A discrete integrable system,, Commun. Math. Phys., 299 (2010), 409.
doi: 10.1007/s00220-010-1075-y. |
[22] |
V. Ovsienko, R. Schwartz and S. Tabachnikov, Liouville-Arnold integrability of the pentagram map on closed polygons,, preprint, (). Google Scholar |
[23] |
A. Postnikov, Total positivity, Grassmannians, and networks,, preprint, (). Google Scholar |
[24] |
O. Schramm, Circle patterns with the combinatorics of the square grid,, Duke Math. J., 86 (1997), 347.
doi: 10.1215/S0012-7094-97-08611-7. |
[25] |
R. Schwartz, The pentagram map,, Experiment. Math., 1 (1992), 71.
|
[26] |
R. Schwartz, The pentagram map is recurrent,, Experiment. Math., 10 (2001), 519.
|
[27] |
R. Schwartz, Discrete monodromy, pentagrams, and the method of condensation,, J. Fixed Point Theory Appl., 3 (2008), 379.
doi: 10.1007/s11784-008-0079-0. |
[28] |
R. Schwartz and S. Tabachnikov, Elementary surprises in projective geometry,, Math. Intelligencer, 32 (2010), 31.
doi: 10.1007/s00283-010-9137-8. |
[29] |
R. Schwartz and S. Tabachnikov, The pentagram integrals on inscribed polygons,, Electron. J. Comb., 18 (2011). Google Scholar |
[30] |
F. Soloviev, Integrability of the pentagram map,, preprint, (). Google Scholar |
[31] |
Yu. Suris, On some integrable systems related to the Toda lattice,, J. Phys. A, 30 (1997), 2235.
doi: 10.1088/0305-4470/30/6/041. |
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