Sign in
Author

Conference

Journal

Organization

Year

DOI
Look for results that meet for the following criteria:
since
equal to
before
between
and
Search in all fields of study
Limit my searches in the following fields of study
Agriculture Science
Arts & Humanities
Biology
Chemistry
Computer Science
Economics & Business
Engineering
Environmental Sciences
Geosciences
Material Science
Mathematics
Medicine
Physics
Social Science
Multidisciplinary
Keywords
(4)
boussinesq equation
Integrable System
Natural Transformation
poisson structure
Subscribe
Academic
Publications
The Pentagram Map: A Discrete Integrable System
The Pentagram Map: A Discrete Integrable System,10.1007/s002200101075y,Communications in Mathematical Physics,Valentin Ovsienko,Richard Schwartz,Se
Edit
The Pentagram Map: A Discrete Integrable System
(
Citations: 6
)
BibTex

RIS

RefWorks
Download
Valentin Ovsienko
,
Richard Schwartz
,
Serge Tabachnikov
The pentagram map is a projectively
natural transformation
defined on (twisted) polygons. A twisted polygon is a map from $${\mathbb Z}$$ into $${{\mathbb{RP}}^2}$$ that is periodic modulo a projective transformation called the monodromy. We find a
Poisson structure
on the space of twisted polygons and show that the pentagram map relative to this
Poisson structure
is completely integrable. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasiperiodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The
Poisson structure
we attach to the pentagram map is a discrete version of the first
Poisson structure
associated with the Boussinesq equation. A research announcement of this work appeared in [16].
Journal:
Communications in Mathematical Physics  COMMUN MATH PHYS
, vol. 299, no. 2, pp. 409446, 2010
DOI:
10.1007/s002200101075y
Cumulative
Annual
View Publication
The following links allow you to view full publications. These links are maintained by other sources not affiliated with Microsoft Academic Search.
(
www.springerlink.com
)
(
www.springerlink.com
)
(
www.math.psu.edu
)
(
adsabs.harvard.edu
)
(
www.springerlink.com
)
(
www.math.brown.edu
)
(
arxiv.org
)
(
math.univlyon1.fr
)
More »
References
(29)
Lattice analogues of Walgebras and classical integrable equations
(
Citations: 17
)
Alexander A. Belov
,
Karen D. Chaltikian
Journal:
Physics Letters B  PHYS LETT B
, vol. 309, no. 34, pp. 268274, 1993
Cluster algebras I: Foundations
(
Citations: 271
)
Sergey Fomin
,
Andrei Zelevinsky
Published in 2001.
Cluster algebras and Poisson geometry
(
Citations: 53
)
Michael Gekhtman
,
Michael Shapiro
,
Alek Vainshtein
Published in 2002.
Determinants and alternating sign matrices
(
Citations: 74
)
D. P. Robbins
,
H. Rumsey
Journal:
Advances in Mathematics  ADVAN MATH
, vol. 62, no. 2, pp. 169184, 1986
Threedimensional geometry and topology
(
Citations: 539
)
William P. Thurston
Published in 1997.
Sort by:
Citations
(6)
The Link Between Erectile and Cardiovascular Health: The Canary in the Coal Mine
(
Citations: 1
)
David R. Meldrum
,
Joseph C. Gambone
,
Marge A. Morris
,
Donald A. N. Meldrum
,
Katherine Esposito
,
Louis J. Ignarro
Journal:
American Journal of Cardiology  AMER J CARDIOL
, vol. 108, no. 4, pp. 599606, 2011
Lagrangian multiform structure for the lattice Gel'fand–Dikii hierarchy
(
Citations: 1
)
S. B. Lobb
,
F. W. Nijhoff
Journal:
Journal of Physics Amathematical and Theoretical  J PHYS AMATH THEOR
, vol. 43, no. 7, 2010
The pentagram map and Ypatterns
(
Citations: 1
)
Max Glick
Published in 2010.
2frieze patterns and the cluster structure of the space of polygons
Sophie MorierGenoud
,
Valentin Ovsienko
,
Serge Tabachnikov
Published in 2010.
The Pentagram Integrals on Inscribed Polygons
Richard Evan Schwartz
,
Serge Tabachnikov
Published in 2010.