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The problem of integrable discretization: Hamiltonian approach
 Progress in Mathematics, Volume 219. Birkhäuser
"... this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the e ..."
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Cited by 71 (2 self)
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this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the equations of motion of the Toda lattice (under the name of a \continuous analogue of the qd algorithm")! The relation of the qd algorithm to integrable systems might have important implications for the numerical analysis, cf. Deift et al. (1991), Nagai and Satsuma (1995).
Transformations of quadrilateral lattices
 J. Math. Phys
"... Abstract. We investigate the τfunction of the quadrilateral lattice using the nonlocal ¯ ∂dressing method, and we show that it can be identified with the Fredholm determinant of the integral equation which naturally appears within that approach. ..."
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Cited by 30 (16 self)
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Abstract. We investigate the τfunction of the quadrilateral lattice using the nonlocal ¯ ∂dressing method, and we show that it can be identified with the Fredholm determinant of the integral equation which naturally appears within that approach.
The focal geometry of circular and conical meshes
, 2006
"... Circular meshes are quadrilateral meshes all of whose faces possess a circumcircle, whereas conical meshes are planar quadrilateral meshes where the faces which meet in a vertex are tangent to a right circular cone. Both are amenable to geometric modeling – recently surface approximation and subdivi ..."
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Cited by 29 (10 self)
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Circular meshes are quadrilateral meshes all of whose faces possess a circumcircle, whereas conical meshes are planar quadrilateral meshes where the faces which meet in a vertex are tangent to a right circular cone. Both are amenable to geometric modeling – recently surface approximation and subdivisionlike refinement processes have been studied. In this paper we extend the original defining property of conical meshes, namely the existence of face/face offset meshes at constant distance, to circular meshes. We study the close relation between circular and conical meshes, their vertex/vertex and face/face offsets, as well as their discrete normals and focal meshes. In particular we show how to construct a twoparameter family of circular (resp., conical) meshes from a given conical (resp., circular) mesh. We further discuss meshes which have both properties and their relation to discrete surfaces of negative Gaussian curvature. The offset properties of special quadrilateral meshes and the threedimensional support structures derived from them are highly relevant for computational architectural design of freeform structures. Another aspect important for design is that both circular and conical meshes provide a discretization of the principal curvature lines of a smooth surface, so the mesh polylines represent principal features of the surface described by the mesh.
Discretization of Surfaces and Integrable Systems
 In: Discrete Integrable Geometry and Physics; Eds.: A.I. Bobenko and
, 1998
"... this paper are analogues of the asymptotic or of the curvature lines on smooth surfaces, one can additionally assume that the number of edges meeting at each vertice is even. The case G = Z ..."
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Cited by 24 (8 self)
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this paper are analogues of the asymptotic or of the curvature lines on smooth surfaces, one can additionally assume that the number of edges meeting at each vertice is even. The case G = Z
On organizing principles of discrete differential geometry. Geometry of spheres
 RUSSIAN MATH. SURVEYS 62:1 1–43
, 2007
"... ..."
Quadratic reductions of quadrilateral lattices
 J. Geom. Phys
, 1999
"... It is shown that quadratic constraints are compatible with the geometric integrability scheme of the multidimensional quadrilateral lattice equation. The corresponding Ribaucourtype reduction of the fundamental transformation of quadrilateral lattices is found as well, and superposition of the Riba ..."
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Cited by 21 (15 self)
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It is shown that quadratic constraints are compatible with the geometric integrability scheme of the multidimensional quadrilateral lattice equation. The corresponding Ribaucourtype reduction of the fundamental transformation of quadrilateral lattices is found as well, and superposition of the Ribaucour transformations is presented in the vectorial framework. Finally, the quadratic reduction approach is illustrated on the example of multidimensional circular lattices.
Quantum geometry of 3dimensional lattices
, 2008
"... We study geometric consistency relations between angles on 3dimensional (3D) circular quadrilateral lattices — lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable “ultralocal” Poisson bracket alge ..."
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We study geometric consistency relations between angles on 3dimensional (3D) circular quadrilateral lattices — lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable “ultralocal” Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure leads to new solutions of the tetrahedron equation (the 3D analog of the YangBaxter equation). These solutions generate an infinite number of nontrivial solutions of the YangBaxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry. The classical geometry of the 3D circular lattices arises as a stationary configuration giving the leading contribution to the partition function in the quasiclassical limit.
The symmetric, Dinvariant and Egorov reductions of the quadrilateral lattice
 J. GEOM. PHYS
, 2000
"... We present a detailed study of the geometric and algebraic properties of the multidimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar; the discrete analogue of a conjugate net) and of its basic reductions. To make this study, we introduce the notions of forward an ..."
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We present a detailed study of the geometric and algebraic properties of the multidimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar; the discrete analogue of a conjugate net) and of its basic reductions. To make this study, we introduce the notions of forward and backward data, which allow us to give a geometric meaning to the τ–function of the lattice, defined as the potential connecting these data. Together with the known circular lattice (a lattice whose elementary quadrilaterals can be inscribed in circles; the discrete analogue of an orthogonal conjugate net) we introduce and study two other basic and independent reductions of the quadrilateral lattice: the symmetric lattice, for which the forward and backward data coincide, and the dinvariant lattice, characterized by the invariance of a certain natural frame along the main diagonal. We finally discuss the Egorov lattice, which is, at the same time, symmetric, circular and dinvariant. The integrability properties of all these lattices are established using geometric, algebraic and analytic means; in particular we present a ¯ ∂ formalism to construct large classes of such lattices. We also discuss quadrilateral hyperplane lattices and the interplay between quadrilateral point and hyperplane lattices in all the above reductions.
DESARGUES MAPS AND THE HIROTA–MIWA EQUATION
"... Abstract. We study the Desargues maps φ: Z N → P M, which generate lattices whose points are collinear with all their nearest (in positive directions) neighbours. The multidimensional consistency of the map is equivalent to the Desargues theorem and its higherdimensional generalizations. The nonlin ..."
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Abstract. We study the Desargues maps φ: Z N → P M, which generate lattices whose points are collinear with all their nearest (in positive directions) neighbours. The multidimensional consistency of the map is equivalent to the Desargues theorem and its higherdimensional generalizations. The nonlinear counterpart of the map is the noncommutative (in general) Hirota–Miwa system. In the commutative case of the complex field we apply the nonlocal ¯ ∂dressing method to construct Desargues maps and the corresponding solutions of the equation. In particular, we identify the Fredholm determinant of the integral equation inverting the nonlocal ¯ ∂dressing problem with the τfunction. Finally, we establish equivalence between the Desargues maps and quadrilateral lattices provided we take into consideration also their Laplace transforms.