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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).
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.
On organizing principles of discrete differential geometry. Geometry of spheres
 RUSSIAN MATH. SURVEYS 62:1 1–43
, 2007
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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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Cited by 15 (9 self)
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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.
The affine Weyl group symmetry of Desargues maps and of the noncommutative HirotaMiwa system
 Phys. Lett. A
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Curvature line parametrized surfaces and orthogonal coordinate systems. Discretization with Dupin cyclides
, 2007
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Lattice geometry of the Hirota equation, [in
 SIDE III – Symmetries and Integrability of Difference Equations
"... Geometric interpretation of the Hirota equation is presented as equation describing the Laplace sequence of twodimensional quadrilateral lattices. Different forms of the equation are given together with their geometric interpretation: (i) the discrete coupled Volterra system for the coefficients of ..."
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Cited by 6 (4 self)
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Geometric interpretation of the Hirota equation is presented as equation describing the Laplace sequence of twodimensional quadrilateral lattices. Different forms of the equation are given together with their geometric interpretation: (i) the discrete coupled Volterra system for the coefficients of the Laplace equation, (ii) the gauge invariant form of the Hirota equation for projective invariants of the Laplace sequence, (iii) the discrete Toda system for the rotation coefficients and (iv) the original form of the Hirota equation for the τfunction of the quadrilateral lattice.
GENERALIZED ISOTHERMIC LATTICES
, 2007
"... Abstract. We study multidimensional quadrilateral lattices satisfying simultaneously two integrable constraints: a quadratic constraint and the projective Moutard constraint. When the lattice is two dimensional and the quadric under consideration is the Möbius sphere one obtains, after the stereogra ..."
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Abstract. We study multidimensional quadrilateral lattices satisfying simultaneously two integrable constraints: a quadratic constraint and the projective Moutard constraint. When the lattice is two dimensional and the quadric under consideration is the Möbius sphere one obtains, after the stereographic projection, the discrete isothermic surfaces defined by Bobenko and Pinkall by an algebraic constraint imposed on the (complex) crossratio of the circular lattice. We derive the analogous condition for our generalized isthermic lattices using Steiner’s projective structure of conics and we present basic geometric constructions which encode integrability of the lattice. In particular we introduce the Darboux transformation of the generalized isothermic lattice and we derive the corresponding Bianchi permutability principle. Finally, we study two dimensional generalized isothermic lattices, in particular geometry of their initial boundary value problem.
Geometric discretization of the Koenigs nets
 J. Math. Phys
"... We introduce the Koenigs lattice, which is a new integrable reduction of the quadrilateral lattice (discrete conjugate net) and provides natural integrable discrete analogue of the Koenigs net. We construct the Darbouxtype transformation of the Koenigs lattice and we show permutability of superposi ..."
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We introduce the Koenigs lattice, which is a new integrable reduction of the quadrilateral lattice (discrete conjugate net) and provides natural integrable discrete analogue of the Koenigs net. We construct the Darbouxtype transformation of the Koenigs lattice and we show permutability of superpositions of such transformations, thus proving integrability of the Koenigs lattice. We also investigate the geometry of the discrete Koenigs transformation. In particular we characterize the Koenigs transformation in terms of an involution determined by a congruence conjugate to the lattice.