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Flat pencils of metrics and Frobenius manifolds., ArXiv: math.DG/9803106
 In: Proceedings of 1997 Taniguchi Symposium ”Integrable Systems and Algebraic Geometry”, editors M.H.Saito, Y.Shimizu and K.Ueno
, 1998
"... Abstracts This paper is based on the author’s talk at 1997 Taniguchi Symposium “Integrable Systems and Algebraic Geometry”. We consider an approach to the theory of Frobenius manifolds based on the geometry of flat pencils of contravariant metrics. It is shown that, under certain homogeneity assumpt ..."
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Cited by 35 (6 self)
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Abstracts This paper is based on the author’s talk at 1997 Taniguchi Symposium “Integrable Systems and Algebraic Geometry”. We consider an approach to the theory of Frobenius manifolds based on the geometry of flat pencils of contravariant metrics. It is shown that, under certain homogeneity assumptions, these two objects are identical. The flat pencils of contravariant metrics on a manifold M appear naturally in the classification of bihamiltonian structures of hydrodynamics type on the loop space L(M). This elucidates the relations between Frobenius manifolds and integrable hierarchies. 1
Computing contour generators of evolving implicit surfaces
 IN SM ’03: PROCEEDINGS OF THE EIGHTH ACM SYMPOSIUM ON SOLID MODELING AND APPLICATIONS, ACM
, 2003
"... The contour generator is an important visibility feature of a smooth object seen under parallel projection. It is the curve on the surface which seperates frontfacing regions from backfacing regions. The apparent contour is the projection of the contour generator onto a plane perpendicular to t ..."
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Cited by 13 (2 self)
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The contour generator is an important visibility feature of a smooth object seen under parallel projection. It is the curve on the surface which seperates frontfacing regions from backfacing regions. The apparent contour is the projection of the contour generator onto a plane perpendicular to the view direction. Both curves play an important role in computer graphics. Our goal
Semisimplicial resolutions and homology of images and discriminants of mappings
 Proc. London Math. Soc
, 1995
"... In [13] the technique of semisimplicial resolutions was applied to the study of the topology of the image of a stable perturbation / of a mapgerm C " —> C p with n
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Cited by 9 (0 self)
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In [13] the technique of semisimplicial resolutions was applied to the study of the topology of the image of a stable perturbation / of a mapgerm C " —> C p with n <p. The crucial role was played there by the multiple point sets D k (f). These are the closures, in (C 1)*, of the sets of /ctuples of pairwise distinct points sharing
A note on first order differential equations of degree greater than one and wavefront evolution
 Bull. London Math. Soc
, 1984
"... Let F be a smooth function of the variables (x, y, p) e IR 3 and consider the first ..."
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Cited by 9 (0 self)
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Let F be a smooth function of the variables (x, y, p) e IR 3 and consider the first
Towards the Legendrian Sturm Theory of Space Curves
, 1998
"... Introduction Sturm theory extends the Morse inequality (minorating the number of critical points of functions on a circle) to the higher derivatives. The Legendrian Morse theory (created by Yu. V. Chekanov in 1986) provides the Morse inequality for the multivalued functions (corresponding to the un ..."
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Cited by 6 (0 self)
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Introduction Sturm theory extends the Morse inequality (minorating the number of critical points of functions on a circle) to the higher derivatives. The Legendrian Morse theory (created by Yu. V. Chekanov in 1986) provides the Morse inequality for the multivalued functions (corresponding to the unknoted Legendrian submanifolds of the space of 1jets of functions). It is a generalization of the Lagrangian intersection theory due to Conley, Zehnder, Chaperon, Floer, Sikorav, Laudenbach, Hofer, Gromov, and others. Below an attempt to extend the Legendrian Morse theory to higher derivatives is presented. It extends the Legendrian Morse theory in the same sense in which the ordinary Sturm theory extends Morse theory. Sturm theory implies the existence of at least four flattening points on every curve in projective 3space having a convex projection. The present paper results imply, for instance, the preservation of at least four flattening points under (not necessary small) deform
Quantum Many–Body Problems and Perturbation Theory”, Russian Journ. of Nuclear Phys
 Physics of Atomic Nuclei 65(6), 11351143 (2002) (English translation) hepth/0108160
, 2002
"... We show that the existence of algebraic forms of exactlysolvable A−B− C −D and G2, F4 OlshanetskyPerelomov Hamiltonians allow to develop the algebraic perturbation theory, where corrections are computed by pure algebraic means. A classification of perturbations leading to such a perturbation theor ..."
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Cited by 4 (0 self)
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We show that the existence of algebraic forms of exactlysolvable A−B− C −D and G2, F4 OlshanetskyPerelomov Hamiltonians allow to develop the algebraic perturbation theory, where corrections are computed by pure algebraic means. A classification of perturbations leading to such a perturbation theory based on representation theory of Lie algebras is given. In particular, this scheme admits an explicit study of anharmonic manybody problems. Some examples are presented.
Perturbations of integrable systems and DysonMehta integrals
, 2002
"... We show that the existence of algebraic forms of quantum, exactlysolvable, completelyintegrable A−B−C −D and G2,F4,E6,7,8 OlshanetskyPerelomov Hamiltonians allow to develop the algebraic perturbation theory, where corrections are computed by pure linear algebra means. A Liealgebraic classificati ..."
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Cited by 3 (1 self)
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We show that the existence of algebraic forms of quantum, exactlysolvable, completelyintegrable A−B−C −D and G2,F4,E6,7,8 OlshanetskyPerelomov Hamiltonians allow to develop the algebraic perturbation theory, where corrections are computed by pure linear algebra means. A Liealgebraic classification of such perturbations is given. In particular, this scheme admits an explicit study of anharmonic manybody problems. The approach also allows to calculate the ratio of a certain generalized DysonMehta integrals algebraically, which are interested by themselves. Invited talk given at the Workshop ”Superintegrable systems in classical and quantum mechanics”,
Hyperbolic Schwarz map of the confluent hypergeometric differential equation, preprint 2007
"... Abstract. The hyperbolic Schwarz map is defined in [SYY1] as a map from the complex projective line to the threedimensional real hyperbolic space by use of solutions of the hypergeometric differential equation. Its image is a flat front ([GMM, KUY, KRSUY]), and generic singularities are cuspidal ed ..."
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Cited by 3 (3 self)
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Abstract. The hyperbolic Schwarz map is defined in [SYY1] as a map from the complex projective line to the threedimensional real hyperbolic space by use of solutions of the hypergeometric differential equation. Its image is a flat front ([GMM, KUY, KRSUY]), and generic singularities are cuspidal edges and swallowtail singularities. In this paper, we study creations/eliminations of the swallowtails on the image surfaces of the twoparameter family of the confluent hypergeometric differential equations, and give a stratification of the parameter space according to types of singularities. Such a study was made for a 1parameter family of hypergeometric differential equation in [NSYY]. Contents
Legendrian graphs generated by tangential families
 Proc. Edinburgh Math. Soc
"... We construct a Legendrian version of Envelope theory. A tangential family is a 1parameter family of rays emanating tangentially from a smooth plane curve. The Legendrian graph of the family is the union of the Legendrian lifts of the family curves in the projectivized cotangent bundle PT ∗ R 2. We ..."
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Cited by 2 (2 self)
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We construct a Legendrian version of Envelope theory. A tangential family is a 1parameter family of rays emanating tangentially from a smooth plane curve. The Legendrian graph of the family is the union of the Legendrian lifts of the family curves in the projectivized cotangent bundle PT ∗ R 2. We study the singularities of Legendrian graphs and their stability under small tangential deformations. We also find normal forms of their projections into the plane. This allows to interprete the beaks perestroika as the apparent contour of a deformation of the Double Whitney Umbrella singularity