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The Encyclopedia of Integer Sequences
"... This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs ..."
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Cited by 879 (15 self)
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This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs to generate the sequence, references, links to relevant web pages, and other
of meanders and train tracks for description of defects and textures in liquid crystals and 2+1 gravity, J.Geom.Phys. 33
, 2000
"... In this work the qualitative analysis of statics and dynamics of defects and textures in liquid crystals is performed with help of meanders and train tracks.It is argued that similar analysis can be applied to 2+1 gravity.More rigorous mathematical justifications are presented in the companion paper ..."
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Cited by 7 (2 self)
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In this work the qualitative analysis of statics and dynamics of defects and textures in liquid crystals is performed with help of meanders and train tracks.It is argued that similar analysis can be applied to 2+1 gravity.More rigorous mathematical justifications are presented in the companion paper (Part II) on quadratic differentials and measured foliations. Meanders were recently introduced by V.Arnold (Siberian J.of Mathematics 29,36 (1988)) and are used originally in the combinatorial problem of finding the number of distinct ways given curve can intersect another curve in prescribed number of points fixed along this auxiliary curve.Train tracks were introduced by W.Thurston (Geometry and Topology of 3Manifolds, Princeton U.Lecture Notes,1979) in connection with description of homeomorphisms of two dimensional surfaces.Train tracks alone are sufficient for the description of statics and dynamics of liquid crystals and gravity.Using train tracks the master equation is obtained which could be used alternatively to the WheelerDeWitt equation for 2+1 gravity.Since solution of this equation is possible but requires large scale numerical work, in this paper we resort to the approximation of train tracks by the meanditic labyrinths. This then allows us to analyse possible phases (and phase transitions) of gravity and liquid crystals using Peierlslike arguments
CONJUGATION SPACES AND 4MANIFOLDS
"... Abstract. We show that 4dimensional conjugation manifolds are all obtained from branched 2fold coverings of knotted surfaces in Z2homology 4spheres. 1. ..."
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Abstract. We show that 4dimensional conjugation manifolds are all obtained from branched 2fold coverings of knotted surfaces in Z2homology 4spheres. 1.
Hyperbolic Carathéodory conjecture
, 2006
"... A quadratic point on a surface in RP 3 is a point at which the surface can be approximated by a quadric abnormally well (up to order 3). We conjecture that the least number of quadratic points on a generic compact nondegenerate hyperbolic surface is 8; the relation between this and the classic Ca ..."
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A quadratic point on a surface in RP 3 is a point at which the surface can be approximated by a quadric abnormally well (up to order 3). We conjecture that the least number of quadratic points on a generic compact nondegenerate hyperbolic surface is 8; the relation between this and the classic Carathéodory conjecture is similar to the relation between the sixvertex and the fourvertex theorems on plane curves. Examples of quartic perturbations of the standard hyperboloid confirm our conjecture. Our main result is a linearization and reformulation of the problem in the framework of 2dimensional Sturm theory; we also define a signature of a quadratic point and calculate local normal forms recovering and generalizing TresseWilczynski’s theorem.
A geometric study of many body systems
, 2008
"... An nbody system is a labelled collection of n point masses in a Euclidean space, and their congruence and internal symmetry properties involve a rich mathematical structure which is investigated in the framework of equivariant Riemannian geometry. Some basic concepts are nconfiguration, configurat ..."
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An nbody system is a labelled collection of n point masses in a Euclidean space, and their congruence and internal symmetry properties involve a rich mathematical structure which is investigated in the framework of equivariant Riemannian geometry. Some basic concepts are nconfiguration, configuration space, internal space, shape space, Jacobi transformation and weighted root system. The latter is a generalization of the root system of SU(n), which provides a bookkeeping for expressing the mutual distances of the point masses in terms of the Jacobi vectors. Moreover, its application to the study of collinear central nconfigurations yields a simple proof of Moulton’s enumeration formula. A major topic is the study of matrix spaces representing the shape space of nbody configurations in Euclidean kspace, the structure of the muniversal shape space and its O(m)equivariant linear model. This also leads to those “orbital fibrations ” where SO(m) or O(m) act on a sphere with a sphere as orbit space. A few of these examples are encountered in the literature, e.g. the special case S 5 /O(2) ≈ S 4 was analyzed independently by Arnold, Kuiper and Massey in the 1970’s. Contents 1
ON AFFINE HYPERSURFACES WITH EVERYWHERE NONDEGENERATE SECOND QUADRATIC FORM
, 2002
"... Abstract. Consider a closed connected hypersurface in R n with constant signature (k, l) of the second quadratic form, and approaching a quadratic cone at infinity. This hypersurface divides R n into two pieces. We prove that one of them contains a kdimensional subspace, and another contains a ldi ..."
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Abstract. Consider a closed connected hypersurface in R n with constant signature (k, l) of the second quadratic form, and approaching a quadratic cone at infinity. This hypersurface divides R n into two pieces. We prove that one of them contains a kdimensional subspace, and another contains a ldimensional subspace, thus proving an affine version of Arnold hypothesis. We construct an example of a surface of negative curvature in R 3 with slightly