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On two conjectures concerning convex curves
 Internat. J. Math. vol
"... Abstract. In this paper we recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the first nontrivial case of curves in RP 3. Namely, we show that i) the tangent developable of any convex curve in RP 3 has degree 4 and ii) construct ..."
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Abstract. In this paper we recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the first nontrivial case of curves in RP 3. Namely, we show that i) the tangent developable of any convex curve in RP 3 has degree 4 and ii) construct an example of 4 tangent lines to a convex curve in RP 3 such that no real line intersects all four of them. The question (discussed in [EG1] and [So4]) whether the second conjecture is true in the special case of rational normal curves still remains open. We start with some important notions. §1. Introduction and results Main definition. A smooth closed curve γ: S 1 → RP n is called locally convex if the local multiplicity of intersection of γ with any hyperplane H ⊂ RP n at any of the intersection points does not exceed n = dim RP n and globally convex or just convex if the above condition holds for the global multiplicity, i.e for the sum of local multiplicities.
HUATYPE INTEGRALS OVER UNITARY GROUPS AND OVER PROJECTIVE LIMITS OF UNITARY GROUPS
, 2001
"... We discuss some natural maps from a unitary group U(n) to a smaller group U(n−m). (These maps are versions of the Livˇsic characteristic function.) We calculate explicitly the direct images of the Haar measure under some maps. We evaluate some matrix integrals over classical groups and some symmetri ..."
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We discuss some natural maps from a unitary group U(n) to a smaller group U(n−m). (These maps are versions of the Livˇsic characteristic function.) We calculate explicitly the direct images of the Haar measure under some maps. We evaluate some matrix integrals over classical groups and some symmetric spaces. (Values of the integrals are products of Ɣfunctions.) These integrals generalize Hua integrals. We construct inverse limits of unitary groups equipped with analogues of Haar measure and evaluate some integrals over these inverse limits. Let K be the real numbers R, the complex numbers C, or the algebra of quaternions H. By U(n, K) = O(n), U(n), Sp(n), we denote the unitary group of the space K n = R n, C n, H n. We also use the notation U ◦ (n, K): = SO(n), U(n), Sp(n) for the connected component of the group U(n, K). By σn, we denote the Haar measure on U ◦ (n, K) normalized by the condition σn(U ◦ (n, K)) = 1. Let Q be a matrix over K. By [Q]p, we denote the upper left block of the matrix Q of size p × p. By {Q}p, we denote the lower right block of size p × p. Let us represent a matrix g ∈ U ◦ (n, K) as an (m + (n − m)) × (m + (n − m)) block matrix () P Q. Consider the map
Two Conjectures on Convex Curves
, 2002
"... In this paper we recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the first nontrivial case of . Namely, we show i) that the tangent developable of any convex curve in has degree 4 and ii) construct an example of 4 tange ..."
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Cited by 4 (0 self)
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In this paper we recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the first nontrivial case of . Namely, we show i) that the tangent developable of any convex curve in has degree 4 and ii) construct an example of 4 tangent lines to a convex curve in RP such that no real line intersects all four of them. The question (discussed in [EG1] and [So4]) whether the second conjecture is true in the special case of rational normal curves still remains open.
A note on Lagrangian loci of quotients
 Canadian Math. Bull
"... Abstract. We study hamiltonian actions of compact groups in the presence of compatible involutions. We show that the lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces corresponding to the conjugacy classes of involut ..."
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Cited by 4 (2 self)
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Abstract. We study hamiltonian actions of compact groups in the presence of compatible involutions. We show that the lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces corresponding to the conjugacy classes of involutions on the group strongly inner to the given one. Our techniques imply that the solution to the additive Thompson problem for a given real form can be relayed to the quasisplit real form in the same inner class. We also consider invariant quotients with respect to the corresponding real form of the complexified group. 1.
Spectral approach to linear programming bounds on codes, Problems of Information Transmission
, 2006
"... ABSTRACT. We give new proofs of asymptotic upper bounds of coding theory obtained within the frame of Delsarte’s linear programming method. The proofs rely on the analysis of eigenvectors of some finitedimensional operators related to orthogonal polynomials. The examples of the method considered in ..."
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ABSTRACT. We give new proofs of asymptotic upper bounds of coding theory obtained within the frame of Delsarte’s linear programming method. The proofs rely on the analysis of eigenvectors of some finitedimensional operators related to orthogonal polynomials. The examples of the method considered in the paper include binary codes, binary constantweight codes, spherical codes, and codes in the projective spaces. 1. Introduction. Let X be a compact metric space with distance function d. A code C is a finite subset of X. Define the minimum distance of C as d(C) = minx,y∈C,x=y d(x,y). A variety of metric spaces that arise from different applications include the binary Hamming space, the binary Johnson space, the sphere in R n, real and complex projective spaces, Grassmann manifolds, etc. Estimating the maximum size of the code with a given value of d is one of the main problems of coding theory. Let M be the cardinality of
Foundations Of Temporal Query Languages
, 1995
"... Temporal Databases are repositories of information dependent on time. The major difference from standard, e.g., relational database systems, is the need of storing possibly infinite objects, e.g., time spans. In recent years, there have been numerous proposals that introduce time into standard relat ..."
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Temporal Databases are repositories of information dependent on time. The major difference from standard, e.g., relational database systems, is the need of storing possibly infinite objects, e.g., time spans. In recent years, there have been numerous proposals that introduce time into standard relational systems. Unfortunately, most of the attempts have been based on adhoc extensions of existing database systems and query languages, e.g., TQUEL and TSQL. Such extensions often create many problems, when precise semantics needs to be developed, if one exists at all. In a recent survey by J. Chomicki, a clean way of defining temporal databases based on logic was proposed. This methodology views temporal databases as multisorted, finitely representable firstorder structures. Query languages then became formulas in suitable logics over the vocabulary of such structures. This method has been quite successful, as most of the existing proposals are subsumed by this approach with only minor ...
Preconditioned Gradient Iterative Methods for Nonlinear Eigenvalue Problems
, 2000
"... The existence of eigenvalues of a finitedimensional symmetric eigenvalue problem with nonlinear entrance of the spectral parameter is studied. Preconditioned gradient iterative methods are suggested for solving the problem. The convergence and the error of these methods for computing eigenvalues ar ..."
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The existence of eigenvalues of a finitedimensional symmetric eigenvalue problem with nonlinear entrance of the spectral parameter is studied. Preconditioned gradient iterative methods are suggested for solving the problem. The convergence and the error of these methods for computing eigenvalues are investigated.
IntegroDifferential Model for Pattern Formation in Bacterial Swarm
"... This paper introduces a onedimensional, partial integrodifferential model for bacterial aggregation. Integrodifferential equation is derived from a chemotactic partial differential system, and stability and numerical analyses are applied to study pattern formation. We demonstrate that an interpla ..."
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This paper introduces a onedimensional, partial integrodifferential model for bacterial aggregation. Integrodifferential equation is derived from a chemotactic partial differential system, and stability and numerical analyses are applied to study pattern formation. We demonstrate that an interplay between positive and negative chemotaxis can generate both periodic patterns and aggregation. In the limit of slow diffusion we use a Lagrangian approach to derive a gradient system and analyze periodic peaklike patterns. We discuss the possibility of existence of other types of patterns, and the applicability of the model to the pattern formation in bacterial swarms, in particular the rippling phenomenon in Myxobacteria. 1 Introduction Pattern formation is an intriguing widespread phenomenon in nature, particularly in biology. In this paper we are concerned with patterns evolving in populations of microorganisms, phenomena that have been studied extensively. The spatiotemporal patte...
Convergence of the Modified Subspace Iteration Method for Nonlinear Eigenvalue Problems
, 1999
"... The existence of eigenvalues of a finitedimensional eigenvalue problem with nonlinear entrance of a spectral parameter is studied. The modified subspace iteration method is suggested for solving the problem. The convergence and the error of this method for computing eigenvalues are investigated. ..."
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The existence of eigenvalues of a finitedimensional eigenvalue problem with nonlinear entrance of a spectral parameter is studied. The modified subspace iteration method is suggested for solving the problem. The convergence and the error of this method for computing eigenvalues are investigated.