An algebraic geometric classification of superintegrable systems in the Euclidean plane

We prove that the set of non-degenerate second order maximally superintegrable systems in the complex Euclidean plane carries a natural structure of a projective variety, equipped with a linear isometry group action. This is done by deriving an explicit system of homogeneous algebraic equations. We then solve these equations and give a detailed analysis of the algebraic geometric structure of the corresponding projective variety. This naturally associates a unique planar line triple arrangement to every superintegrable system, providing a geometric realisation of this variety and an intrinsic labelling scheme. In particular, our results confirm the known classification by independent, purely algebraic means.     

Interpolation between Brezis–Vázquez and Poincaré inequalities on nonnegatively curved spaces: sharpness and rigidities

This paper is devoted to investigate an interpolation inequality between the Brezis–Vázquez and Poincar33; inequalities (shortly, BPV inequality) on nonnegatively curved spaces. As a model case, we first prove that the BPV inequality holds on any Minkowski space, by fully characterizing the existence and shape of its extremals. We then prove that if a complete Finsler manifold with nonnegative Ricci curvature supports the BPV inequality, then its flag curvature is identically zero. In particular, we deduce that a Berwald space of nonnegative Ricci curvature supports the BPV inequality if and only if it is isometric to a Minkowski space. Our arguments explore fine properties of Bessel functions, comparison principles, and anisotropic symmetrization on Minkowski spaces. As an application, we characterize the existence of nonzero solutions for a quasilinear PDE involving the Finsler–Laplace operator and a Hardy-type singularity on Minkowski spaces where the sharp BPV inequality plays a crucial role. The results are also new in the Riemannian/Euclidean setting.     

Unbounded symmetric operators in K-homology and the Baum-Connes conjecture

Using the unbounded picture of analytical K-homology, we associate a well-defined K-homology class to an unbounded symmetric operator satisfying certain mild technical conditions. We also establish an “addition formula” for the Dirac operator on the circle and for the Dolbeault operator on closed surfaces. Two proofs are provided, one using topology and the other one, surprisingly involved, sticking to analysis, on the basis of the previous result. As a second application, we construct, in a purely analytical language, various homomorphisms linking the homology of a group in low degree, the K-homology of its classifying space and the analytic K-theory of its C^*-algebra, in close connection with the Baum-Connes assembly map. For groups classified by a 2-complex, this allows to reformulate the Baum-Connes conjecture.     

q-Derivative Operators and Basic Hypergeometric Series

By means of q-derivative operators, we investigate formal power series expansions. Two main expansion formulae in terms of q-derivative operators are established which can be considered as extensions of the corresponding results due to Carlitz (1973) and Liu (2002). Their applications to basic hypergeometric evaluations and transformations are discussed through series compositions and their q-derivative operations. Direct verification of the two main theorems are also presented.     

On the largest principal angle between random subspaces

Formulas are derived for the probability density function and the probability distribution function of the largest canonical angle between two p-dimensional subspaces of Rⁿ chosen from the uniform distribution on the Grassmann manifold (which is the unique distribution invariant by orthogonal transformations of Rⁿ). The formulas involve the gamma function and the hypergeometric function of a matrix argument.     

Central Delannoy Numbers and Balanced Cohen-Macaulay Complexes

We introduce a new join operation on colored simplicial complexes that preserves the Cohen-Macaulay property. An example of this operation puts the connection between the central Delannoy numbers and Legendre polynomials in a wider context. On leave from the R33;nyi Mathematical Institute of the Hungarian Academy of Sciences. Received April 18, 2005     

A non-commutative binomial formula

We prove a binomial formula for variables x and y satisfying a quadratic relation xy = ax² + qyx + by². Such relations are important in quantum group theory and non-commutative geometry.     

螺旋测微器实验教学中两个问题的研究

本文提出采用“回零法”能有效的防止螺旋测微器读数不准的问题 ,并对螺旋测微器因保养不当造成测微螺杆旋不动提供了解决的方法     

Research on the gain of excimer-like (N2)2 dimers in a microwave discharge

The experimental measurements and theoretical estimation for the gain of a (N₂)₂ dimer at 336.21 nm for the a¹ B _2g?a¹ B _3u excimer-like transition in a microwave discharge are reported. The maximum experimental gain for (N₂)₂ dimers at 336.21 nm is 1.19 % cm^-1 in pure N₂ gas at 9.23 Torr at 500 W of input microwave power. Received: 21 July 2000 / Revised version: 30 October 2000 / Published online: 27 April 2001     

Opial-type inequalities for differential operators

Some continuous and discrete versions of Opial-type inequalities which are readily applicable to differential and difference operators are established. These generalize earlier results of Anastassiou and Pe?ari?, and of Koliha and Pe?ari?.