Geometry,topology, and vacuum energy

Quantum corrections are investigated in multidimensional models for which the internal manifold is a squashed three-dimensional sphere orS ³/Z _p . It is shown that the divergences which arise in the summation over topologies disappear when quantum matter is taken into account.Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 1, pp. 12–20, January, 1992.     

Euler数，Bernoulli数及有序Bell数的渐近估计

利用特殊函数的发生函数为亚纯函数的特点,给出Euler数、Bernoulli数、有序Bell数等几个组合数的带有余项的渐进估计．     

Electrostatics, Hyperbolic Geometry and Wandering Vectors

A family of planar discrete electrostatic systems on the unitcircle with finitely atomic external fields is considered. Thegeometry of particles in the external field yielding a givenminimum energy configuration is studied. As an application,the wandering vectors of the shift operator in the Dirichletspaces associated with finitely atomic measures are also studied.In particular, the zero locus of a wandering vector is discussed.     

Cubic Curves,Finite Geometry and Cryptography

Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational points are also surveyed. A possible strengthening of the security of elliptic curve cryptography is proposed using a ‘shared secret’ related to the group law. Cubic curves are also used in a new way to construct sets of points having various combinatorial and geometric properties that are of particular interest in finite Desarguesian planes.     

Quasigroups, Geometry and Nonlinear Geometric Algebra

A survey of the methods of the theory of quasigroups and loops in algebra and geometry is presented in order to attract the attention of mathematicians and physicists to promising applications of this new branch of mathematics in applied sciences.     

PBW Degeneration: Algebra,Geometry, and Combinatorics

We review the recent progress in the theory of Poincaré–Birkhoff–Witt degenerations of irreducible representations of simple Lie algebras. We describe algebraic, geometric, and combinatorial aspects of the theory.     

Unit Balls, Lorentz Boosts, and Hyperbolic Geometry

The bounded symmetric spaces naturally associated with the Poincaré and Beltrami-Klein models of hyperbolic geometry on the open unit ball B in ${\mathbb{R}^n}$ and with the automorphism group of biholomorphic maps on the open ball in ${\mathbb{C}^n}$ give rise by a standard construction to specialized loop structures (nonassociative groups), which we use to define canonical metrics, called rapidity metrics. We show that this rapidity metric agrees with the classical Poincaré metric resp. the Cayley-Klein metric resp. the Bergman metric. We introduce the Lorentz boost of vectors in B, which turns out to be a loop isomorphism. It induces a similarity of metrics between the rapidity metric of the Einstein or Möbius loop and the trace metric on positive definite matrices restricted to the Lorentz boosts.     

圈钱偏好、非流通股与IPO定价

在传统信号理论基础上，本文建立了一个用于解释股票市场上的圈钱偏好与IPO定价问题的模型。模型主要基于一个新的假设——发行者对从一级和二级市场得到的资金总额具有特定的偏好，并且发行者出于某种考虑将保留一定股份作为非流通股。本文研究了该假设下，如果投资者自始至终无法知道项目的真实价值，而只能根据发行份额和折价程度来判断项目的真实情况时的IPO定价问题。发行者的这种偏好的实际意义，就是通常所说的圈钱偏好。     

两种四角系统的Wiener数和Hyper-Wiener数(英文)

我们推导出两类四角系统的Wiener数和Hyper-Wiener数的计算公式.     

Combinatorial Veronese Structures, Their Geometry, and Problems of Embeddability

Finite generalized Veronese spaces of special type (associated with single-line geometry) are studied, and problems concerning embeddability of the resulting configurations are discussed. Received: October 17, 2005. Revised: February 13, 2006.     

3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients

Using 3-Sasakian reduction techniques we obtain infinite familiesof new 3-Sasakian manifolds M (p ₁,p ₂, p ₃) andM (p ₁,p ₂, p ₃, p ₄) in dimension 11 and 15 respectively. The metric cone on (p ₁,p ₂, p ₃) is a generalization ofthe Kronheimer hyperkähler metric on the regular maximalnilpotent orbit of sl (3, C)whereas the cone on M (p ₁,p ₂, p ₃, p ₄)generalizes the hyperkähler metric onthe 16-dimensional orbit of so(6, C).These are the first examples of 3-Sasakian metrics which are neither homogeneous nor toric. In addition we consider some further U(1)-reductions of M(p ₁,p ₂, p ₃).These yield examples of nontoric 3-Sasakian orbifold metrics in dimensions 7. As a result we obtain explicit families O() of compact self-dual positivescalar curvature Einstein metrics with orbifoldsingularities and with only one Killing vector field.     

Jones Invariant,Cantorian Geometry and Quantum Spacetime

The relation between Jones knot polynominals and statistical mechanics is discussed in the light of Cantorian geometry. It is further shown that von Neumanns continuous geometry may be regarded as being a quantum spacetime akin to Cantorian space E ^(∞) and noncommutative geometry.     

Gaussian Beams,the Hamilton-Jacobi Equations,and Finsler Geometry

Relationships between Gaussian beams and geometry are considered in the paper. We show that the main properties of the Gaussian beam solutions are determined by the natural geometry related to the problem under consideration. The geometry is determined by the Hamilton-Jacobi equation and the corresponding Hamiltonian. In particular, we find a geometric interpretation of the Riccati equation for the quadratic form of the phase function corresponding to a Gaussian beam in the case of Finsler geometry. Bibliography: 2 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 66–92.     

Jacobi Cohomology, Local Geometry of Moduli Spaces, and Hitchin Connections

We develop some cohomological tools for the study of the localgeometry of moduli and parameter spaces in complex AlgebraicGeometry. Notably, we develop canonical formulae for the differentialoperators of arbitrary order and their natural action on suitable‘natural’ modules (for example, functions); in particular,we obtain a formula, in terms of the moduli problem, for theLie bracket of vector fields on a moduli space. As an application,we obtain another construction and proof of flatness for thefamiliar KZW or Hitchin connection on moduli spaces of curves.2000 Mathematics Subject Classification 14D15, 32G05.