辛轨形群胚的辛邻域定理

本文给出一种几何的子轨形群胚的定义,还给出了判定子轨形群胚的依据,并证明了紧子轨形群胚的轨形管状邻域、紧辛子轨形群胚的辛邻域和紧Lagrangian子轨形群胚的Lagrangian邻域的存在性.     

Symplectic Spreads

We construct an infinite family of symplectic spreads in spaces of odd rank and characteristic.     

Symplectic modules

A symplectic module is a finite group with a regular antisymmetric form. The paper determines sufficient conditions for the invariants of the maximal isotropic subgroups (Lagrangians), and asymptotic values for a lower bound of a group which contains Lagrangians of all symplectic modules of a fixed orderp ⁿ. These results have application to the splitting fields of universal division algebras.     

Symplectic meanders

Analogous to the 𝔰𝔩(n) case, we address the computation of the index of seaweed subalgebras of 𝔰𝔭(2n) by introducing graphical representations called symplectic meanders. Formulas for the algebra’s index may be computed by counting the connected components of its associated meander. In certain cases, formulas for the index can be given in terms of elementary functions.     

Symplectic Matroids

A symplectic matroid is a collection B of k-element subsets of J = {1, 2, ..., n, 1*, 2*, ...; n*}, each of which contains not both of i and i* for every i n, and which has the additional property that for any linear ordering of J such that i j implies j* i* and i j* implies j i* for all i, j n, B has a member which dominates element-wise every other member of B. Symplectic matroids are a special case of Coxeter matroids, namely the case where the Coxeter group is the hyperoctahedral group, the group of symmetries of the n-cube. In this paper we develop the basic properties of symplectic matroids in a largely self-contained and elementary fashion. Many of these results are analogous to results for ordinary matroids (which are Coxeter matroids for the symmetric group), yet most are not generalizable to arbitrary Coxeter matroids. For example, representable symplectic matroids arise from totally isotropic subspaces of a symplectic space very similarly to the way in which representable ordinary matroids arise from a subspace of a vector space. We also examine Lagrangian matroids, which are the special case of symplectic matroids where k = n, and which are equivalent to Bouchet's symmetric matroids or 2-matroids.     

关于流形上鞅的内向爆发

在本文中,我们构造了一个２维完备黎曼流形;其上的布朗运动是非内向爆发的,且存在一个对应于负Ｌｅｖｉ－Ｃｉｖｉｔａ联络的内向爆发鞅。此外,我们也考虑了布朗运动的非内向爆发。     

Symplectic Geometry

Book Review

Symplectic GeometryA. T. Fomenko: Advanced Studies in Contemporary Mathematics, Volume 5, Gordon and Breach Science Publishers, 1988, 387 pp     

Symplectic enumeration

Every complex projective space of odd dimension carries a natural contact structure. We give first steps towards the enumeration of curves in ℙ³ tangent to the contact structure. Such a curve is involutive in the sense that its homogeneous ideal is closed under Poisson bracket. Involutive curves in ℙ³ contained in a plane split as a union of concurrent lines. We give a formula for the number of plane involutive curves of a given degree in ℙ³ meeting the appropriate number of lines. We also discuss strategies to deal with the enumeration of involutive rational curves.     

The Symplectic Monoid

In this paper, we describe explicitly the symplectic monoid ? and its Renner monoid ? using elementary methods. We refine the Bruhat–Renner decomposition of ? and analyze in detail the length function on ?. We then show that every element of ? has a unique canonical form decomposition, which is an analogue of the canonical form of elements in Chevalley groups. We also compute the order of ? over a finite field, and as a consequence we obtain a new combinatorial identity.     

Symplectic RK Methods and Symplectic PRK Methods with Real Eigenvalues

Properties of symplectic Runge-Kutta (RK) methods and symplectic partitioned Runge-Kutta (PRK) methods with real eigenvalues are discussed in this paper. It is shown that an s stage such method can‘t reach order more than s 1. Particularly, we prove that no symplectic RK method with real eigenvalues exists in stage s of order s 1 when s is even. But an example constructed by using the W-transformation shows that PRK method of this type does not necessarily meet this order barrier. Another useful way other than W-transformation to construct symplectic PRK method with real eigenvalues is then presented. Finally, a class of efficient symplectic methods is recommended.     

Symplectic orthogonality spaces

Let E be a finite dimensional vector space over the Galois field GF(2). Let lin(E) denote the set of one-dimensional subspaces of E. Let Φ be the symplectic inner product on E. Consider the elements of lin(E) as vertices of a graph, two vertices being connected exactly when they are distinct and orthogonal with respect to Φ. This graph is characterized abstractly.     

Symplectic curvature tensors

In the paper, one establishes the decomposition of the space of tensors which have the symmetries of the curvature of a torsionless symplectic connection into Sp (n)-irreducible components. This leads to three interesting classes of symplectic connections: flat, Ricci flat, and similar to the Levi-Civita connections of Kähler manifolds with constant holomorphic sectional curvature (we call them connections with reducible curvature). A symplectic manifold with two transversal polarizations has a canonical symplectic connection, and we study properties that are encountered if this canonical connection belongs to the classes mentioned above. For instance, in the reducible case we can compute the Pontrjagin classes, and these will be zero if the polarizations are real, etc. If the polarizations are real and there exist points where they are either singular or nontransversal, one has residues in the sense ofLehmann [L], which should be expected to play an interesting role in symplectic geometry.     

Symplectic Difference Schemes for Hamiltonian Systems in General Symplectic Structure

We consider the construction of phase flow generating functions and symplectic difference schemes for Hamiltonian systems in general symplectic structure with variable coefficients.     

局部环上辛变换关于辛平延的分解

研究了局部环上辛变换的辛平延,利用亏失数、剩余数的理论,讨论了局部环上辛变换关于辛平延的分解．     

Symplectic half-flat solvmanifolds

We classify solvable Lie groups admitting left invariant symplectic half-flat structure. When the Lie group has a compact quotient by a lattice, we show that these structures provide solutions of supersymmetric equations of type IIA.     

Polar Symplectic Representations

We study polar representations in the sense of Dadok and Kac which are symplectic. We show that such representations are coisotropic and use this fact to give a classification. We also study their moment maps and prove that they separate closed orbits. Our work can also be seen as a specialization of some of the results of Knop on multiplicity free symplectic representations to the polar case.