*-Polynomial Identities of Matrices with the Symplectic Involution: The Low Degrees

Abstract

The *-polynomial identities of minimal degree of M _n (F) are determined for n = 2, 4, * the symplectic involution.     

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．     

辛代数的不变双线性型

考察了辛代数和与它相联系的李三系的双线性型之间的关系,并证明了辛代数的反对称不变双线性型可以唯一扩张到与它相联系的李三系中.作为这种关系的一个应用,得到了二次辛代数是单辛代数的一个充要条件,并证明二次辛代数的唯一分解定理.     

Poisson Brackets of Even Symplectic Forms on the Algebra of Differential Forms

Given a symplectic form and a pseudo-Riemannian metric on a manifold, a nondegenerate even Poisson bracket on the algebra of differential forms is defined and its properties are studied. A comparison with the Koszul–Schouten bracket is established.     

Symplectic resolutions, Lefschetz property and formality

We introduce a method to resolve a symplectic orbifold(M,ω) into a smooth symplectic manifold . Then we study how the formality and the Lefschetz property of are compared with that of (M,ω). We also study the formality of the symplectic blow-up of (M,ω) along symplectic submanifolds disjoint from the orbifold singularities. This allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the Lefschetz property but is not formal, therefore giving a counter-example to a conjecture of Babenko and Taimanov.     

中间逻辑中的良构范式

本文在中间逻辑中引入了良构范式的概念,为该类范式的研究提出了一个通用方法,并应用该方法证明了HT逻辑是存在一般蕴含范式的最弱中间逻辑,经典命题逻辑CPL是存在限制蕴含范式的仅有中间逻辑.     

Embedded Surfaces for Symplectic Circle Actions

The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces.More precisely, it is shown that(1) if(M, ω) admits a Hamiltonian S~1-action, then there exists a two-sphere S in M with positive symplectic area satisfying c1(M, ω), [S] 0,and(2) if the action is non-Hamiltonian, then there exists an S~1-invariant symplectic2-torus T in(M, ω) such that c1(M, ω), [T] = 0. As applications, the authors give a very simple proof of the following well-known theorem which was proved by Atiyah-Bott,Lupton-Oprea, and Ono: Suppose that(M, ω) is a smooth closed symplectic manifold satisfying c1(M, ω) = λ· [ω] for some λ∈ R and G is a compact connected Lie group acting effectively on M preserving ω. Then(1) if λ 0, then G must be trivial,(2) if λ = 0, then the G-action is non-Hamiltonian, and(3) if λ 0, then the G-action is Hamiltonian.     

Skew-Hadamard Matrices and the Smith Normal Form

We show that the Smith normal form of every skew-Hadamard matrix of order 4m is diag[1,2,...,2, 2m,...,2m,4m]     

SYMPLECTIC SCHEMES FOR NONAUTONOMOUS HAMILTONIAN SYSTEM

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四元数矩阵的Jordan标准形

本文是在四元数矩阵的重行列式理论的基础上，直接利用四元数的乘法证明了。任意一个四元数矩阵都相似于特征主值表征的Ｊｏｒｄａｎ标准形及其唯一性．     

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.     

Error Analysis of the Symplectic Lanczos Method for the Symplectic Eigenvalue Problem

A rounding error analysis of the symplectic Lanczos algorithm for the symplectic eigenvalue problem is given. It is applicable when no break down occurs and shows that the restriction of preserving the symplectic structure does not destroy the characteristic feature of nonsymmetric Lanczos processes. An analog of Paige's theory on the relationship between the loss of orthogonality among the Lanczos vectors and the convergence of Ritz values in the symmetric Lanczos algorithm is discussed. As to be expected, it follows that (under certain assumptions) the computed J-orthogonal Lanczos vectors loose J-orthogonality when some Ritz values begin to converge.     

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 geometry and dissipative differential operators

Everitt and Markus characterized the domains of self-adjoint differential operators in terms of Lagrangian subspaces of complex symplectic spaces. In this paper we define Dissipative and strictly Dissipative subspaces for complex symplectic spaces and characterize the domains of dissipative and strictly dissipative differential operators in terms of these subspaces.     

矩阵对的相似标准形

设A,B,C,D都是n阶方阵,矩阵对(A,B)相似于矩阵对(C,D),如果存在n阶可逆矩阵P,使得P-1AP=C,P-1BP=D.本文借助Belitskii约化算法,提供一种在相似变化下化任一n阶矩阵对为标准形的有效方法,该方法可以看作Jordan标准形的推广.     

Differential Posets and Smith Normal Forms

We conjecture a strong property for the up and down maps U and D in an r-differential poset: DU + tI and UD + tI have Smith normal forms over . In particular, this would determine the integral structure of the maps U, D, UD, DU, including their ranks in any characteristic. As evidence, we prove the conjecture for the Young-Fibonacci lattice Y F studied by Okada and its r-differential generalizations Z(r), as well as verifying many of its consequences for Young’s lattice Y and the r-differential Cartesian products Y ^r .     

Comparison of Normal Linear Experiments by Quadratic Forms

Let X and Y be observation vectors in normal linear experiments =N(A, V) and F = N(B, W). We write > Fif for any quadratic form YGY there exists a quadratic formXHX such that E(XHX) = E(Y'GY) and var(X'HX) var(Y'GY).The relation > is characterized by the matrices A, B, V and W. Moreoversome connections with known orderings of linear experiments are given.     

精细辛几何算法的误差估计

该文讨论了精细辛几何算法的计算误差,先展开二阶和四阶精细辛几何算法的表达式得到误差同精细剖分数目的关系,然后分析了任意阶精细辛几何算法的误差,得到了一致简洁的结果,总的误差可近似表示为单个精细步长的误差乘以剖分数目,最后讨论了在要求控制精度下剖分数目的选取,该方法克服了算法精度对积分时间步长的依赖性.     

Root Systems In Finite Symplectic Vector Spaces

We study realizations of root systems in possibly degenerate symplectic vector spaces over finite fields, up to symplectic isomorphisms. The main result of this article is the classification of such realizations for the field 𝔽₂. Thereby, each root system requires a specific degree of degeneracy of the symplectic vector space. Our main motivation for this article is that for each such realization of a root system we can construct a Nichols algebra over a nonabelian group.