Symmetries in finite order variational sequences

We refer to Krupka's variational sequence, i.e. the quotient of the de Rham sequence on a finite order jet space with respect to a variationally trivial subsequence. Among the morphisms of the variational sequence there are the Euler-Lagrange operator and the Helmholtz operator.In this note we show that the Lie derivative operator passes to the quotient in the variational sequence. Then we define the variational Lie derivative as an operator on the sheaves of the variational sequence. Explicit representations of this operator give us some abstract versions of Noether's theorems, which can be interpreted in terms of conserved currents for Lagrangians and Euler-Lagrange morphisms.     

On Finite Morphisms of Ruled Surfaces

We prove that a surjective morphism :XX between ruled surfaces is finite and it descends to a finite morphism :CC between base curves of X and X. When is restricted to the fibres of X, it has a constant degree, say a, and then deg=a deg. In addition, we have several properties on the inverse image of a minimal section and a fibre of X as well as on the direct images. We also investigate precisely the case when both C and C are elliptic and X is the fibre product C ×_C X especially.     

Components of the Fundamental Category

In this article we study the fundamental category (Goubault and Raussen, 2002; Goubault, 2000) of a partially ordered topological space (Nachbin, 1965; Johnstone, 1982), as arising in, e.g., concurrency theory (Fajstrup et al., 1999). The algebra of dipaths modulo dihomotopy (the fundamental category) of such a po-space is essentially finite in a number of situations: We define a component category of a category of fractions with respect to a suitable system, which contains all relevant information. Furthermore, some of these simpler invariants are conjectured to also satisfy some form of a van Kampen theorem, as the fundamental category does (Goubault, 2002; Grandis, 2001). We end up by giving some hints about how to carry out some computations in simple cases.     

Initial Morphisms versus Embeddings

There is a usual practice in topology of completing an object via an initial morphism XY and then using some ad hoc method of transforming that initial morphism into an embedding XY. In the setting of categorical topology we show that there is a bicoreflective general process available for carrying out such constructions. We further show that this bicoreflector can be adapted to respect a closure operator when the topological construct is endowed with such.     

具有泛分解态射的广义(I,…,j)逆

给出了范畴中具有泛分解态射的广义 (i,… ,j)逆存在的一些充要条件 ,证明了态射的广义Moore Penrose逆的充要条件及其表达式 ,推广了关于态射的(i,… ,j) 逆的一些结果。     

关于范畴中态射的广义Moore-Penrose逆

给出了范畴中态射的广义（i,…,j)逆存在的某些充要条件，证明了态射的广义Moore-Penrose逆的表达式，推广了态射的Moore-Penrose逆的相应结果。     

G2型单代数群的单限制模的G1-扩张

设G是特征p的代数闭域K上单连通半单代数群，G1是第一次Frobenius态射的核，即G1＝KerF。要确定单G模的G－扩张必须先确定限制单模的G1－扩张。本文就是确定出特征p=5的代数闭域K上G2型单代数群的所有单G1－模的扩张群。     

具有广义分解态射的加权Moore-Penrose逆

研究范畴中态射的加权Moore-Penrose逆,利用态射广义分解的性质给出了态射加权Moore-Penrose逆存在的一些充要条件,导出了态射的加权Moore-Penrose逆的表达式,推广了态射Moore-Penrose逆的相应结果.