scholarly journals On Darboux Theorem for symplectic forms on direct limits of symplectic Banach manifolds

2018 ◽  
Vol 15 (12) ◽  
pp. 1850206
Author(s):  
Fernand Pelletier
Keyword(s):  
Direct Limit ◽  
Symplectic Forms ◽  
Banach Manifolds ◽  
Finite Dimensional ◽  
Direct Limits ◽  
Darboux Theorem

Given an ascending sequence of weak symplectic Banach manifolds on which the Darboux Theorem is true, we can ask about conditions under which the Darboux Theorem is also true on the direct limit. We will show that, in general, without very strong conditions, the answer is negative. In particular, we give an example of an ascending symplectic Banach manifolds on which the Darboux Theorem is true but not on the direct limit. In the second part, we illustrate this discussion in the context of an ascending sequence of Sobolev manifolds of loops in symplectic finite-dimensional manifolds. This context gives rise to an example of direct limit of weak symplectic Banach manifolds on which the Darboux Theorem is true around any point.

scholarly journals K-loop Structures Raised by the Direct Limits of Pseudo Unitary $U(p, a_{n})$ and Pseudo Orthogonal $O(p, a_{n})$ Groups

2017 ◽  
Vol 9 (5) ◽  
pp. 37
Author(s):  
ALPER BULUT
Keyword(s):  
Relativistic Theory ◽  
Direct Limit ◽  
Infinite Dimensional ◽  
Dimensional Group ◽  
Inverse Property ◽  
Finite Dimensional ◽  
Direct Limits ◽  
Bol Loop

A left Bol loop satisfying the automorphic inverse property is called a K-loop or a gyrocommutative gyrogroup. K-loops have been in the centre of attraction since its first discovery by A.A. Ungar in the context of Einstein's 1905 relativistic theory. In this paper some of the infinite dimensional K-loops are built from the direct limit of finite dimensional group transversals.

scholarly journals Real Dimension Groups

2013 ◽  
Vol 56 (3) ◽  
pp. 551-563 ◽  
Author(s):  
David Handelman
Keyword(s):  
Direct Limit ◽  
Polynomial Rings ◽  
Vector Spaces ◽  
Real Vector ◽  
Product Decomposition ◽  
Finite Dimensional ◽  
Dimension Groups ◽  
Ordered Vector Spaces ◽  
Direct Limits ◽  
Directed Set

AbstractDimension groups (not countable) that are also real ordered vector spaces can be obtained as direct limits (over directed sets) of simplicial real vector spaces (finite dimensional vector spaces with the coordinatewise ordering), but the directed set is not as interesting as one would like; for instance, it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over a countable directed set. It turns out this is the case when the group is additionally simple, and it is shown that the latter have an ordered tensor product decomposition. In an appendix, we provide a huge class of polynomial rings that, with a pointwise ordering, are shown to satisfy interpolation, extending a result outlined by Fuchs.

Inverse limit stability for semiflows

2000 ◽  
Vol 20 (2) ◽  
pp. 453-471 ◽  
Author(s):  
HIROSHI IKEDA
Keyword(s):  
Inverse Limit ◽  
Compact Manifolds ◽  
Banach Manifolds ◽  
Finite Dimensional

On semiflows on Banach manifolds, Quandt (1988 and 1989) stated that some special semiflows were inverse limit stable. However, the concepts and proofs used to show inverse limit stability of semiflows are rough and ambiguous. We consider semiflows on finite-dimensional compact manifolds or on finite-dimensional compact non-singular branched manifolds. In this paper we show that more special semiflows are inverse limit stable. That is, Anosov semiflows are inverse limit stable.

scholarly journals On Direct Limit Closed Classes of Algebras

2016 ◽  
Vol 45 (3/4) ◽  
Author(s):  
Emília Halušková
Keyword(s):  
Direct Limit ◽  
Direct Limits

Axiomatic classes of algebras of a given type which are closed with respect to direct limits are studied in this paper.

scholarly journals On the generalization of the Darboux theorem

2019 ◽  
Vol 12 (2) ◽  
Author(s):  
Kaveh Eftekharinasab
Keyword(s):  
Symplectic Form ◽  
Vector Fields ◽  
Projective Limit ◽  
Smoothness Condition ◽  
General Context ◽  
Classifying Space ◽  
Banach Manifolds ◽  
Dependent Vector ◽  
Darboux Theorem ◽  
General Vector

Darboux theorem to more general context of Frechet manifolds we face an obstacle:  in general vector fields do not have local flows. Recently, Fr\'{e}chet geometry has been developed in terms of projective limit of Banach manifolds. In this framework under an appropriate Lipchitz condition The Darboux theorem asserts that a symplectic  manifold $(M^{2n},\omega)$ is locally symplectomorphic to $(R^{2n}, \omega_0)$, where $\omega_0$  is the standard symplectic form on  $R^{2n}$. This theorem was proved by Moser in 1965, the idea of proof, known as the Moser’s trick, works in many situations. The Moser tricks is to construct an appropriate isotopy $ \ff_t $  generated by a time-dependent vector field $ X_t  $ on $M$ such that $ \ff_1^{*} \omega = \omega_0$. Nevertheless,  it was showed by Marsden that Darboux theorem is not valid for weak symplectic Banach manifolds. However, in 1999 Bambusi showed that if we  associate to each point of a Banach manifold a suitable Banach space (classifying space) via a given symplectic form then the Moser trick can be applied to obtain the theorem if the  classifying space does not depend on the point of the manifold and a suitable smoothness condition holds.  If we want to try to generalize the local flows exist and with some restrictive conditions the Darboux theorem was proved by Kumar.  In this paper we consider the category of so-called bounded Fr\'{e}chet manifolds and prove that in this category vector fields have local flows and following the idea of Bambusi we associate to each point of a manifold a Fr\'{e}chet space independent of the choice of the point and with the assumption of bounded smoothness on vector fields  we prove the Darboux theorem.

scholarly journals SPECIALIZATIONS OF DIRECT LIMITS AND OF LOCAL COHOMOLOGY MODULES

2007 ◽  
Vol 50 (2) ◽  
pp. 459-475 ◽  
Author(s):  
Dam Van Nhi
Keyword(s):  
Local Cohomology ◽  
Direct Limit ◽  
Finitely Generated ◽  
Direct System ◽  
Local Cohomology Modules ◽  
Direct Limits ◽  
Cohomology Modules

AbstractIn this paper we introduce the specialization of an $S$-module which is a direct limit of a direct system of finitely generated $S$-modules indexed by $\mathbb{N}$. This specialization preserves the Buchsbaum property, the generalized Cohen–Macaulay property and the Castelnuovo–Mumford regularity of a module.

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