Integrability on Direct Limits of Banach Manifolds

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.

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Reflective subcategories

Glasgow Mathematical Journal ◽

10.1017/s0017089500010120 ◽

2000 ◽

Vol 42 (1) ◽

pp. 97-113 ◽

Cited By ~ 4

Author(s):

Juan Rada ◽

Manuel Saorín ◽

Alberto del Valle

Keyword(s):

Category Theory ◽

Full Subcategory ◽

Subject Classification ◽

Adjoint Functor ◽

Direct Limits ◽

The Relationship ◽

Good Behaviour

Given a full subcategory [Fscr ] of a category [Ascr ], the existence of left [Fscr ]-approximations (or [Fscr ]-preenvelopes) completing diagrams in a unique way is equivalent to the fact that [Fscr ] is reflective in [Ascr ], in the classical terminology of category theory.In the first part of the paper we establish, for a rather general [Ascr ], the relationship between reflectivity and covariant finiteness of [Fscr ] in [Ascr ], and generalize Freyd's adjoint functor theorem (for inclusion functors) to not necessarily complete categories. Also, we study the good behaviour of reflections with respect to direct limits. Most results in this part are dualizable, thus providing corresponding versions for coreflective subcategories.In the second half of the paper we give several examples of reflective subcategories of abelian and module categories, mainly of subcategories of the form Copres (M) and Add (M). The second case covers the study of all covariantly finite, generalized Krull-Schmidt subcategories of {\rm Mod}_{R}, and has some connections with the “pure-semisimple conjecture”.1991 Mathematics Subject Classification 18A40, 16D90, 16E70.

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First Direct Limits on Lightly Ionizing Particles with Electric Charge Less than e/6

Physical Review Letters ◽

10.1103/physrevlett.114.111302 ◽

2015 ◽

Vol 114 (11) ◽

Cited By ~ 10

Author(s):

R. Agnese ◽

A. J. Anderson ◽

D. Balakishiyeva ◽

R. Basu Thakur ◽

D. A. Bauer ◽

...

Keyword(s):

Electric Charge ◽

Direct Limits

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Equalization of finite flowers

Journal of Symbolic Logic ◽

10.1017/s0022481200028966 ◽

1988 ◽

Vol 53 (1) ◽

pp. 105-123

Author(s):

Stefano Berardi

Keyword(s):

Set Theory ◽

Forthcoming Paper ◽

Direct Limits

A dilator D is a functor from ON to itself commuting with direct limits and pull-backs. A dilator D is a flower iff D(x) is continuous. A flower F is regular iff F(x) is strictly increasing and F(f)(F(z)) = F(f(z)) (for f ϵ ON(x,y), z ϵ X).Equalization is the following axiom: if F, G ϵ Flr (class of regular flowers), then there is an H ϵ Flr such that F ° H = G ° H. From this we can deduce that if ℱ is a set ⊆ Flr, then there is an H ϵ Flr which is the smallest equalizer of ℱ (it can be said that H equalizes ℱ iff for every F, G ϵ ℱ we have F ° H = G ° H). Equalization is not provable in set theory because equalization for denumerable flowers is equivalent to -determinacy (see a forthcoming paper by Girard and Kechris).Therefore it is interesting to effectively find, by elementary means, equalizers even in the simplest cases. The aim of this paper is to prove Girard and Kechris's conjecture: “ is the (smallest) equalizer for Flr < ω” (where Flr < ω denotes the set of finite regular flowers). We will verify that is an equalizer of Flr < ω; we will sketch the proof that it is the smallest one at the end of the paper. We will denote by H.

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Banach manifolds and the Gelfand representation theorem

Indagationes Mathematicae (Proceedings) ◽

10.1016/s1385-7258(78)80006-7 ◽

1978 ◽

Vol 81 (1) ◽

pp. 157-164

Author(s):

S.T.M. Ackermans

Keyword(s):

Representation Theorem ◽

Banach Manifolds

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Noncommutative G-semihereditary rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498818500147 ◽

2018 ◽

Vol 17 (01) ◽

pp. 1850014 ◽

Cited By ~ 2

Author(s):

Jian Wang ◽

Yunxia Li ◽

Jiangsheng Hu

Keyword(s):

Associative Ring ◽

Sufficient Condition ◽

Projective Modules ◽

The Third ◽

Flat Modules ◽

Gorenstein Projective Modules ◽

Direct Limits ◽

Gorenstein Flat ◽

Finitely Presented ◽

Semihereditary Rings

In this paper, we introduce and study left (right) [Formula: see text]-semihereditary rings over any associative ring, and these rings are exactly [Formula: see text]-semihereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. Some new characterizations of left [Formula: see text]-semihereditary rings are given. Applications go in three directions. The first is to give a sufficient condition when a finitely presented right [Formula: see text]-module is Gorenstein flat if and only if it is Gorenstein projective provided that [Formula: see text] is left coherent. The second is to investigate the relationships between Gorenstein flat modules and direct limits of finitely presented Gorenstein projective modules. The third is to obtain some new characterizations of semihereditary rings, [Formula: see text]-[Formula: see text] rings and [Formula: see text] rings.

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Darboux chart on projective limit of weak symplectic Banach manifold

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887815500723 ◽

2015 ◽

Vol 12 (07) ◽

pp. 1550072 ◽

Cited By ~ 1

Author(s):

Pradip Mishra

Keyword(s):

Banach Space ◽

Symplectic Structure ◽

Reflexive Banach Space ◽

Projective Limit ◽

Fréchet Space ◽

Frechet Space ◽

Banach Manifold ◽

Boundedness Condition ◽

Projective System ◽

Banach Manifolds

Suppose M be the projective limit of weak symplectic Banach manifolds {(Mi, ϕij)}i, j∈ℕ, where Mi are modeled over reflexive Banach space and σ is compatible with the projective system (defined in the article). We associate to each point x ∈ M, a Fréchet space Hx. We prove that if Hx are locally identical, then with certain smoothness and boundedness condition, there exists a Darboux chart for the weak symplectic structure.

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