Vector fields defined only over a part of a manifold give rise to indexes and to transfers. These local vector fields form a topological space whose relation to configuration spaces was studied by Dusa McDuff, and whose higher dimensional homotopy and homology promise invariants of parametrized families of local vector fields. We show that the assignment of the transfer to the vector field give...

We clarify the relation between 2-form Einstein gravity and its topological version. The physical space of the topological version is contained in that of the Einstein gravity. Moreover a new vector field is introduced into 2-form Einstein gravity to restore the large symmetry of its topological version. The wave function of the universe is obtained for each model. † Talk given at the Workshop ...

Sunada’s work on topological crystallography emphasizes the role of the ‘maximal abelian cover’ of a graph X. This is a covering space of X for which the group of deck transformations is the first homology group H1(X,Z). An embedding of the maximal abelian cover in a vector space can serve as the pattern for a crystal: atoms are located at the vertices, while bonds lie on the edges. We prove th...

We prove, in great detail, holomorphy E ⊓ C (I ,Π ) → C (I ,Π ) of the map (x , y) 7→ x ◦ [ id , y ] where [ id , y ] : I ∋ t 7→ (t , y (t)) for a real compact interval I , and where Π is a complex Banach space and E is a certain locally convex space of continuous functions x : I× (υsΠ ) → υsΠ for which x (t , ·) is holomorphic for all t ∈ I . We also discuss aspects of the application of this ...

For a new class of topological vector spaces, namely κ-normed spaces, an associated quasisemilinear topological preordered space is defined and investigated. This structure arise naturally from the consideration of a κ-norm, that is a distance function between a point and a G δ-subset. For it, analogs of the Hahn-Banach theorem are proved.

The terminology and notation used in this paper have been introduced in the following articles: [29], [16], [12], [13], [11], [1], [2], [32], [18], [30], [24], [25], [26], [27], [3], [9], [34], [35], [33], [28], [15], [21], [37], [10], [31], [20], [23], [5], [14], [6], [22], [8], [4], [19], [36], and [7]. Let I be a set and let J be a relational structure yielding many sorted set indexed by I. ...

on a hypergroupoid one can define a topology such that the hyperoperationis pseudocontinuous or continuous. in this paper we extend thisconcepts to the fuzzy case. we give a connection between the classical and thefuzzy (pseudo)continuous hyperoperations.