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modality”) and a differential combinator, satisfying a number of coherence conditions. In

Let G be a Lie group which is the union of an ascending sequence G1 ⊆ G2 ⊆ · · · of Lie groups (all of which may be infinite-dimensional). We study the question when G = lim Gn in the category of Lie groups, topological groups, smooth manifolds, resp., topological spaces. Full answers are obtained for G the group Diffc(M) of compactly supported C∞-diffeomorphisms of a σ-compact smooth manifold M; and for test function groups C ∞ c (M,H) of compactly supported smooth maps with values in a finite-dimensional Lie group H. We also discuss the cases where G is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie group-valued analytic maps, or a weak direct product of Lie groups.

Abstract. In the first part of the paper I investigate categorical models of multiplicative biadditive intuitionistic linear logic, and note that in them some surprising coherence laws arise. The thesis for the second part of the paper is that these models provide the right framework for investigating differential structure in the context of linear logic. Consequently, within this setting, I introduce a notion of creation operator (as considered by physicists for bosonic Fock space in the context of quantum field theory), provide an equivalent description of creation operators in terms of creation maps, and show that they induce a differential operator satisfying all the basic laws of differentiation (the product and chain rules, the commutation relations, etc.). 1

Let u ̸ ≡ const satisfy an elliptic equation L 0u ≡ Σai,jDiju + ΣbjDju = 0 with smooth coefficients in a domain in R 3. It is shown that the critical set |∇u | −1 {0} has locally finite 1-dimensional Hausdorff measure. This implies in particular that for a solution u ̸ ≡ 0 of (L0 + c)u = 0, with c ∈ C ∞ , the critical zero set u −1 {0} ∩ |∇u | −1 {0} has locally finite 1-dimensional Hausdorff measure.

The rate distortion manifold is considered as a carrier for elements of the theory of information proposed by C. E. Shannon combined with the semantic precepts of F. Dretske’s theory of communication. This type of information space was suggested by R. Wallace as a possible geometric–topological descriptive model for incorporating a dynamic information based treatment of the Global Workspace theory of B. Baars. We outline a more formal mathematical description for this class of information space and further clarify its structural content and overall interpretation within prospectively a broad range of cognitive situations that apply to individuals, human institutions, distributed cognition and massively parallel intelligent machine design. Povzetek: Predstavljena je formalna definicija prostora za opisovanje kognitivnih procesov. 1

The article provides an introduction to infinite-dimensional differential calculus over topological fields and surveys some of its applications, notably in the areas of infinitedimensional Lie groups and dynamical systems.

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tension cohomology of my Advances paper to these algebras. During the months of June and July I have been studying Homological Perturbation Theory, created by Shih, Ronald Brown, Gugenheim, Stasheff, Moore and Cartan for studying the cohomology of fibrations, algebras, nilpotent groups and K(ß;n) spaces. The reason I find it imperative to study this theory is that my work on the combinatorial extension cohomology of groups, Lie algebras and associative algebras is a special case of this theory. I derived the results for groups in a very self-contained and ad hoc manner, and to say that this is a special case of Homological Perturbation Theory is not to say that it will be a trivial matter to sort out all the details in the case of group extensions, which has never been done. In fact, it will be possible to give an independent derivation of the Hochschild-Serre spectral sequence for group extensions. This I hope to start writing on this this Type

Résumé. Cet article développe une théorie générale des structures géométriques sur des variétés, basée sur la théorie des catégories. De nombreuses généralisations connues des variétés et des variétés de Riemann rentrent dans le cadre de cette théorie générale. On donne aussi