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Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach
, 2006
"... Table of contents ..."
Restriction Categories I
 Categories of Partial Maps, Theoret. Comput. Sci
, 2006
"... modality”) and a differential combinator, satisfying a number of coherence conditions. In ..."
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modality”) and a differential combinator, satisfying a number of coherence conditions. In
Direct limits of infinitedimensional Lie groups compared to direct limits in related categories
 J. Funct. Anal
, 2007
"... Let G be a Lie group which is the union of an ascending sequence G1 ⊆ G2 ⊆ · · · of Lie groups (all of which may be infinitedimensional). We study the question when G = lim Gn in the category of Lie groups, topological groups, smooth manifolds, resp., topological spaces. Full answers are obtaine ..."
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Let G be a Lie group which is the union of an ascending sequence G1 ⊆ G2 ⊆ · · · of Lie groups (all of which may be infinitedimensional). 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 σcompact smooth manifold M; and for test function groups C ∞ c (M,H) of compactly supported smooth maps with values in a finitedimensional 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 groupvalued analytic maps, or a weak direct product of Lie groups.
Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic (Extended Abstract)
"... 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 investigati ..."
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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
Rate distortion manifolds as model spaces for cognitive information
 In preparation
, 2007
"... 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 desc ..."
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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
Critical Sets of Smooth SOLUTIONS TO ELLIPTIC EQUATIONS in Dimension 3
, 2001
"... 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 1dimensional Hausdorff measure. This implies in particular that for a solution u ̸ ≡ 0 of (L0 + c)u = 0, with c ..."
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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 1dimensional 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 1dimensional Hausdorff measure.
Aspects of padic nonlinear functional analysis
 VOLOVICH (EDS.), PADIC MATHEMATICAL PHYSICS. 2ND INTERNATIONAL CONFERENCE (BELGRADE, 2005), AIP CONF. PROC. 826, AMER. INST. PHYSICS
, 2006
"... The article provides an introduction to infinitedimensional 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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The article provides an introduction to infinitedimensional differential calculus over topological fields and surveys some of its applications, notably in the areas of infinitedimensional Lie groups and dynamical systems.
Calculus of smooth functions between convenient vector spaces
 Aarhus Preprint Series 1984/85
"... We give an exposition of some basic results concerning remainders in Taylor expansions for smooth functions between convenient vector spaces, in the sense of Frölicher and Kriegl, cf. [2], [11], [3], [13]. We needed such results in [9], but could not find them in the works quoted. 1 The method we em ..."
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We give an exposition of some basic results concerning remainders in Taylor expansions for smooth functions between convenient vector spaces, in the sense of Frölicher and Kriegl, cf. [2], [11], [3], [13]. We needed such results in [9], but could not find them in the works quoted. 1 The method we employ is very puristic: we never have to consider limits, or any other analytic tools, except for finite dimensional vector spaces R n. In this sense, we carry Frölicher’s program of considering mutually balancing sets of curves R → X and functions X → R to the extreme. (Also, the puristic aspect makes it easy to transfer the theory to ”synthetic ” contexts, like [7].) Besides the introduction, where we recall some of the existing theory, the paper contains two sections: 1) on the general theory of Taylor remainders for smooth maps X → Y, where X and Y are convenient vector spaces; and 2) a more refined theory for the case where X = R n. First, we recall the notion of convenient vector space in the formulation of [3]: it is a vector space X over R, equipped with a linear subspace X ′ ∗This is a retyping of a preprint [8] with the same title, Aarhus Preprint Series 1984/85 No. 18. The bibliography has been updated, since [9] and [10] in the meantime have been published. Also, [4] has been published (1988). The numbering of the equations have changed, but the numbering of Propositions, Theorems, etc. is unchanged compared to the Preprint Version. 1 [4] does have some of these results; [8] is quoted there (Section 4.4) in connection with
Research Report
, 1995
"... 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) s ..."
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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 selfcontained 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 HochschildSerre spectral sequence for group extensions. This I hope to start writing on this this Type