There Is Only One Notion of Differentiation

Differential linear logic was introduced as a syntactic proof-theoretic approach to the analysis of differential calculus. Differential categories were subsequently introduce to provide a categorical model theory for differential linear logic. Differential categories used two different approaches for defining differentiation abstractly: a deriving transformation and a coderiliction. While it was thought that these notions could give rise to distinct notions of differentiation, we show here that these notions, in the presence of a monoidal coalgebra modality, are completely equivalent

A polyocular framework for research on multifunctional farming and rural development

The paradox of multifunctionality is that, on the one hand, the specialized functionalities of agriculture only arise because of the functional differentiation of social systems and scientific disciplines and, on the other hand, multifunctionality can only enter as a way to mediate between conflicts, interests and fragmented knowledge when different functions and observations of functions combine. The aim of this paper is to contribute to a theoretical and methodological platform for multidisciplinary research on multifunctional farming. With the notions of polyocular cognition and polyocular communication we introduce a second order, interdisciplinary communication process that can meet the challenge of creating a shared view on multifunctional farming. Polyocular communication must be based on other rules than the rules of the involved disciplines. Whereas disciplinary communication is about providing consistent, efficient and precise knowledge in the context of a sharply delimited research world, polyocular communication is about extending a multidimensional space of understanding

A semiotic polyocular framework for multidisciplinary research in relation to multifunctional farming and rural development

The concept of multifunctional farming rises out of a problematization of the role of agriculture in society and, in particular, in relation to rural development. Hitherto multifunctional farming has primarily been used as a notion on the relationship between agriculture and society concerning the range of commodity and non-commodity goods that farms provide for society. But the agro-economic achievements together with societal development have led to a point where praxis is questioned and discourse potentially reopened. In an indirect way, the notion of multifunctionality reflects, that aspects not captured by the distinction between commodity and non-commodity need to be reintroduced. This paper offers a new framework (theoretical and methodical) suggesting a poly-ocular multidisciplinary approach and constructivist semiotic understanding of multifunctionality, which supports dialogue and interactions between the approaches, involved. Each research perspective has its own construction of the object of ‘farming’ and the ‘environment’ of farming; and thereby also its own perception of the functions and problems of farming. It therefore comes as no surprise that problems of communication are experienced between different perspectives, or that confusion on shared notions can cause frustrations and difficulties for multidisciplinary studies of multifunctionality. The present framework introduces a notion of multifunctionality, which enables the explicit handling of different perspectives by way of a distinction between the ‘immediate object’, as it appears to the observer, and the ‘dynamical object’, which represents the potentiality of the object in itself. From such semiotic point of view, the notion of multifunctionality becomes genuinely multidisciplinary. Multifunctionality cannot be reduced and included in one perspective, but has to be observed as a second order observation that involves reflexive communication between different perspectives and disciplines

Tensor Distributions in the Presence of Degenerate Metrics

Tensor distributions and their derivatives are described without assuming the presence of a metric. This provides a natural framework for discussing tensor distributions on manifolds with degenerate metrics, including in particular metrics which change signature.Comment: REVTeX, 19 pages; submitted to IJMP

Parametric Manifolds II: Intrinsic Approach

A parametric manifold is a manifold on which all tensor fields depend on an additional parameter, such as time, together with a parametric structure, namely a given (parametric) 1-form field. Such a manifold admits natural generalizations of Lie differentiation, exterior differentiation, and covariant differentiation, all based on a nonstandard action of vector fields on functions. There is a new geometric object, called the deficiency, which behaves much like torsion, and which measures whether a parametric manifold can be viewed as a 1-parameter family of orthogonal hypersurfaces.Comment: Plain TeX, 13 pages, no figure

On the Index and the Order of Quasi-regular Implicit Systems of Differential Equations

This paper is mainly devoted to the study of the differentiation index and the order for quasi-regular implicit ordinary differential algebraic equation (DAE) systems. We give an algebraic definition of the differentiation index and prove a Jacobi-type upper bound for the sum of the order and the differentiation index. Our techniques also enable us to obtain an alternative proof of a combinatorial bound proposed by Jacobi for the order. As a consequence of our approach we deduce an upper bound for the Hilbert-Kolchin regularity and an effective ideal membership test for quasi-regular implicit systems. Finally, we prove a theorem of existence and uniqueness of solutions for implicit differential systems

On Absolute Equivalence and Linearization I

In this paper, we study the absolute equivalence between Pfaffian systems with a degree 1 independence condition and obtain structural results, particularly for systems of corank 3. We apply these results to understanding dynamic feedback linearization of control systems with 2 inputs.Comment: 32 page