Journal ArticleDOI
Derivative Strings, Differential Strings and Semi-Holonomic Jets
TLDR
In this paper, the derivative strings of Barndorff-Nielsen and the differential strings of Blaesild & Mora are considered from the coordinate-free viewpoint. And it is shown that the derivative string of given length and degree over a differentiable manifold form a vector bundle associated to a principal bundle of higher-order frames and that there is an analogous result for differential strings.Abstract:
The derivative strings of Barndorff-Nielsen and the differential strings of Blaesild & Mora are considered here from the coordinate-free viewpoint. It is shown that the derivative strings of given length and degree over a differentiable manifold form a vector bundle associated to a principal bundle of higher-order frames and that there is an analogous result for differential strings. Bundles of derivative strings are identified with vector bundles obtained from 0-truncated versions of Ehresmann's semi-holonomic jets by dualization and by taking tensor products. Similarly, bundles of differential strings are identified with vector bundles obtained from semiholonomic jets of certain tensor fields.read more
Citations
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Journal ArticleDOI
Connections on non-parametric statistical manifolds by Orlicz space geometry
Paolo Gibilisco,Giovanni Pistone +1 more
TL;DR: In this paper, it was shown that the manifold structure on ℳμ is the natural domain for the mixture connection and for its dual, the exponential connection, and that the bundle connection pair (ℱα, ∇α) is simply (isomorphic to) the pull-back of the Amari embedding.
Journal ArticleDOI
Stochastic calculus, statistical asymptotics, Taylor strings and phyla
TL;DR: In this paper, the authors provide an exposition without proofs of the theory of higher order calculus known as (Taylor or statistical) string theory, which has a potentially wider role to play in geometric approaches in asymptotic studies.
Journal ArticleDOI
Higher-Order Tensors, Strings and New Tensors
Alan L. Carey,Michael K. Murray +1 more
TL;DR: In this article, a generalization of tensors such as strings and new-tensors is considered and their properties can be described by the representation theory of an infinite-dimensional group.
Journal ArticleDOI
Finite-dimensional algebraic representations of the infinite phylon group
TL;DR: The concept of phylon is introduced as a generalisation of derivative strings, differential strings and new tensors in this article, and the behaviour of phyla under change of coordinates is given by finite-dimensional algebraic representations of a very large group, the infinite phylon group.
References
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Book
Lectures on Differential Geometry
TL;DR: In this article, the authors present an algebraic model of transitive differential geometry and the integrability problem for geometrical structures on manifolds, which they call integral calculus on manifold.
Book
The Geometry of Jet Bundles
TL;DR: In this article, the authors present a glossary of symbols for linear operations on general bundles, including first-order and second-order jet bundles, as well as higher order and infinite jet bundles.
Journal ArticleDOI
Foundations of Differential Geometry, Volume I.
Book
Parametric statistical models and likelihood
TL;DR: In this paper, the authors present a generalization of the p*-model to include ancillary statistics, such as partial sufficiency, partial ancillarity, and conditionality.