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The Garden of Knowledge as a Knowledge Manifold  A Conceptual Framework for Computer Supported Subjective Education
 CID17, TRITANAD9708, DEPARTMENT OF NUMERICAL ANALYSIS AND COMPUTING SCIENCE
, 1997
"... This work presents a unied patternbased epistemological framework, called a Knowledge Manifold, for the description and extraction of knowledge from information. Within this framework it also presents the metaphor of the Garden Of Knowledge as a constructive example. Any type of KM is defined in te ..."
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This work presents a unied patternbased epistemological framework, called a Knowledge Manifold, for the description and extraction of knowledge from information. Within this framework it also presents the metaphor of the Garden Of Knowledge as a constructive example. Any type of KM is defined in terms of its objective calibration protocols  procedures that are implemented on top of the participating subjective knowledgepatches. They are the procedures of agreement and obedience that characterize the coherence of any type of interaction, and which are used here in order to formalize the concept of participator consciousness in terms of the inversedirect limit duality of Category Theory.
A comparative review of recent researches in geometry
, 1872
"... gen, A. Deichert), had but a limited circulation at first. With this I could be satisfied more easily, as the views developed in the Programme could not be expected at first to receive much attention. But now that the general development of mathematics has taken, in the meanwhile, the direction corr ..."
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gen, A. Deichert), had but a limited circulation at first. With this I could be satisfied more easily, as the views developed in the Programme could not be expected at first to receive much attention. But now that the general development of mathematics has taken, in the meanwhile, the direction corresponding precisely to these views, and particularly since Lie has begun the publication in extended form of his Theorie der Transformationsgruppen ([23]), it seems proper to give a wider circulation to the expositions in my Programme. An Italian translation by M. Gina Fano was recently published in the Annali di Matematica, ser. 2, vol. 17. A kind reception for the English translation, for which I am much indebted to Mr. Haskell, is likewise desired. The translation is an absolutely literal one; in the two or three places where a few words are changed, the new phrases are enclosed in square brackets []. In the same way are indicated a number of additional footnotes which it seemed desirable to append, most of them having already appeared in the Italian translation. F. KLEIN.
LECTURES ON CHAINLET GEOMETRY – NEW TOPOLOGICAL METHODS IN GEOMETRIC MEASURE THEORY
"... Abstract. These draft notes are from a graduate course given by the author in Berkeley during the spring semester of 2005. They cover the basic ideas of a new, geometric approach to geometric measure theory. They begin with a new theory of exterior calculus at a single point. This infinitesimal theo ..."
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Abstract. These draft notes are from a graduate course given by the author in Berkeley during the spring semester of 2005. They cover the basic ideas of a new, geometric approach to geometric measure theory. They begin with a new theory of exterior calculus at a single point. This infinitesimal theory extends, by linearity, to a discrete exterior theory, based at finitely many points. A general theory of calculus culminates by taking limits in Banach spaces, and is valid for domains called “chainlets ” which are defined to be elements of the Banach spaces. Chainlets include manifolds, rough domains (e.g., fractals), soap films, foliations, and Euclidean space. Most of the work is at the level of the infinitesimal calculus, at a single point. The number of limits needed to get to the full theory is minimal. Tangent spaces are not used in these notes, although they can be defined within the theory. This new approach is made possible by giving the Grassmann algebra more geometric structure. As a result, much of geometric measure theory is simplified. Geometry is restored and significant results from the classical theory are expanded. Applications include existence of solutions to a problem of Plateau, an optimal GaussGreen theorem and new models for Maxwell’s equations. 1. Polyhedral chains Chainlet geometry is first developed for domains in Euclidean space (e.g., submanifolds), and later expanded to abstract manifolds ∗.
Superspace: a Comfortably Vast Algebraic Variety 1
, 901
"... Supersymmetry has been studied for over three decades by physicists, its superset even longer by mathematicians, and superspace has proven to be very useful both conceptually and in facilitating computations. However, the (1) necessary ..."
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Supersymmetry has been studied for over three decades by physicists, its superset even longer by mathematicians, and superspace has proven to be very useful both conceptually and in facilitating computations. However, the (1) necessary
Hopf Maps, Lowest Landau Level, and Fuzzy Spheres ⋆
"... doi:10.3842/SIGMA.2010.071 Abstract. This paper is a review of monopoles, lowest Landau level, fuzzy spheres, and their mutual relations. The Hopf maps of division algebras provide a prototype relation between monopoles and fuzzy spheres. Generalization of complex numbers to Clifford algebra is exac ..."
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doi:10.3842/SIGMA.2010.071 Abstract. This paper is a review of monopoles, lowest Landau level, fuzzy spheres, and their mutual relations. The Hopf maps of division algebras provide a prototype relation between monopoles and fuzzy spheres. Generalization of complex numbers to Clifford algebra is exactly analogous to generalization of fuzzy twospheres to higher dimensional fuzzy spheres. Higher dimensional fuzzy spheres have an interesting hierarchical structure made of “compounds ” of lower dimensional spheres. We give a physical interpretation for such particular structure of fuzzy spheres by utilizing Landau models in generic even dimensions. With Grassmann algebra, we also introduce a graded version of the Hopf map, and discuss its relation to fuzzy supersphere in context of supersymmetric Landau model.
GeoMAP Unification
"... The aim of this chapter is to contribute to David Hestenes ’ vision formulated on his website ..."
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The aim of this chapter is to contribute to David Hestenes ’ vision formulated on his website
Physics Reports 589 (2015) 1–71 Contents lists available at ScienceDirect Physics Reports
"... journal homepage: www.elsevier.com/locate/physrep ..."
MATTHEW O’BRIEN: AN INVENTOR OF VECTOR ANALYSIS
"... Abstract. In the midnineteenth century, several mathematicians were engaged in a search for a new symbolism and methodology to express and solve physical problems in three dimensions. Hamilton’s quaternions promised much but ultimately proved unequal to the task. Around 1880, the work of Gibbs and ..."
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Abstract. In the midnineteenth century, several mathematicians were engaged in a search for a new symbolism and methodology to express and solve physical problems in three dimensions. Hamilton’s quaternions promised much but ultimately proved unequal to the task. Around 1880, the work of Gibbs and Heaviside led to the modern formulation of vector analysis. But decades earlier an essentially equivalent approach was formulated by Matthew O’Brien and applied to problems in geometry, statics, dynamics and optics. Although his work deserved greater attention, it was ignored by his contemporaries. One reason was that he was overshadowed by the towering and influential figure of Hamilton. 1.