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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
, 2009
"... 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.
Spacetime algebra as a powerful tool for electromagnetism
 PHYSICS REPORTS 589 (2015) 1–71
, 2015
"... ..."
AN ALGEBRA OF PIECES OF SPACE —
, 904
"... Abstract. We sketch the outlines of Gian Carlo Rota’s interaction with the ideas that Hermann Grassmann developed in his Ausdehnungslehre[13, 15] of 1844 and 1862, as adapted and explained by Giuseppe Peano in 1888. This leads us past what Gian Carlo variously called GrassmannCayley algebra and Pea ..."
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Abstract. We sketch the outlines of Gian Carlo Rota’s interaction with the ideas that Hermann Grassmann developed in his Ausdehnungslehre[13, 15] of 1844 and 1862, as adapted and explained by Giuseppe Peano in 1888. This leads us past what Gian Carlo variously called GrassmannCayley algebra and Peano spaces to the Whitney algebra of a matroid, and finally to a resolution of the question “What, really, was Grassmann’s regressive product?”. This final question is the subject of ongoing joint work with Andrea Brini, Francesco Regonati, and William Schmitt. 1. Almost ten years later We are gathered today in order to renew and deepen our recollection of the ways in which our paths intersected that of Gian Carlo Rota. We do this in poignant sadness, but with a bittersweet touch: we are pleased to have this opportunity to meet and to discuss his life and work, since we know how Gian Carlo transformed us through his friendship and his love of mathematics. We will deal only with the most elementary of geometric questions; how to represent pieces of space of various dimensions, in their relation to one another. It’s a simple story, but one that extends over a period of some 160 years. We’ll start and finish with Hermann Grassmann’s project, but the trail will lead us by Giuseppe Peano, Hassler Whitney, to Gian Carlo Rota and his colleagues. Before I start, let me pause for a moment to recall a late afternoon at the Accademia Nazionale dei Lincei, in 1973, on the eve of another talk I was petrified to give, when Gian Carlo decided to teach me how to talk, so I wouldn’t make a fool of myself the following day. The procedure was for me to start my talk, with an audience of one, and he would interrupt whenever there was a problem. We were in that otherwise empty conference hall for over two hours, and I never got past my first paragraph. It was terrifying, but it at least got me through the first battle with my fears and apprehensions, disguised as they usually are by timidity, selfeffacement, and other forms of apologetic behavior. 2. Synthetic Projective Geometry Grassmann’s plan was to develop a purely formal algebra to model natural (synthetic) operations on geometric objects: flat, or linear pieces of space of all possible dimensions. His approach was to be synthetic, so that the symbols in his algebra
A case study in geometric algebra: Fitting room models to 3D point clouds Author:
, 2011
"... Many geometrical problems exist which have been researched thoroughly, but always using classical methods such as linear algebra as a framework for the problem. As linear algebra is an algebra based on coordinates and numbers as basic elements of computation, this leads to longwinded and nonunivers ..."
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Many geometrical problems exist which have been researched thoroughly, but always using classical methods such as linear algebra as a framework for the problem. As linear algebra is an algebra based on coordinates and numbers as basic elements of computation, this leads to longwinded and nonuniversal code. Geometric algebra is an alternative formalism in which geometric objects are the basic elements of computation. Using this formalism to represent geometrical problems can often yield more readable and more compact code. In this paper we present a case study of such a problem – specifically fitting room models to 3D point clouds – and the advantages geometric algebra has over classical methods in solving