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Incidence loops and their geometry
1992Publisher Summary This chapter discusses the concept of incidence loops and their geometry. An incidence group (P, L,·) is a group (P,·) together with a structure (P, L) of an incidence space such that both structures are compatible. The notion of incidence group can be generalized by weakening the assumptions concerning the algebraic structure of P;
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Incidence problems in discrete geometry
2017Over the past decade, discrete geometry research has flourished with clever uses of algebraic methods. The polynomial method has had a deep impact on a wide collection of results in combinatorics, such as tight asymptotic lower bounds on finite field Kakeya and Nikodym sets, near optimal lower bound for Erdos' distinct distances problem, and improved ...
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Affine Geometry: Incidence with Parallelism (IP)
2015This brief chapter introduces the notion of parallelism, discusses the two forms of the parallel axiom, defines affine geometry, and proves five elementary theorems relating to intersecting planes and parallel lines.
Keith G. Calkins +3 more
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Formalization of Hilbert's geometry of incidence and parallelism
Synthese, 1997The author first describes how \textit{D. Hilbert} changed the phrasing of his axioms of incidence in the various early editions of his Grundlagen der Geometrie [(Teubner, Leipzig) (1899; JFM 30.0424.01); second edition (1903; JFM 34.0523.01); seventh edition (1930; JFM 56.0481.01)], in which ``bestimmen'' gave way to ``es gibt''.
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CHARACTERIZATIONS OF THE LIE INCIDENCE GEOMETRIES
1983INTRODUCTION A yery famous theorem (associated with the names Hilbert, von Staudt, Veblen and Young) characterizes projective spaces of dimension greater than 2 as linear incidence systems satisfying a certain (variously named) axiom. By the term “characterization”, one means a complete classification in terms of division rings.
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Geometries for Grazing Incidence Mirrors
2023Michael J. Pivovaroff, Takashi Okajima
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