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Geometric Hyperplanes of Lie Incidence Geometries
Geometriae Dedicata, 1997Let \(\Gamma=({\mathcal P},{\mathcal L})\) be a geometry of points and lines. A subspace of \(\Gamma\) is a set of points which contains every line that meets it in at least two points. An embedding \(\mu\) of \(\Gamma\) in a finite-dimensional vector space \(V\) consists of a map \(\mu_1\) of \({\mathcal P}\) into the set of 1-subspaces of \(V\) and a
Cooperstein, Bruce N., Shult, Ernest E.
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On the foundations of incidence geometry
Geometriae Dedicata, 1988Diagram geometries and chamber systems of various types have been used and investigated intensively in recent years - not only in finite group theory, but in other areas as well. This development has led to a need for some clarification of the variations and generalizations introduced by the many authors, and for a discussion of the different axiomatic
Buekenhout, Francis, Buset, Dominique
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1981
In this section we shall define the notions of an abstract geometry and an incidence geometry. These are given by listing a set of axioms that must be satisfied. After the definitions are made, we will give a number of examples which will serve as models for these geometries.
Richard S. Millman, George D. Parker
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In this section we shall define the notions of an abstract geometry and an incidence geometry. These are given by listing a set of axioms that must be satisfied. After the definitions are made, we will give a number of examples which will serve as models for these geometries.
Richard S. Millman, George D. Parker
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Modal logics for incidence geometries
Journal of Logic and Computation, 1997Let \(S=(P,L,\text{in})\) be an incidence plane by the well-known axioms: \(P,L\neq\emptyset\); \(\text{in}\subseteq P\times L\); \(P\cap L=\emptyset\); two points are together incident with one and only one line; each line contains at least two different points; each point belongs at least to two different lines.
Balbiani, Philippe +3 more
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On the Incidence Geometry of Grassmann Spaces
Geometriae Dedicata, 1999The main result is a characterization of the Grassmann space \({\mathbf G}\) of a projective space \(\mathcal P\). By definition, the point set \(P\) of \({\mathbf G}\) is the set of lines of \(\mathcal P\), the line set \(\mathcal L\) of \({\mathbf G}\) consists of all plane line pencils in \(\mathcal P\).
FERRARA DENTICE, Eva, MELONE N.
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Two incidence propositions in chain geometries
Monatshefte für Mathematik, 2011In the first half of the paper, the authors deal with Jordan systems of an associative \(F\)-algebra \(A\) with \(1\); \(F\) is a field and \(A^{*}\) denotes the group of units of \(A\). An element \(a\in A\) is called \textit{quadratic}, if there exists an \(\bar a\in A\) with \(a+{\bar a}\in F\) and \(a\cdot{\bar a}\in F\); \(a\cdot{\bar a}=:N(a ...
Özcan, Münevver, Herzer, Armin
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Preliminaries and Incidence Geometry (I)
2015This chapter contains a brief summary of several types of mathematical knowledge needed to read this book, including the elements of logic, set theory, mapping theory, and algebraic structures such as number systems and vector spaces. Definitions of basis, dimension, linear mappings, isomorphism, matrices and determinants are given; there is also ...
Edward John Specht +3 more
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