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Projective and Affine Quadrics
2020Dupin ring cyclide with a three-web consisting of Villarceau circles of both kinds and of isogonal trajectories of the circular curvature lines. In a conformal model of elliptic geometry, the cyclide represents a Clifford surface, where each kind of Villarceau circles is a family of Clifford parallels, while the isogonal trajectories play the role of ...
Boris Odehnal +2 more
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2013
In Definitions 5.2.1 and 5.1.1, projective and affine spaces were introduced by means of axioms, and in Propositions 5.2.2 and 5.1.3, the spaces ℙ(V) and \(\mathbb {A}(V)\), where V is a vector space, were shown to be examples. In this chapter we show that by and large there are no further examples.
Francis Buekenhout, Arjeh M. Cohen
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In Definitions 5.2.1 and 5.1.1, projective and affine spaces were introduced by means of axioms, and in Propositions 5.2.2 and 5.1.3, the spaces ℙ(V) and \(\mathbb {A}(V)\), where V is a vector space, were shown to be examples. In this chapter we show that by and large there are no further examples.
Francis Buekenhout, Arjeh M. Cohen
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Affine and Projective Geometry
1977In this chapter we shall introduce two different (but closely related) geometrical languages. The first of these, the language of affine geometry, is the one which appeals most closely to our intuitive ideas of geometry. In this language the subspaces of a vector space of dimensions 0, 1 and 2 are called “points”, “lines” and “planes”, respectively ...
K. W. Gruenberg, A. J. Weir
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Affine and Projective Transformations
2010In addition to isometries, there are two kinds of mappings that preserve lines: affine (Section 3.1) and projective (Section 3.2) transformations. Affine transformations f of \({\mathbb{R}}^{n}\) have the following property: If l is a line then f(l) is also a line, and if l ∥ k then f(l) ∥ f(k). A line in \({\mathbb{R}}^{n}\) means a set of the form {r
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Affine and Projective Geometries
1997In 0.1.6 we have defined the real linear spaces L n. If in this definition we replace the field ℝ of real numbers by the field ℂ of complex numbers or by the skew field ℍ of quaternions, we obtain the complex linear space ℂL n or quaternionic linear space ℍL n. We have mentioned the space ℂL n in 1.6.1.
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2015
The normalized least-mean-squares (NLMS) algorithm has a problem that the convergence slows down for correlated input signals. The reason for this phenomenon is explained by looking at the algorithm from a geometrical point of view. This observation motivates the affine projection algorithm (APA) as a natural generalization of the NLMS algorithm.
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The normalized least-mean-squares (NLMS) algorithm has a problem that the convergence slows down for correlated input signals. The reason for this phenomenon is explained by looking at the algorithm from a geometrical point of view. This observation motivates the affine projection algorithm (APA) as a natural generalization of the NLMS algorithm.
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2020
This chapter gives fundamental results on finite affine and projective planes. It provides detailed proofs on various counting results concerning these planes such as the number of points, lines, points on a line, and lines through a point. It describes the canonical relation between affine planes and mutually orthogonal Latin squares.
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This chapter gives fundamental results on finite affine and projective planes. It provides detailed proofs on various counting results concerning these planes such as the number of points, lines, points on a line, and lines through a point. It describes the canonical relation between affine planes and mutually orthogonal Latin squares.
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1997
This chapter and the following two are in a relationship similar to the three parts of Chapter 2: the combinatorial part will be covered here automorphisms (collineations) follow in Chapter 4, and constructions in Chapter 5.
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This chapter and the following two are in a relationship similar to the three parts of Chapter 2: the combinatorial part will be covered here automorphisms (collineations) follow in Chapter 4, and constructions in Chapter 5.
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Robust hybrid affine projection filtering algorithm under α-stable environment
Signal Processing, 2023Xingli Zhou +2 more
exaly
Affine and Projective Varieties
1992In this book we will be dealing with varieties over a field K, which we will take to be algebraically closed throughout. Algebraic geometry can certainly be done over arbitrary fields (or even more generally over rings), but not in so straightforward a fashion as we will do here; indeed, to work with varieties over nonalgebraically closed fields the ...
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