Geometry and Its Applications, Second Edition by Walter A. Meyer

By Walter A. Meyer

Meyer's Geometry and Its functions, moment Edition, combines conventional geometry with present principles to give a latest technique that's grounded in real-world purposes. It balances the deductive method with discovery studying, and introduces axiomatic, Euclidean geometry, non-Euclidean geometry, and transformational geometry. The textual content integrates functions and examples all through and comprises historic notes in lots of chapters.

The moment version of Geometry and Its Applications is an important textual content for any university or college that makes a speciality of geometry's usefulness in different disciplines. it really is specifically applicable for engineering and technological know-how majors, in addition to destiny arithmetic teachers.

  • Realistic purposes built-in in the course of the textual content, together with (but no longer restricted to):
    • Symmetries of inventive patterns
    • Physics
    • Robotics
    • Computer vision
    • Computer graphics
    • Stability of architectural structures
    • Molecular biology
    • Medicine
    • Pattern recognition
  • Historical notes integrated in lots of chapters

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Extra info for Geometry and Its Applications, Second Edition

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1 (Tate). Let K be a non-Archimedean field of arbitrary characteristic and let q ∈ K × with 0 < |q| < 1. Then the field of meromorphic q-periodic © Springer International Publishing Switzerland 2016 W. Lütkebohmert, Rigid Geometry of Curves and Their Jacobians, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. , F (q) is finitely generated field of transcendence degree 1 over K and of genus 1. More precisely, F (q) = K(℘, ℘), ˜ where n∈Z q nξ − 2 · s1 , (1 − q n ξ )2 n∈Z q 2n ξ 2 + s1 (1 − q n ξ )3 ℘ (ξ ) = ℘(ξ ˜ )= with s := m≥1 m qm 1 − qm for ∈ N.

It suffices to look at a proper algebraic A-scheme Y where A is an affinoid algebra. If Y is projective over A, then we can assume that Y = PAn is already the projective space. For c ∈ |K × | consider the subsets Yi (c) := x ∈ PAn , ξj (x) ≤ c · ξi (x) for j = 0, . . , n for i = 1, . . , n. Then Y (c) := {Y0 (c), . . , Yn (c)} is an admissible affinoid covering of PAn for c ≥ 1 with Yi (1) A Yi (c) for i = 0, . . , n and c > 1. 3. 2] there exists a surjective Amorphism f : Z → Y from a projective A-scheme Z to Y .

In the next section we will study more general group actions than Tate’s action M × P1K → P1K ; (q, z) −→ q · z, on the projective line. The group M is only a special case of a Schottky group; cf. 3, and so Tate’s curves are a special case of Mumford curves; cf. 1. In the following sections we will present much more general results. 1. In this section we will study the structure of those finitely generated subgroups of the projective linear group PGL(2, K), which are free of torsion and act discontinuously on a non-empty open subdomain of the projective line.

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