| Description |
1 online resource |
| Series |
Anneli Lax New Mathematical Library ; v. 41 |
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Anneli Lax new mathematical library.
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| Contents |
Cover -- Title Page -- Contents -- Preface -- Part I Lattice Points and Number Theory -- 1 Lattice Points and Straight Lines -- 1.1 The Fundamental Lattice -- 1.2 Lines in Lattice Systems -- 1.3 Lines with Rational Slope -- 1.4 Lines with Irrational Slope -- 1.5 Broadest Paths without Lattice Points -- 1.6 Rectangles on Paths without Lattice Points -- Problem Set for Chapter 1 -- References -- 2 Counting Lattice Points -- 2.1 The Greatest Integer Function, [x] -- Problem Set for Section 2.1 -- 2.2 Positive Integral Solutions of ax + by = n |
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Problem Set for Section 2.22.3 Lattice Points inside a Triangle -- Problem Set for Section 2.3 -- References -- 3 Lattice Points and the Area of Polygons -- 3.1 Points and Polygons -- 3.2 Pick's Theorem -- Problem Set for Section 3.2 -- 3.3 A Lattice Point Covering Theorem for Rectangles -- Problem Set for Section 3.3 -- References -- 4 Lattice Points in Circles -- 4.1 How Many Lattice Points Are There? -- 4.2 Sums of Two Squares -- 4.3 Numbers Representable as a Sum of Two Squares -- Problem Set for Section 4.3 |
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4.4 Representations of Prime Numbers as Sums of TwoSquares4.5 A Formula for R(n) -- Problem Set for Section 4.5 -- References -- Part II An Introduction to the Geometry of Numbers -- 5 Minkowski's Fundamental Theorem -- 5.1 Minkowski's Geometric Approach -- Problem Set for Section 5.1 -- 5.2 Minkowski M-Sets -- Problem Set for Section 5.2 -- 5.3 Minkowski's Fundamental Theorem -- Problem Set for Section 5.3 -- 5.4 (Optional) Minkowski's Theorem in n Dimensions -- References -- 6 Applications of Minkowski's Theorems -- 6.1 Approximating Real Numbers |
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6.2 Minkowski's First TheoremProblem Set for Section 6.2 -- 6.3 Minkowski's Second Theorem -- Problem for Section 6.3 -- 6.4 Approximating Irrational Numbers -- 6.5 Minkowski's Third Theorem -- 6.6 Simultaneous Diophantine Approximations -- Reading Assignment for Chapter 6 -- References -- 7 Linear Transformations and Integral Lattices -- 7.1 Linear Transformations -- Problem Set for Section 7.1 -- 7.2 The General Lattice -- 7.3 Properties of the Fundamental Lattice -- Problem Set for Section 7.3 -- 7.4 Visible Points |
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8 Geometric Interpretations of Quadratic Forms8.1 Quadratic Representation -- 8.2 An Upper Bound for the Minimum Positive Value -- 8.3 An Improved Upper Bound -- 8.4 (Optional) Bounds for the Minima of Quadratic Formsin More Than Two Variables -- 8.5 Approximating by Rational Numbers -- 8.6 Sums of Four Squares -- References -- 9 A New Principle in the Geometry of Numbers -- 9.1 Blichfeldt's Theorem -- 9.2 Proof of Blichfeldt's Theorem -- 9.3 A Generalization of Blichfeldt's Theorem -- 9.4 A Return to Minkowski's Theorem |
| Summary |
The Geometry of Numbers presents a self-contained introduction to the geometry of numbers, beginning with easily understood questions about lattice-points on lines, circles, and inside simple polygons in the plane. Little mathematical expertise is required beyond an acquaintance with those objects and with some basic results in geometry. The reader moves gradually to theorems of Minkowski and others who succeeded him. On the way, he or she will see how this powerful approach gives improved approximations to irrational numbers by rationals, simplifies arguments on ways of representing integers as sums of squares, and provides a natural tool for attacking problems involving dense packings of spheres. An appendix by Peter Lax gives a lovely geometric proof of the fact that the Gaussian integers form a Euclidean domain, characterizing the Gaussian primes, and proving that unique factorization holds there. In the process, he provides yet another glimpse into the power of a geometric approach to number theoretic problems |
| Notes |
Title from publishers bibliographic system (viewed on 30 Jan 2012) |
| Subject |
Geometry of numbers.
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Number theory.
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Geometry of numbers.
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Number theory.
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| Form |
Electronic book
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| Author |
Olds, C. D
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Lax, Anneli
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Davidoff, Giuliana P
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| ISBN |
9780883859551 |
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0883859556 |
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