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Olds C.D., Davidoff G. — Geometry of Numbers
Olds C.D., Davidoff G. — Geometry of Numbers



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Название: Geometry of Numbers

Авторы: Olds C.D., Davidoff G.

Аннотация:

This book 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.

The geometry of numbers originated with the publication of Minkowski's seminal work in 1896 and ultimately established itself as an important field in its own right. By resetting various problems into geometric contexts, it sometimes allows difficult questions in arithmetic or other areas of mathematics to be answered more easily; inevitably, it lends a larger, richer perspective to the topic under investigation. Its principal focus is the study of lattice points, or points in n-dimensional space with integer coordinates-a subject with an abundance of interesting problems and important applications. Advances in the theory have proved highly significant for modern science and technology, yielding new developments in crystallography, superstring theory, and the design of error-detecting and error-correcting codes by which information is stored, compressed for transmission, and received.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2001

Количество страниц: 168

Добавлена в каталог: 05.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$C(\sqrt{n})$      45
$R(n) = R(n = p^2 + q^2)$      48 56
$u \equiv v (mod m)$      56
$\Gamma(x)$      105 121
$\infty$      46
$\lim_{n\rightarrow\infty}$      46
$\mathbb{R}^{n+1}$      83
Absolute value of a complex number      133
Additive structure of plane      133
Admissible lattice      145
Affine transformation      73—74
Approximating irrationals      105—107
Arithmetic-geometric mean inequality      80
Associate primes      135
Axes, major and minor      100
Axes, rotating      113
Bertrand, J.      152
Binary quadratic form      97 104
Blichfeldt, Hans Frederik, life of      153—154
Blichfeldt, Hans Frederik, life of, a packing theorem      149
Blichfeldt, Hans Frederik, life of, on approximating irrationals      124—129 131
Blichfeldt, Hans Frederik, life of, on quadratic minima      105 121
Blichfeldt, Hans Frederik, life of, proof of Minkowski’s Fundamental Theorem      72—73 117
Blichfeldt’s Theorem      113—116 119—121
Blichfeldt’s Theorem, generalization of      116
Bo1yai, Janos      141
Bolzano — Weierstrass theorem      15
Boundary of a polygon      36
Center of symmetry      66
Checkerboard lattice      148—149
Cluster points      15—16
Common divisor      55
complex numbers      54 133—134
Congruence classes      140—141
Congruence notation      56 108 140—141
Congruent numbers      140
Conjugate of z      134
Consecutive sides      35
Contracting an M-set      66
Convex point set      65
Covering, of a lattice point      114
Critical determinant      146
Critical lattice      146
Davenport, Harold      108
de M’eziriac, Claude Bachet      107
Degree of approximation      105
Determinant $\Delta$ of a lattice      73—74 85
Determinant $\Delta$ of a linear transformation      85
Determinant, critical      146
Difference points      116
Dilating an M-set      67
Diophantine approximations, simultaneous      82
Diophantine equation      28 54
Diophantine inequalities      63
Diophantus      51 54 107
Discriminant d      98
Doubly connected polygon      37
Einstein, Albert      152
Eisenstein, F.      152
ellipse      23 100—101
Equivalent lattices      89 91—92 95
Equivalent primes      135
Euclid’s algorithm      10 136
Euler, Leonhard      97 107
Euler, Leonhard, on representations of n      52 54—56
Expanding an M-set      66
Exterior of a polygon      36
Face-centered cubic (fee) lattice      148
Fermat, Pierre de      52 54 107
Fibonacci sequence      32
Fibonacci, Leonardo      53
Field      140
Four-dimensional sphere      109
Fractional part of x      26
Frederick the Great, Prussian Academy of      107
Fundamental lattice L      3—4 88
Fundamental parallelogram      89—90
Fundamental parallelopiped      74
Fundamental point-lattice Lambda      3—4 18 23 88 90
Fundamental Theorem of Arithmetic      8 135
Fundamental Theorem of Complex Arithmetic      136 138—139
Gamma function      105 I2l
Gauss, Carl Friedrich      152
Gauss, Carl Friedrich, on lattice points in circles      4 45—46 48 57—58
Gauss, Carl Friedrich, on packing      148
Gauss, Carl Friedrich, on quadratic forms      97 105
Gaussian integers      133—134
Gaussian integers, factorization of      134—135
Gaussian integers, unique factorization of      138—139
Gaussian primes      139—134
General affine transformation      73—74
General lattice      88
Geometry of numbers      3 63 75 85 113
Girard, Albert      52
Greatest common divisor      5 7
Greatest integer function      14 25—28
Harriot, Thomas      148
Hermite, Charles      63 105 108 123 152
Hexagon, regular      146
Hilbert, David      151
Hurwitz, Adolf      151
Infimum      146
Integers, divisibility property      7
Integers, expressing, in standard form      52
Integers, gaussian      133—134
Integers, Gaussian, factorization of      134—135
Integers, Gaussian, unique factorization of      138—139
Integers, relatively prime      5 95
Integral part of x      25
Integral solutions      28—31
Intercepts      30
Interior of a polygon      36
Inverse transformation      86—87
Irrational numbers      5
Irrational numbers, approximating      80 105 123—131
Jacobi, Carl Gustav Jacob      57 58—59 63
Jordan, Camille      152
Kepler conjecture      148
Kepler.Johannes      148
Kirchhoff.R.      151
Korkine.A.      101 104 105 149
Kronecker, Leopold      151
Kummer, E.      151
Lagrange, Joseph Louis      97 107
Lagrange’s Theorem      107—110
Lattice path      32
Lattice point      3—4 88
Lattice point, covering property      38 114
Lattice point, covering theorem      38
Lattice point, visible      95—96
Lattice square      72
Lattice systems      3 4—10
Lattice, admissible      145
Lattice, checkerboard      148—149
Lattice, critical      146
Lattice, face-centered cubic (fcc)      148
Lattice, general      88
Lattices, equivalent      89 91—92 95
Lcgendre, Adrien Marie      56 97
Least common multiple      8
Lie, Sophus      153
Linear transformation      85 87
Liouville.Joseph      58
Liouville’s identity      58
Lower bound of |f(x, y)\      98
M-set      65—67 113
Mersenne.Marin      54
Minimum of |f(x, y)|      98 104
Minkowski Theorem, A (for approximating irrationals)      123—129
Minkowski, Hermann      151—153
Minkowski, Hermann, his geometric point of view      3 63—64 75 101 109 113
Minkowski, Hermann, on quadratic minima      105
Minkowski’s First Theorem      76—80
Minkowski’s fundamental theorem      67
Minkowski’s Fundamental Theorem, applications of      75—76 83—84 98 100—101 113 125
Minkowski’s Fundamental Theorem, in y-space      74
Minkowski’s Fundamental Theorem, proofs of      67—73 117—119
Minkowski’s General Theorem      73 109 131
Minkowski’s Second Theorem      79—81
Minkowski’s Third Theorem      82
Mitchell, H.L.      III 46
Multiplicative inverses      134
Multiplicative structure of the plane      133
n!      27
N(n)      45
Niven.Ivan      42
Nontrivial factorization      135
Numbers, complex      54 133—134
Numbers, congruent      140
Numbers, irrational      5
Numbers, rational      5 8
Numbers, real      5
Packing of circles      146—147
Packing of circles, of lattices      145—146
Packing of circles, of spheres      145 147—149
Parallel displacement      113
Parallelogram, fundamental      89—90
Parallelogram, primitive      89—90
Parallelopiped.fundamental      74
Path of maximum width      17—20
Path of width d      18
Path, lattice point-free      17—23
Pick.Georg      36
Pick’s Theorem      36
Plane, structure of the      133
Point-lattice, construction of      88—89
Point-lattice, transformation of      91—95
Point-lattices, equivalent      89
Points, difference      116
POLYGONS      35—36
Polygons, doubly connected      37
Positive definite quadratic form      98 102 104
Prime Gaussian integer      135
Primitive parallelogram      89—90
prism      124
Pythagorean Theorem      133 137
Quadratic form, binary      97 104
Quadratic form, positive definite      98 102 104
Quadratic representation      97—98
Quadratic residues      108
Raleigh, Sir Walter      148
Rational numbers      5 8
Rational numbers, approximating irrationals by      105—107
Real numbers      5
Relatively prime Gaussian integers      135
Relatively prime integers      5—6 95
Representation of an integer n      48—53 110;
Representation, quadratic      97—98
Representations of prime numbers      54—56
Rhombus      37
Ring      134
Rotating axes      113
Sides of a polygon      35
Simple polygon      35
Simultaneous Diophantine approximations      82
Slope formula      11
Slope, irrational      5 10—17
Slope, rational      5 6—10
Smith, H.J.S.      152
Sphere packing      145 147—149
Sphere, four-dimensional      109
Standard form for integers      52
Steinhaus, Hugo      36
Symmetry about the origin      22 64
Symmetry, center of      66
T(n)      50
Tchebychev, P.L.      123 129
Thue, Alex      54
Transformation, affine      73
Transformation, inverse      86
Transformation, linear      85—87 90—95
Transformation, point-lattice      90—95
translating      113
Translation of an M-set      68
Unique factorization theorem      52
Unit area      20 95
Unit square      72 90
Units of a ring      134
Vectors      74 89
Vertices      35
Visible points      95—96
Voigt, W.      151
Volume of a four-dimensional sphere      109
von Helmholtz, H.L.E      151
Weber, H.      151
Weierstrass, Karl      151
Wilson’s Theorem      140
y-space      74
Zolotareff.E.I.      102 104 105 149
Zuckerman, Herbert      42
[x]      14 25—28
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