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Stein S.K., Szabo S. — Algebra and Tiling: Homomorphisms in the Service of Geometry
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Название: Algebra and Tiling: Homomorphisms in the Service of GeometryАвторы: Stein S.K., Szabo S. Аннотация: Often questions about tiling space or a polygon lead to other questions. For instance, tiling by cubes raises questions about finite abelian groups. Tiling by triangles of equal areas soon involves Sperner's lemma from topology and valuations from algebra. The first six chapters of Algebra and Tiling form a self-contained treatment of these topics, beginning with Minkowski's conjecture about lattice tiling of Euclidean space by unit cubes, and concluding with Laczkowicz's recent work on tiling by similar triangles. The concluding chapter presents a simplified version of Rédei's theorem on finite abelian groups: if such a group is factored as a direct product of subsets, each containing the identity element, and each of prime order, than at least one of them is a subgroup. Algebra and Tiling is accessible to undergraduate mathematics majors, as most of the tools necessary to read the book are found in standard upper division algebra courses, but teachers, researchers and professional mathematicians will find the book equally appealing.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1995Количество страниц: 224Добавлена в каталог: 11.04.2008Операции: Положить на полку |
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Abrams, A. vii Adjacent clusters 38 Adjacent cubes 37 Alkhadjandi, Abu Mohammed 1 2 Bachet, C.G. 2 Bachman, G. 133 Banach, S. vi Barnes, F.W. vi 200 Ben Alhocain, Abu Djafar Mohammed 1 Berger, R. 54 200 Boltanskii, V.G. 200 Brouwer's fixed point theorem 109 Centrally symmetric set 12 Chakerian, G.D. vii 120 Character (of finite abelian group) 163 193 Character group 193 Character principal- 163 Chrisman, M. vii 148 Clusters 35 Clusters equivalent 38 Complete polygon 110 Complete triangle with respect to a labeling 110 Complete triangle with respect to a valuation 118 Conway, J.H. vi 31 32 53 54 105 200 Corradi, K. 28 32 185 Covering 19 Covering by crosses 97 Covering by semicrosses 99 Covering n-covering (of a group) 100 102 Covering set (of a group) 100 102 cross 58 Cross, covering by 97 Cross, packing by 87 Cross, tiling by 60—62 Cyclic subset 27 Cyclotomic polynomial 198 Dad-del, A. 101 105 de Braijn, N.G. vii 48 54 200 Dehn, M.W. vii 140 Dekking, F.M. 200 Density of a covering 86 Density of a family 85 Density of a lattice covering 86 Density of a lattice packing 86 Density of a packing 86 Density of a Z-lattice covering 86 Density of a Z-lattice packing 86 Diophantos 1 2 Dummit, D.S. 185 Equidissection 119 Equidissection of centrally symmetric polygon 130—132 Equidissection of cube 120 Equidissection of regular hexagon 128—130 Equidissection of regular n-gon 120 Equidissection of square 118 Equidissection of trapezoid 121—126 Equivalent clusters 38 Equivalent vectors 38 Euler phi-function 198 Euler, L. 2 162 Everett, H. 105 Eves, H. 153 Exact sequence 67 194 195 Factorization (of a group) 26 Factorization normalized 156 Factorization replaceable, factor of 157 Fbote, R.M 185 Fejer, L. 155 Fejes-Toth, L. 201 Fermat, P. 2 Forcade, R.W. 78 82 Formal sum 196 Fourier, J.B.J 155 Freiling, C. vii Freller, M. 80 Frenicle, B. 2 Furtwangler's conjecture 155 Furtwangler, P. 29 32 33 Gale, D. vii 200 Galovich, S. 82 Gauss, C.F. 5 6 32 Gobel, F. 201 Golomb, S. 53 54 80 82 200 Group ring 167 Grunbaum, B. 200 201 Hajos's theorem 26 Hajos's theorem for p-groups 160 Hajos, G. vi 22 23 26—29 32 152 155 156 159—161 Hales, A.W. 109 133 Hamaker, W. 80 82 105 Harmonic brick 52 Harmonic brick multiple of 52 Hermite, C. 9 10—12 15 Hickerson, D.R. vii 65 72 76 82 97 Hilbert, D. 10 15 Hocking, J.C. 55 Hsiang, W.Y. 86 Hurwitz, A. 10 Ireland, K. 133 Joy, K. 95 k-factorization 29 Karteszi, F. 80 82 Kasimatis, E.A. 81 120 128 130 133 Keller, O.H. 28 33 Kenyon, R. vii 201 Kepler, J. 85 Klarner, D. A. 55 201 Klee, V. vii 108 109 133 Korchmaros, G. 80 Laczkovich, M. vii 135 151—153 201 Lagarias, J.C. 28 31 33 53 54 200 Lagrange, J.L. 2 Lattice 3 189 Lattice integral 30 Lattice logarithm on S(k) 78 Lattice standard 5 Lattice, basis of 4 189 Lattice, determinant of 5 8 Lattice, dimension of 4 Lattice, fundamental parallelepiped of 5 189 Lattice, fundamental parallelogram of 5 Lenstra, A.K. 31 33 Lenstra, H.W. 31 33 Lovdsz, L. 31 33 m-equidissection 118 Mackinnon, N. 53 55 Mead, D.G. 120 133 201 Medyanik, A.I. 80 Mersenne, M. 2 Mertens' conjecture 31 32 Mertens, F. 31 32 Meschkowski, H. vi 201 Mills, W.H. 79 82 Minkowski's conjecture 22 Minkowski's conjecture algebraic form 26 27 Minkowski's conjecture generalization of 28 29 Minkowski, H. v vi 1 10—12 15—19 22 23 26—31 33 36 152 155 156 Molnar, E. 80 82 Monotonic matrix 94 Monsky, P. 108 109 118 131—133 Multiplier set 67 Odlyzko, A.M. 31 33 p-component (of finite abelian group) 74 P-dimension 74 p-group 74 p-part 180 Packing 19 Packing by crosses 97—99 Packing by semicrosses 88—97 Packing n-packing (of a group) 89 Packing set (of a group) 89 Pascal, B. 2 Penrose, R. 201 Perron, O. 28 33 Pollington, A.D. 78 82 Posa, L. vii 135 147 Pythagoras 1 Pythagorean Theorem 1 Q-lattice-tiling 35 Q-tiling 35 Quadratic form determinant of 8 Quadratic form positive definite 2 Quadratic form relation to lattice 5 Redei's theorem 28 156 182 Redei's theorem for cyclic p-groups 160 Redei's theorem for p-groups 165 176 Redei's theorem for type (p,p) 173 Redei's theorem general 182 Redei, L. vi vii 28 32 33 36 152 155—157 161—165 169 172 173 176 178 180—182 185 188 Replacement principle fifth 181 Replacement principle first 158 Replacement principle fourth 180 Replacement principle general 169 Replacement principle second 169 Replacement principle third 170 Richman, F. vii 107—109 113 118 131 133 Riemann hypothesis 31 Riemann, B. 31 Rinne, D. vii Robinson, R.M. 29 33 54 201 Rogers, С.A. 105 Root of unity 198 Root of unity primitive 198 Rosen, M. 133 Schattschneider, D. 201 Schmidt, T. 22 34 35 55 Semicross 58 Semicross, covering by 99—102 Semicross, packing by 88—97 Semicross, tiling by 59—66 Senechal, M. vii Set centrally symmetric 12 Set closed 12 Set convex 12 Setstar (also star body) 58 Shephard, G.C. 200 201 Shift 87 Shor, P. 28 33 Simplicial dissection 109 Sloane, N.J.A. 31 32 105 Spectrum 120 Spectrum principal 120 Sperner's Lemma 107 109 110 Sperner's lemma oriented version 130 Sperner, E. v 107 109—111 118 120 122 130 132 Splitting (of a group) 62 67 Splitting (of a group) and exact sequence 69 70 Splitting (of a group) coset- 77 Splitting (of a group) multipler set of 67 Splitting (of a group) nonsingular 70 Splitting (of a group) purely singular 70 Splitting (of a group) set 62 67 Splitting (of a group) singular 70 Star body 58 Stein, S.K 34 80—82 105 106 120 130 131 133 201 Strauss, E.G. 109 133 Szabo, S. 28 33 34 47 55 80—82 106 185 201 Szekeres, G. vii 152 201 Tarski, A. vi te Riele, H.J.J. 31 33 Thomas, J. vii 107 109 118 131 133 Thompson, T. 106 Thurston, W.P. vi 55 201 Tiling 19 Tiling integer 35 Tiling k-fold 29 Tiling lattice 35 Tiling rational 35 Translate (of a set) 11 Twin (cubes) 22 Type (of a finite abelian group) 162 Ulrich, W. 80 82 Valuations (of a field) 111 Valuations (of a field) and dissections 117 Valuations (of a field) equivalent 111 Valuations (of a field) extension of 113—116 Valuations (of a field) p-adic 112 Van der Waerden, B.L. 133 Wagon, S. 201 Welch, L. 80 82 Woepcke, M.E. 34 Woldar, A.J. 83 Young, G.S. 55 Z-lattice-tiling 35 Z-tiling 35
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