Curves and Fractal DimensionSpringer Science & Business Media, 18. stu 1994. - Broj stranica: 324 A mathematician, a real one, one for whom mathematical objects are abstract and exist only in his mind or in some remote Platonic universe, never "sees" a curve. A curve is infinitely narrow and invisible. Yet, we all have "seen" straight lines, circles, parabolas, etc. when many years ago (for some of us) we were taught elementary geometry at school. E. Mach wanted to suppress from physics everything that could not be perceived: physics and metaphysics must not exist together. Many a scientist was deeply influenced by his philosophy. In his book Claude Tricot tells us that a curve has a non-vanishing width. Its width is that of the pencil or of the pen on the paper, or of the chalk on the blackboard. The abstract curve which cannot be seen and which does not really concern us here is the intersection of all those thick curves that contain it. For Claude Tricot it is only the thick curves that are pertinent. He describes in detail the way bumps, peaks, and irregularities appear on the curve as its width decreases. This is not a new point of view. Indeed Hausdorff and Bouligand initiated the idea at the beginning of this century. However, Claude Tricot manages to refine the theory extensively and interestingly. His approach is both realistic and mathematically rigorous. Mathematicians who only feed on abstractions as well as engineers who tackle tangible problems will enjoy reading this book. |
Sadržaj
III | xi |
V | xii |
VI | xiv |
VII | 1 |
VIII | 4 |
IX | 5 |
X | 8 |
XI | 9 |
XCI | 148 |
XCII | 151 |
XCIII | 154 |
XCIV | 157 |
XCV | 160 |
XCVI | 162 |
XCVII | 164 |
XCVIII | 168 |
XIII | 11 |
XIV | 12 |
XV | 13 |
XVI | 14 |
XVII | 17 |
XVIII | 19 |
XIX | 20 |
XX | 21 |
XXI | 23 |
XXII | 27 |
XXIII | 28 |
XXV | 30 |
XXVI | 34 |
XXVII | 37 |
XXIX | 38 |
XXX | 39 |
XXXI | 40 |
XXXII | 41 |
XXXV | 44 |
XXXVI | 45 |
XXXVII | 48 |
XXXVIII | 50 |
XXXIX | 51 |
XL | 53 |
XLI | 54 |
XLIII | 56 |
XLIV | 61 |
XLV | 64 |
XLVI | 65 |
XLVII | 67 |
XLVIII | 69 |
XLIX | 72 |
L | 74 |
LI | 75 |
LII | 76 |
LIII | 79 |
LV | 80 |
LVI | 83 |
LVII | 85 |
LVIII | 87 |
LIX | 91 |
LX | 92 |
LXI | 94 |
LXII | 95 |
LXIII | 96 |
LXIV | 99 |
LXVI | 100 |
LXVII | 106 |
LXVIII | 108 |
LXIX | 109 |
LXX | 110 |
LXXII | 113 |
LXXIII | 114 |
LXXIV | 117 |
LXXV | 122 |
LXXVI | 124 |
LXXVII | 127 |
LXXVIII | 128 |
LXXIX | 129 |
LXXXI | 131 |
LXXXII | 133 |
LXXXIII | 135 |
LXXXIV | 137 |
LXXXVI | 138 |
LXXXVII | 139 |
LXXXVIII | 142 |
LXXXIX | 144 |
XC | 146 |
XCIX | 171 |
CI | 173 |
CII | 174 |
CIII | 176 |
CIV | 177 |
CV | 179 |
CVI | 181 |
CVII | 184 |
CVIII | 186 |
CIX | 189 |
CX | 193 |
CXI | 195 |
CXII | 197 |
CXIII | 200 |
CXIV | 202 |
CXV | 203 |
CXVI | 207 |
CXVII | 208 |
CXVIII | 214 |
CXIX | 215 |
CXXI | 216 |
CXXII | 217 |
CXXIII | 219 |
CXXIV | 221 |
CXXV | 226 |
CXXVI | 228 |
CXXVII | 230 |
CXXVIII | 234 |
CXXIX | 236 |
CXXX | 237 |
CXXXII | 239 |
CXXXIII | 240 |
CXXXIV | 241 |
CXXXV | 245 |
CXXXVI | 247 |
CXXXVII | 249 |
CXXXVIII | 250 |
CXXXIX | 252 |
CXL | 253 |
CXLII | 254 |
CXLIII | 257 |
CXLIV | 258 |
CXLV | 260 |
CXLVI | 263 |
CXLVII | 265 |
CXLVIII | 266 |
CXLIX | 269 |
CL | 271 |
CLI | 274 |
CLII | 275 |
CLIII | 277 |
CLV | 279 |
CLVI | 280 |
CLVII | 281 |
CLVIII | 283 |
CLIX | 285 |
CLX | 288 |
CLXI | 293 |
CLXIII | 294 |
CLXIV | 297 |
CLXV | 302 |
CLXVI | 303 |
CLXVII | 304 |
CLXVIII | 306 |
CLXIX | 307 |
CLXX | 311 |
Ostala izdanja - Prikaži sve
Uobičajeni izrazi i fraze
angle assume balls belongs Bibliographical notes Borel bounded bounded set breadth(K c₁ Cantor set Chap closed set compute the dimension constant construct contains contiguous intervals continuous function converges convex hull convex set coordinates deduce defined denote dense set deviation diam diameter disjoint disk dist domain E₁ endpoints equal equivalent example exists expansive curve exponent finite length fractal curve fractal dimension function z(t geometrical graph Hausdorff distance homothety inequality infinite integer intervals of length intervals of rank L(Pn lim sup limit method Minkowski sausage notion null measure obtain order of growth P₁ packing dimension parameter parameterization plane polygonal approximation polygonal curve properties prove ratio real line real number rectangle rectifiable rectifiable curve satisfies self-similar curve sequence sets of null similarities simple curve straight line subarcs symmetrical perfect set theorem Tricot union velocity
Popularni odlomci
Stranica vii - Hausdorff dimension, an excellent mathematical tool, but we believe that, as it stands, it has no practical application in the study of curves originated in other sciences: physics, biology, or engineering.