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Singularity spectrum

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25: 542: 95: 330: 476: 391: 362: 226: 420: 255: 54: 440: 105: 583: 523: 607: 76: 163: 135: 263: 120: 142: 192:
to describe the fractal dimension of a subset of points of a function belonging to a group of points that have the same
612: 149: 576: 496: 131: 37: 47: 41: 33: 196:. Intuitively, the singularity spectrum gives a value for how "fractal" a set of points are in a function. 58: 569: 189: 445: 501: 479: 156: 367: 338: 202: 193: 602: 519: 396: 231: 553: 425: 596: 112: 94: 541: 549: 491: 88: 18: 325:{\displaystyle D(\alpha )=D_{F}\{x,\alpha (x)=\alpha \}} 557: 116: 448: 428: 399: 370: 341: 266: 234: 205: 470: 434: 414: 385: 356: 324: 249: 220: 364:is the function describing the Hölder exponent, 46:but its sources remain unclear because it lacks 577: 8: 465: 459: 319: 292: 121:introducing citations to additional sources 584: 570: 453: 447: 427: 398: 369: 340: 286: 265: 233: 204: 77:Learn how and when to remove this message 199:More formally, the singularity spectrum 111:Relevant discussion may be found on the 7: 538: 536: 14: 540: 104:relies largely or entirely on a 93: 23: 471:{\displaystyle D_{F}\{\cdot \}} 409: 403: 380: 374: 351: 345: 310: 304: 276: 270: 244: 238: 215: 209: 1: 556:. You can help Knowledge by 514:van den Berg, J. C. (2004), 608:Mathematical analysis stubs 629: 535: 497:Fractional Brownian motion 386:{\displaystyle \alpha (x)} 357:{\displaystyle \alpha (x)} 221:{\displaystyle D(\alpha )} 32:This article includes a 61:more precise citations. 552:–related article is a 472: 436: 416: 387: 358: 326: 251: 222: 188:is a function used in 132:"Singularity spectrum" 473: 437: 417: 388: 359: 327: 252: 223: 190:multifractal analysis 16:Mathematical function 446: 426: 415:{\displaystyle f(x)} 397: 368: 339: 264: 250:{\displaystyle f(x)} 232: 203: 186:singularity spectrum 117:improve this article 516:Wavelets in Physics 502:Hausdorff dimension 480:Hausdorff dimension 613:Chaos theory stubs 468: 432: 412: 383: 354: 322: 247: 218: 34:list of references 565: 564: 525:978-0-521-53353-9 435:{\displaystyle x} 257:, is defined as: 182: 181: 167: 87: 86: 79: 620: 586: 579: 572: 544: 537: 528: 482:of a point set. 477: 475: 474: 469: 458: 457: 441: 439: 438: 433: 421: 419: 418: 413: 392: 390: 389: 384: 363: 361: 360: 355: 331: 329: 328: 323: 291: 290: 256: 254: 253: 248: 227: 225: 224: 219: 177: 174: 168: 166: 125: 97: 89: 82: 75: 71: 68: 62: 57:this article by 48:inline citations 27: 26: 19: 628: 627: 623: 622: 621: 619: 618: 617: 593: 592: 591: 590: 533: 526: 513: 510: 488: 449: 444: 443: 424: 423: 395: 394: 366: 365: 337: 336: 282: 262: 261: 230: 229: 228:of a function, 201: 200: 194:Hölder exponent 178: 172: 169: 126: 124: 110: 98: 83: 72: 66: 63: 52: 38:related reading 28: 24: 17: 12: 11: 5: 626: 624: 616: 615: 610: 605: 595: 594: 589: 588: 581: 574: 566: 563: 562: 545: 531: 530: 524: 509: 506: 505: 504: 499: 494: 487: 484: 467: 464: 461: 456: 452: 431: 411: 408: 405: 402: 382: 379: 376: 373: 353: 350: 347: 344: 333: 332: 321: 318: 315: 312: 309: 306: 303: 300: 297: 294: 289: 285: 281: 278: 275: 272: 269: 246: 243: 240: 237: 217: 214: 211: 208: 180: 179: 115:. Please help 101: 99: 92: 85: 84: 42:external links 31: 29: 22: 15: 13: 10: 9: 6: 4: 3: 2: 625: 614: 611: 609: 606: 604: 601: 600: 598: 587: 582: 580: 575: 573: 568: 567: 561: 559: 555: 551: 546: 543: 539: 534: 527: 521: 518:, Cambridge, 517: 512: 511: 507: 503: 500: 498: 495: 493: 490: 489: 485: 483: 481: 462: 454: 450: 429: 422:at the point 406: 400: 377: 371: 348: 342: 316: 313: 307: 301: 298: 295: 287: 283: 279: 273: 267: 260: 259: 258: 241: 235: 212: 206: 197: 195: 191: 187: 176: 165: 162: 158: 155: 151: 148: 144: 141: 137: 134: â€“  133: 129: 128:Find sources: 122: 118: 114: 108: 107: 106:single source 102:This article 100: 96: 91: 90: 81: 78: 70: 60: 56: 50: 49: 43: 39: 35: 30: 21: 20: 558:expanding it 547: 532: 515: 334: 198: 185: 183: 173:January 2012 170: 160: 153: 146: 139: 127: 103: 73: 67:January 2012 64: 53:Please help 45: 59:introducing 597:Categories 508:References 143:newspapers 463:⋅ 372:α 343:α 317:α 302:α 274:α 213:α 113:talk page 603:Fractals 486:See also 550:fractal 492:Fractal 478:is the 157:scholar 55:improve 522:  335:Where 159:  152:  145:  138:  130:  548:This 164:JSTOR 150:books 40:, or 554:stub 520:ISBN 184:The 136:news 442:. 393:of 119:by 599:: 44:, 36:, 585:e 578:t 571:v 560:. 529:. 466:} 460:{ 455:F 451:D 430:x 410:) 407:x 404:( 401:f 381:) 378:x 375:( 352:) 349:x 346:( 320:} 314:= 311:) 308:x 305:( 299:, 296:x 293:{ 288:F 284:D 280:= 277:) 271:( 268:D 245:) 242:x 239:( 236:f 216:) 210:( 207:D 175:) 171:( 161:· 154:· 147:· 140:· 123:. 109:. 80:) 74:( 69:) 65:( 51:.

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"Singularity spectrum"
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multifractal analysis
Hölder exponent
Hausdorff dimension
Fractal
Fractional Brownian motion
Hausdorff dimension
ISBN
978-0-521-53353-9
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fractal
stub
expanding it

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