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: –
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:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.