22:
140:
1098:
804:
1958:
4947:
4707:
Cantor function has derivative 0 almost everywhere, current research focusses on the question of the size of the set of points where the upper right derivative is distinct from the lower right derivative, causing the derivative to not exist. This analysis of differentiability is usually given in terms of
4404:; the infinitely distant "leaves" on the tree correspond to the points on the Cantor set, and so, the monoid also represents the self-symmetries of the Cantor set. In fact, a large class of commonly occurring fractals are described by the dyadic monoid; additional examples can be found in the article on
4963:
loosely resembles the Cantor function visually, appearing as a "smoothed out" form of the latter; it can be constructed by passing from a continued fraction expansion to a binary expansion, just as the Cantor function can be constructed by passing from a ternary expansion to a binary expansion. The
4706:
As mentioned above, the Cantor function is also the cumulative distribution function of a measure on the Cantor set. Different Cantor functions, or Devil's
Staircases, can be obtained by considering different atom-less probability measures supported on the Cantor set or other fractals. While the
1093:{\displaystyle c(x)={\begin{cases}\sum _{n=1}^{\infty }{\frac {a_{n}}{2^{n}}},&x=\sum _{n=1}^{\infty }{\frac {2a_{n}}{3^{n}}}\in {\mathcal {C}}\ \mathrm {for} \ a_{n}\in \{0,1\};\\\sup _{y\leq x,\,y\in {\mathcal {C}}}c(y),&x\in \smallsetminus {\mathcal {C}}.\end{cases}}}
64:. Though it is continuous everywhere and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument reaches from 0 to 1. Thus, in one sense the function seems very much like a constant one which cannot grow, and in another, it does indeed
2396:
4711:, with the Hausdorff dimension the most popular choice. This line of research was started in the 1990s by Darst, who showed that the Hausdorff dimension of the set of non-differentiability of the Cantor function is the square of the dimension of the Cantor set,
4771:
2604:
6099:
3947:
3097:
1851:
3015:
1942:
4646:
708:
515:
has the ternary representation 0.01210121... The digits after the first 1 are replaced by 0s to produce 0.01000000... This is not rewritten since it has no 2s. This is the binary representation of
4395:
4536:
1369:
595:
482:
4072:
2187:
628:
has the ternary representation 0.21102 (or 0.211012222...). The digits after the first 1 are replaced by 0s to produce 0.21. This is rewritten as 0.11. This is the binary representation of
3301:
3754:
3696:
3638:
3580:
3523:
3460:
2844:
4241:
3402:
3353:
4762:
4452:
626:
4949:
Later, Troscheit obtain a more comprehensive picture of the set where the derivative does not exist for more general normalized Gibb's measures supported on self-conformal and
4294:
4120:
3192:
2761:
1321:
1158:
655:
542:
513:
429:
402:
has the ternary representation 0.02020202... There are no 1s so the next stage is still 0.02020202... This is rewritten as 0.01010101... This is the binary representation of
400:
1280:
1129:
736:
2947:
2888:
4340:
2171:
converges pointwise to the Cantor function defined above. Furthermore, the convergence is uniform. Indeed, separating into three cases, according to the definition of
3995:
3148:
1588:
2664:
2644:
3812:
3792:
3215:
1710:
1675:
1236:
1201:
796:
201:
4177:
4150:
2699:
4942:{\displaystyle \dim _{H}\left\{x:f'(x)=\lim _{h\to 0^{+}}{\frac {\mu ()}{h}}{\text{ does not exist}}\right\}=\left(\dim _{H}\operatorname {supp} (\mu )\right)^{2}}
364:
282:
2415:
1418:
1256:
329:
306:
221:
1508:
200000...), and every point not in the Cantor set is in one of these intervals, so its derivative is 0 outside of the Cantor set. On the other hand, it has no
253:
5134:
1438:
3820:
1613:
zero over any interval will give positive probability to some interval to which this distribution assigns probability zero. In particular, as
1631:
but, as mentioned above, is not absolutely continuous. However, every absolutely continuous function is continuous with bounded variation.
1440:
goes from 0 to 1, and takes on every value in between. The Cantor function is the most frequently cited example of a real function that is
4408:. Other fractals possessing self-similarity are described with other kinds of monoids. The dyadic monoid is itself a sub-monoid of the
3026:
4400:
The dyadic monoid itself has several interesting properties. It can be viewed as a finite number of left-right moves down an infinite
1783:
1131:, the ternary expansion is repeating with trailing 2's and there is an alternative non-repeating expansion ending in 1. For example,
2958:
1856:
6117:
5083:
Falconer, Kenneth J. (2004-01-01). "One-sided multifractal analysis and points of non-differentiability of devil's staircases".
4568:
5456:
5394:
5279:
5266:
Reprinted in: E. Zermelo (Ed.), Gesammelte
Abhandlungen Mathematischen und Philosophischen Inhalts, Springer, New York, 1980.
5224:
4960:
4457:
660:
53:
4348:
1598:. This is why there are no jump discontinuities in the function; any such jump would correspond to an atom in the measure.
2670:
with (uncountably) infinitely many points (zero-dimensional volume), but zero length (one-dimensional volume). Only the
4474:
2391:{\displaystyle \max _{x\in }|f_{n+1}(x)-f_{n}(x)|\leq {\frac {1}{2}}\,\max _{x\in }|f_{n}(x)-f_{n-1}(x)|,\quad n\geq 1.}
1326:
547:
434:
5563:
5475:
5596:
4008:
1617:
pointed out, the function is not the integral of its derivative even though the derivative exists almost everywhere.
5048:
Darst, Richard (1993-09-01). "The
Hausdorff Dimension of the Nondifferentiability Set of the Cantor Function is 2".
4765:
1524:
3220:
112:
6123:
5516:
3707:
3649:
3591:
3528:
3471:
3413:
2777:
5979:
5936:
4964:
question mark function has the interesting property of having vanishing derivatives at all rational numbers.
4186:
3358:
3309:
4122:
Thus, every dyadic rational is in one-to-one correspondence with some self-symmetry of the Cantor function.
5852:
1606:
1602:
6057:
1642:
showed that the arc length of its graph is 2. Note that the graph of any nondecreasing function such that
6139:
5623:
4714:
4414:
602:
6182:
5504:
4246:
4077:
3159:
2708:
5389:. Vol. 181 (2nd ed.). Providence, Rhode Island: American Mathematical Society. p. 734.
1285:
5789:
3759:
These operations can be stacked arbitrarily. Consider, for example, the sequence of left-right moves
2622:
can be defined as the set of those numbers in the interval that do not contain the digit 1 in their
1134:
631:
518:
489:
405:
376:
1261:
1110:
828:
717:
5645:
4342:
as a general element of the monoid, there is a corresponding self-symmetry of the Cantor function:
2896:
2861:
116:
4699:) looks like the Cantor function turned on its side, with the width of the steps getting wider as
4319:
6192:
6177:
5148:
Troscheit, Sascha (2014-03-01). "Hölder differentiability of self-conformal devil's staircases".
3303:
and likewise for the other cases. For the left and right magnifications, write the left-mappings
6131:
6082:
5690:
5556:
5333:
Fleron, Julian F. (1994-04-01). "A Note on the
History of the Cantor Set and Cantor Function".
4768:
showed that this squaring relationship holds for all
Ahlfors's regular, singular measures, i.e.
3959:
41:
5238:[The power of perfect sets of points: Extract from a letter addressed to the editor].
3112:
6187:
5916:
5608:
5236:"De la puissance des ensembles parfaits de points: Extrait d'une lettre adressée à l'éditeur"
5128:
4973:
4559:
2649:
2629:
2623:
1595:
1537:
1384:
61:
5022:
3797:
3762:
3200:
2599:{\displaystyle \max _{x\in }|f(x)-f_{n}(x)|\leq 2^{-n+1}\,\max _{x\in }|f_{1}(x)-f_{0}(x)|.}
1680:
1645:
1601:
However, no non-constant part of the Cantor function can be represented as an integral of a
1206:
1171:
745:
150:
6077:
6072:
5862:
5794:
5325:
5289:
5167:
5092:
4978:
4155:
4128:
2677:
1441:
4460:. In particular, it obeys the exact same symmetry relations, although in an altered form.
1445:
340:
258:
8:
5835:
5812:
5695:
5680:
5613:
2702:
1591:
1453:
1380:
49:
45:
5747:
5171:
5096:
107:) introduced the Cantor function and mentioned that Scheeffer pointed out that it was a
6062:
6042:
6006:
6001:
5764:
5358:
5191:
5157:
5116:
5065:
1241:
314:
291:
206:
65:
1394:
226:
6172:
6105:
6067:
5991:
5899:
5804:
5710:
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5601:
5591:
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5549:
5524:
5471:
5452:
5435:
5400:
5390:
5350:
5313:
5296:
Dovgoshey, O.; Martio, O.; Ryazanov, V.; Vuorinen, M. (2006). "The Cantor function".
5275:
5257:
5220:
5183:
5120:
5108:
4956:
4708:
1740:
1635:
1628:
1621:
1610:
1528:
1388:
5527:
5512:
5195:
3106:. This is exhibited by defining several helper functions. Define the reflection as
6022:
5889:
5872:
5700:
5485:
Vitali, A. (1905), "Sulle funzioni integrali" [On the integral functions],
5425:
5342:
5305:
5247:
5175:
5100:
5057:
4558:∈ {0,1}. This expansion is discussed in greater detail in the article on the
3953:
3794:
Adding the subscripts C and D, and, for clarity, dropping the composition operator
1721:
1423:
4125:
Some notational rearrangements can make the above slightly easier to express. Let
6037:
5974:
5635:
5321:
5309:
5285:
5212:
4950:
4542:
3942:{\displaystyle L_{D}R_{D}L_{D}L_{D}R_{D}\circ c=c\circ L_{C}R_{C}L_{C}L_{C}R_{C}}
1725:
1107:
base 3 representation that only contains the digits 0 or 2. (For some members of
5732:
1634:
The Cantor function is non-decreasing, and so in particular its graph defines a
6052:
5984:
5955:
5911:
5894:
5877:
5830:
5774:
5759:
5727:
5665:
1712:
has length not greater than 2. In this sense, the Cantor function is extremal.
108:
5996:
5179:
5104:
6166:
5906:
5882:
5752:
5722:
5705:
5670:
5655:
5439:
5404:
5354:
5317:
5261:
5187:
5112:
4409:
4405:
4309:
3103:
26:
2140:
The three definitions are compatible at the end-points 1/3 and 2/3, because
6151:
6146:
6047:
6027:
5784:
5717:
1973:} of functions on the unit interval that converges to the Cantor function.
1957:
100:
6112:
6032:
5742:
5737:
4401:
3952:
Arbitrary finite-length strings in the letters L and R correspond to the
1513:
1103:
This formula is well-defined, since every member of the Cantor set has a
33:
2626:, except if the 1 is followed by zeros only (in which case the tail 1000
5965:
5950:
5945:
5926:
5660:
5430:
5413:
5362:
5252:
5235:
5069:
4456:
Note that the Cantor function bears more than a passing resemblance to
4316:
is then the monoid of all such finite-length left-right moves. Writing
2615:
1758: > 0 there are finitely many pairwise disjoint intervals (
1728:
1517:
1509:
739:
57:
4981:, a function that is continuous everywhere but differentiable nowhere.
56:
in analysis, because it challenges naive intuitions about continuity,
5921:
5867:
5779:
5630:
5532:
5470:. Wiley series in probability and statistics. John Wiley & sons.
5447:
Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008) .
5384:
5346:
5061:
2952:
and a pair of magnifications, one on the left and one on the right:
331:
contains a 1, replace every digit strictly after the first 1 with 0.
5822:
2855:
21:
5270:
Darst, Richard B.; Palagallo, Judith A.; Price, Thomas E. (2010),
5162:
3092:{\displaystyle c\left({\frac {x+2}{3}}\right)={\frac {1+c(x)}{2}}}
1627:
The Cantor function is also a standard example of a function with
139:
5769:
5572:
5416:[General investigations on rectification of the curves].
5372:
Leçons sur l'intégration et la recherche des fonctions primitives
2667:
1747:
over which the Cantor function cumulatively rises more than
5840:
5295:
5150:
Mathematical
Proceedings of the Cambridge Philosophical Society
5085:
Mathematical
Proceedings of the Cambridge Philosophical Society
4180:
1846:{\displaystyle \sum \limits _{k=1}^{M}(y_{k}-x_{k})<\delta }
3010:{\displaystyle c\left({\frac {x}{3}}\right)={\frac {c(x)}{2}}}
1594:, has no discrete part. That is, the corresponding measure is
5274:, Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd.,
1387:; though it is continuous everywhere and has zero derivative
2666:
to get rid of any 1). It turns out that the Cantor set is a
1937:{\displaystyle \sum \limits _{k=1}^{M}(c(y_{k})-c(x_{k}))=1}
5541:
1086:
4179:
stand for L and R. Function composition extends this to a
3643:
The two sides can be mirrored one onto the other, in that
2767:. We may define the Cantor function alternatively as the
5414:"Allgemeine Untersuchungen über Rectification der Curven"
5376:
Lessons on integration and search for primitive functions
4641:{\displaystyle C_{z}(y)=\sum _{k=1}^{\infty }b_{k}z^{k}.}
5522:
5446:
5219:(Second ed.). Createspace Independent Publishing.
5009:
3956:, in that every dyadic rational can be written as both
2405:
denotes the limit function, it follows that, for every
337:
Interpret the result as a binary number. The result is
119:. The Cantor function was discussed and popularized by
3102:
The magnifications can be cascaded; they generate the
1540:
1426:
1397:
1379:
The Cantor function challenges naive intuitions about
1139:
703:{\displaystyle c({\tfrac {200}{243}})={\tfrac {3}{4}}}
689:
671:
636:
607:
576:
558:
523:
494:
463:
445:
410:
381:
4774:
4717:
4571:
4477:
4417:
4390:{\displaystyle \gamma _{D}\circ c=c\circ \gamma _{C}}
4351:
4322:
4249:
4189:
4158:
4131:
4080:
4011:
3962:
3823:
3800:
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1683:
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1329:
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1113:
807:
748:
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663:
634:
605:
550:
521:
492:
437:
408:
379:
343:
317:
294:
261:
229:
209:
153:
18:
Continuous function that is not absolutely continuous
5269:
1520:containing the interval endpoints described above.
4941:
4756:
4640:
4531:{\displaystyle y=\sum _{k=1}^{\infty }b_{k}2^{-k}}
4530:
4446:
4389:
4334:
4288:
4235:
4171:
4144:
4114:
4066:
3989:
3941:
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3396:
3347:
3295:
3209:
3186:
3142:
3091:
3009:
2941:
2882:
2838:
2771:-dimensional volume of sections of the Cantor set
2755:
2693:
2658:
2638:
2598:
2390:
1936:
1845:
1704:
1669:
1582:
1432:
1412:
1364:{\displaystyle x\in \smallsetminus {\mathcal {C}}}
1363:
1315:
1274:
1250:
1230:
1195:
1152:
1123:
1092:
790:
730:
702:
649:
620:
590:{\displaystyle c({\tfrac {1}{5}})={\tfrac {1}{4}}}
589:
536:
507:
477:{\displaystyle c({\tfrac {1}{4}})={\tfrac {1}{3}}}
476:
423:
394:
358:
323:
300:
276:
247:
215:
195:
5487:Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur.
5133:: CS1 maint: DOI inactive as of September 2024 (
1620:The Cantor function is the standard example of a
6164:
5050:Proceedings of the American Mathematical Society
4820:
2515:
2420:
2291:
2192:
1000:
4067:{\displaystyle y=0.b_{1}b_{2}b_{3}\cdots b_{m}}
5451:(Second ed.). ClassicalRealAnalysis.com.
4682:) is the Cantor function. In general, for any
4655: = 1/3, the inverse of the function
2614:The Cantor function is closely related to the
1715:
5557:
1456:. It is constant on intervals of the form (0.
4106:
4094:
3296:{\displaystyle (r\circ c)(x)=r(c(x))=1-c(x)}
3153:The first self-symmetry can be expressed as
1590:. This probability distribution, called the
1525:cumulative probability distribution function
1523:The Cantor function can also be seen as the
1452: = log 2/log 3) but not
989:
977:
143:Iterated construction of the Cantor function
5564:
5550:
5246:. International Press of Boston: 381–392.
4545:(binary) expansion of the real number 0 ≤
1947:
1743:sub-intervals with total length <
5429:
5411:
5251:
5161:
5147:
4434:
3749:{\displaystyle L_{C}\circ r=r\circ R_{C}}
3691:{\displaystyle L_{D}\circ r=r\circ R_{D}}
3633:{\displaystyle R_{D}\circ c=c\circ R_{C}}
3575:{\displaystyle R_{C}(x)={\frac {2+x}{3}}}
3518:{\displaystyle R_{D}(x)={\frac {1+x}{2}}}
3455:{\displaystyle L_{D}\circ c=c\circ L_{C}}
2513:
2289:
1952:
1639:
1016:
120:
5424:. International Press of Boston: 49–82.
5369:
5082:
3465:Similarly, define the right mappings as
2839:{\displaystyle f(x)=H_{D}(C\cap (0,x)).}
138:
124:
25:The graph of the Cantor function on the
20:
6118:List of fractals by Hausdorff dimension
5465:
4997:
4236:{\displaystyle g_{010}=g_{0}g_{1}g_{0}}
3397:{\displaystyle L_{C}(x)={\frac {x}{3}}}
3348:{\displaystyle L_{D}(x)={\frac {x}{2}}}
3217:denotes function composition. That is,
1168:is a member of the Cantor set). Since
6165:
5484:
5468:The theory of measures and integration
5332:
5233:
2854:The Cantor function possesses several
1614:
128:
104:
5545:
5523:
5382:
5047:
5010:Thomson, Bruckner & Bruckner 2008
5211:
5035:
4670:) is the Cantor function. That is,
4312:of such strings. The dyadic monoid
2162:, by induction. One may check that
1739:, there exists a finite sequence of
5217:Real analysis for graduate students
4757:{\displaystyle (\log 2/\log 3)^{2}}
4447:{\displaystyle SL(2,\mathbb {Z} ).}
1861:
1788:
621:{\displaystyle {\tfrac {200}{243}}}
13:
5341:(2). Informa UK Limited: 136–140.
4610:
4500:
4463:
4458:Minkowski's question-mark function
4296:for some binary strings of digits
3814:in all but a few places, one has:
2890:, there is a reflection symmetry
2849:
1956:
1356:
1267:
1116:
1075:
1025:
957:
954:
951:
942:
903:
847:
723:
14:
6204:
6100:How Long Is the Coast of Britain?
5498:
4289:{\displaystyle g_{A}g_{B}=g_{AB}}
4115:{\displaystyle b_{k}\in \{0,1\}.}
3187:{\displaystyle r\circ c=c\circ r}
2756:{\displaystyle D=\log(2)/\log(3)}
2609:
334:Replace any remaining 2s with 1s.
5386:A first course in Sobolev spaces
1316:{\displaystyle 0\leq c(x)\leq 1}
311:If the base-3 representation of
308:in base 3, using digits 0, 1, 2.
5508:at Encyclopaedia of Mathematics
3407:Then the Cantor function obeys
2378:
1153:{\displaystyle {\tfrac {1}{3}}}
650:{\displaystyle {\tfrac {3}{4}}}
537:{\displaystyle {\tfrac {1}{4}}}
508:{\displaystyle {\tfrac {1}{5}}}
424:{\displaystyle {\tfrac {1}{3}}}
395:{\displaystyle {\tfrac {1}{4}}}
113:fundamental theorem of calculus
6124:The Fractal Geometry of Nature
5517:Wolfram Demonstrations Project
5378:], Paris: Gauthier-Villars
5141:
5076:
5041:
5029:
5015:
5003:
4991:
4925:
4919:
4872:
4869:
4851:
4848:
4827:
4813:
4807:
4745:
4718:
4588:
4582:
4549:≤ 1 in terms of binary digits
4438:
4424:
3548:
3542:
3491:
3485:
3378:
3372:
3329:
3323:
3290:
3284:
3269:
3266:
3260:
3254:
3245:
3239:
3236:
3224:
3125:
3119:
3080:
3074:
2998:
2992:
2936:
2924:
2909:
2903:
2830:
2827:
2815:
2806:
2790:
2784:
2750:
2744:
2730:
2724:
2705:) takes a finite value, where
2589:
2585:
2579:
2563:
2557:
2543:
2537:
2525:
2487:
2483:
2477:
2461:
2455:
2448:
2442:
2430:
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2367:
2361:
2339:
2333:
2319:
2313:
2301:
2272:
2268:
2262:
2246:
2240:
2220:
2214:
2202:
2015:) will be defined in terms of
1925:
1922:
1909:
1900:
1887:
1881:
1834:
1808:
1693:
1687:
1658:
1652:
1577:
1574:
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1544:
1407:
1401:
1348:
1336:
1304:
1298:
1275:{\displaystyle {\mathcal {C}}}
1219:
1213:
1184:
1178:
1124:{\displaystyle {\mathcal {C}}}
1067:
1055:
1041:
1035:
817:
811:
785:
773:
770:
767:
755:
742:on , then the Cantor function
731:{\displaystyle {\mathcal {C}}}
682:
667:
569:
554:
456:
441:
353:
347:
271:
265:
242:
230:
190:
178:
175:
172:
160:
147:To define the Cantor function
1:
5205:
4562:. Then consider the function
4005:and as finite length of bits
2942:{\displaystyle c(x)=1-c(1-x)}
2883:{\displaystyle 0\leq x\leq 1}
1534:supported on the Cantor set:
1374:
134:
5571:
5310:10.1016/j.exmath.2005.05.002
4335:{\displaystyle \gamma \in M}
2763:is the fractal dimension of
2158:(1) = 1 for every
1964:Below we define a sequence {
1607:probability density function
1603:probability density function
81:Lebesgue's singular function
7:
6140:Chaos: Making a New Science
5023:"Cantor Staircase Function"
4967:
1716:Lack of absolute continuity
1605:; integrating any putative
10:
6209:
5412:Scheeffer, Ludwig (1884).
2624:base-3 (triadic) expansion
6091:
6015:
5964:
5935:
5851:
5821:
5803:
5644:
5579:
5298:Expositiones Mathematicae
5180:10.1017/S0305004113000698
5105:10.1017/S0305004103006960
3990:{\displaystyle y=n/2^{m}}
2087:) = 1/2, when
93:Cantor staircase function
5449:Elementary real analysis
5383:Leoni, Giovanni (2017).
5304:(1). Elsevier BV: 1–37.
4985:
4183:, in that one can write
3143:{\displaystyle r(x)=1-x}
1994:Then, for every integer
1583:{\textstyle c(x)=\mu ()}
284:by the following steps:
97:Cantor–Lebesgue function
5515:by Douglas Rivers, the
5506:Cantor ternary function
5107:(inactive 2024-09-26).
2659:{\displaystyle \ldots }
2646:can be replaced by 0222
2639:{\displaystyle \ldots }
1948:Alternative definitions
1731:is 0, for any positive
111:to an extension of the
73:Cantor ternary function
6132:The Beauty of Fractals
5466:Vestrup, E.M. (2003).
5213:Bass, Richard Franklin
4961:question mark function
4943:
4758:
4642:
4614:
4532:
4504:
4448:
4391:
4336:
4290:
4237:
4173:
4146:
4116:
4068:
3991:
3943:
3808:
3807:{\displaystyle \circ }
3788:
3787:{\displaystyle LRLLR.}
3750:
3692:
3634:
3576:
3519:
3456:
3398:
3349:
3297:
3211:
3210:{\displaystyle \circ }
3188:
3144:
3093:
3011:
2943:
2884:
2840:
2757:
2695:
2660:
2640:
2600:
2409: ≥ 0,
2392:
1961:
1953:Iterative construction
1938:
1880:
1847:
1807:
1735: < 1 and
1706:
1705:{\displaystyle f(1)=1}
1671:
1670:{\displaystyle f(0)=0}
1584:
1434:
1414:
1365:
1317:
1276:
1252:
1232:
1231:{\displaystyle c(1)=1}
1197:
1196:{\displaystyle c(0)=0}
1154:
1125:
1094:
907:
851:
792:
791:{\displaystyle c:\to }
732:
704:
651:
622:
591:
538:
509:
478:
425:
396:
360:
325:
302:
278:
249:
217:
197:
196:{\displaystyle c:\to }
144:
85:Cantor–Vitali function
71:It is also called the
29:
5370:Lebesgue, H. (1904),
4974:Dyadic transformation
4944:
4759:
4686: < 1/2,
4643:
4594:
4560:dyadic transformation
4533:
4484:
4449:
4392:
4337:
4308:is just the ordinary
4291:
4238:
4174:
4172:{\displaystyle g_{1}}
4147:
4145:{\displaystyle g_{0}}
4117:
4069:
3992:
3944:
3809:
3789:
3751:
3693:
3635:
3577:
3520:
3457:
3399:
3350:
3298:
3212:
3189:
3145:
3094:
3012:
2944:
2885:
2841:
2758:
2696:
2694:{\displaystyle H_{D}}
2661:
2641:
2601:
2393:
1960:
1939:
1860:
1848:
1787:
1707:
1672:
1585:
1454:absolutely continuous
1435:
1415:
1366:
1318:
1277:
1253:
1233:
1198:
1155:
1126:
1095:
887:
831:
793:
733:
705:
652:
623:
592:
539:
510:
479:
426:
397:
361:
326:
303:
279:
250:
218:
198:
142:
50:absolutely continuous
24:
6078:Lewis Fry Richardson
6073:Hamid Naderi Yeganeh
5863:Burning Ship fractal
5795:Weierstrass function
5335:Mathematics Magazine
4979:Weierstrass function
4883: does not exist
4772:
4715:
4569:
4475:
4415:
4349:
4320:
4247:
4187:
4156:
4129:
4078:
4009:
3960:
3821:
3798:
3763:
3708:
3650:
3592:
3529:
3472:
3414:
3359:
3310:
3221:
3201:
3160:
3113:
3027:
2959:
2897:
2862:
2778:
2709:
2678:
2674:-dimensional volume
2650:
2630:
2416:
2188:
2001:, the next function
1857:
1784:
1726:uncountably infinite
1681:
1646:
1538:
1442:uniformly continuous
1424:
1420:goes from 0 to 1 as
1395:
1327:
1286:
1262:
1242:
1207:
1172:
1135:
1111:
805:
746:
718:
661:
632:
603:
548:
519:
490:
435:
406:
377:
359:{\displaystyle c(x)}
341:
315:
292:
277:{\displaystyle c(x)}
259:
227:
207:
151:
52:. It is a notorious
40:is an example of a
5836:Space-filling curve
5813:Multifractal system
5696:Space-filling curve
5681:Sierpinski triangle
5234:Cantor, G. (1884).
5172:2014MPCPS.156..295T
5097:2004MPCPS.136..167F
2701:(in the sense of a
1754:In fact, for every
1592:Cantor distribution
1512:at any point in an
1323:also holds for all
1282:, it is clear that
6063:Aleksandr Lyapunov
6043:Desmond Paul Henry
6007:Self-avoiding walk
6002:Percolation theory
5646:Iterated function
5587:Fractal dimensions
5525:Weisstein, Eric W.
5431:10.1007/bf02421552
5253:10.1007/bf02418423
4939:
4841:
4754:
4638:
4528:
4444:
4387:
4332:
4286:
4233:
4169:
4142:
4112:
4064:
3987:
3939:
3804:
3784:
3746:
3688:
3630:
3572:
3515:
3452:
3394:
3345:
3293:
3207:
3184:
3140:
3089:
3007:
2939:
2880:
2836:
2753:
2691:
2656:
2636:
2596:
2541:
2446:
2388:
2317:
2218:
2114:1/2 + 1/2 ×
1962:
1934:
1843:
1702:
1667:
1580:
1444:(precisely, it is
1430:
1410:
1361:
1313:
1272:
1248:
1228:
1193:
1150:
1148:
1121:
1090:
1085:
1031:
798:can be defined as
788:
728:
700:
698:
680:
647:
645:
618:
616:
587:
585:
567:
534:
532:
505:
503:
474:
472:
454:
421:
419:
392:
390:
356:
321:
298:
274:
245:
213:
193:
145:
30:
6183:Special functions
6160:
6159:
6106:Coastline paradox
6083:Wacław Sierpiński
6068:Benoit Mandelbrot
5992:Fractal landscape
5900:Misiurewicz point
5805:Strange attractor
5686:Apollonian gasket
5676:Sierpinski carpet
5528:"Cantor Function"
5458:978-1-4348-4367-8
5396:978-1-4704-2921-8
5281:978-981-4291-28-6
5226:978-1-4818-6914-0
4957:Hermann Minkowski
4951:self-similar sets
4884:
4879:
4819:
4709:fractal dimension
4703:approaches zero.
3585:Then, likewise,
3570:
3513:
3392:
3343:
3197:where the symbol
3087:
3053:
3005:
2977:
2703:Hausdorff-measure
2618:. The Cantor set
2514:
2419:
2290:
2287:
2191:
2149:(0) = 0 and
1772:) (1 ≤
1741:pairwise disjoint
1636:rectifiable curve
1629:bounded variation
1622:singular function
1611:almost everywhere
1529:Bernoulli measure
1446:Hölder continuous
1413:{\textstyle c(x)}
1389:almost everywhere
1251:{\displaystyle c}
1147:
999:
963:
949:
935:
874:
714:Equivalently, if
697:
679:
644:
615:
584:
566:
531:
502:
471:
453:
418:
389:
324:{\displaystyle x}
301:{\displaystyle x}
223:be any number in
216:{\displaystyle x}
89:Devil's staircase
77:Lebesgue function
6200:
6023:Michael Barnsley
5890:Lyapunov fractal
5748:Sierpiński curve
5701:Blancmange curve
5566:
5559:
5552:
5543:
5542:
5538:
5537:
5494:
5481:
5462:
5443:
5433:
5418:Acta Mathematica
5408:
5379:
5366:
5329:
5292:
5265:
5255:
5240:Acta Mathematica
5230:
5200:
5199:
5165:
5145:
5139:
5138:
5132:
5124:
5080:
5074:
5073:
5045:
5039:
5033:
5027:
5026:
5019:
5013:
5007:
5001:
4995:
4948:
4946:
4945:
4940:
4938:
4937:
4932:
4928:
4909:
4908:
4890:
4886:
4885:
4882:
4880:
4875:
4843:
4840:
4839:
4838:
4806:
4784:
4783:
4763:
4761:
4760:
4755:
4753:
4752:
4734:
4647:
4645:
4644:
4639:
4634:
4633:
4624:
4623:
4613:
4608:
4581:
4580:
4537:
4535:
4534:
4529:
4527:
4526:
4514:
4513:
4503:
4498:
4453:
4451:
4450:
4445:
4437:
4396:
4394:
4393:
4388:
4386:
4385:
4361:
4360:
4341:
4339:
4338:
4333:
4295:
4293:
4292:
4287:
4285:
4284:
4269:
4268:
4259:
4258:
4242:
4240:
4239:
4234:
4232:
4231:
4222:
4221:
4212:
4211:
4199:
4198:
4178:
4176:
4175:
4170:
4168:
4167:
4151:
4149:
4148:
4143:
4141:
4140:
4121:
4119:
4118:
4113:
4090:
4089:
4073:
4071:
4070:
4065:
4063:
4062:
4050:
4049:
4040:
4039:
4030:
4029:
3996:
3994:
3993:
3988:
3986:
3985:
3976:
3954:dyadic rationals
3948:
3946:
3945:
3940:
3938:
3937:
3928:
3927:
3918:
3917:
3908:
3907:
3898:
3897:
3873:
3872:
3863:
3862:
3853:
3852:
3843:
3842:
3833:
3832:
3813:
3811:
3810:
3805:
3793:
3791:
3790:
3785:
3755:
3753:
3752:
3747:
3745:
3744:
3720:
3719:
3697:
3695:
3694:
3689:
3687:
3686:
3662:
3661:
3639:
3637:
3636:
3631:
3629:
3628:
3604:
3603:
3581:
3579:
3578:
3573:
3571:
3566:
3555:
3541:
3540:
3524:
3522:
3521:
3516:
3514:
3509:
3498:
3484:
3483:
3461:
3459:
3458:
3453:
3451:
3450:
3426:
3425:
3403:
3401:
3400:
3395:
3393:
3385:
3371:
3370:
3354:
3352:
3351:
3346:
3344:
3336:
3322:
3321:
3302:
3300:
3299:
3294:
3216:
3214:
3213:
3208:
3193:
3191:
3190:
3185:
3149:
3147:
3146:
3141:
3098:
3096:
3095:
3090:
3088:
3083:
3063:
3058:
3054:
3049:
3038:
3016:
3014:
3013:
3008:
3006:
3001:
2987:
2982:
2978:
2970:
2948:
2946:
2945:
2940:
2889:
2887:
2886:
2881:
2845:
2843:
2842:
2837:
2805:
2804:
2762:
2760:
2759:
2754:
2737:
2700:
2698:
2697:
2692:
2690:
2689:
2665:
2663:
2662:
2657:
2645:
2643:
2642:
2637:
2605:
2603:
2602:
2597:
2592:
2578:
2577:
2556:
2555:
2546:
2540:
2512:
2511:
2490:
2476:
2475:
2451:
2445:
2397:
2395:
2394:
2389:
2374:
2360:
2359:
2332:
2331:
2322:
2316:
2288:
2280:
2275:
2261:
2260:
2239:
2238:
2223:
2217:
2181:, one sees that
2136:
2128:
2094:
2069:
2061:
2000:
1943:
1941:
1940:
1935:
1921:
1920:
1899:
1898:
1879:
1874:
1852:
1850:
1849:
1844:
1833:
1832:
1820:
1819:
1806:
1801:
1722:Lebesgue measure
1711:
1709:
1708:
1703:
1676:
1674:
1673:
1668:
1640:Scheeffer (1884)
1589:
1587:
1586:
1581:
1439:
1437:
1436:
1431:
1419:
1417:
1416:
1411:
1370:
1368:
1367:
1362:
1360:
1359:
1322:
1320:
1319:
1314:
1281:
1279:
1278:
1273:
1271:
1270:
1258:is monotonic on
1257:
1255:
1254:
1249:
1237:
1235:
1234:
1229:
1202:
1200:
1199:
1194:
1159:
1157:
1156:
1151:
1149:
1140:
1130:
1128:
1127:
1122:
1120:
1119:
1099:
1097:
1096:
1091:
1089:
1088:
1079:
1078:
1030:
1029:
1028:
973:
972:
961:
960:
947:
946:
945:
936:
934:
933:
924:
923:
922:
909:
906:
901:
875:
873:
872:
863:
862:
853:
850:
845:
797:
795:
794:
789:
737:
735:
734:
729:
727:
726:
709:
707:
706:
701:
699:
690:
681:
672:
656:
654:
653:
648:
646:
637:
627:
625:
624:
619:
617:
608:
596:
594:
593:
588:
586:
577:
568:
559:
543:
541:
540:
535:
533:
524:
514:
512:
511:
506:
504:
495:
483:
481:
480:
475:
473:
464:
455:
446:
430:
428:
427:
422:
420:
411:
401:
399:
398:
393:
391:
382:
365:
363:
362:
357:
330:
328:
327:
322:
307:
305:
304:
299:
283:
281:
280:
275:
254:
252:
251:
248:{\displaystyle }
246:
222:
220:
219:
214:
202:
200:
199:
194:
121:Scheeffer (1884)
101:Georg Cantor
6208:
6207:
6203:
6202:
6201:
6199:
6198:
6197:
6163:
6162:
6161:
6156:
6087:
6038:Felix Hausdorff
6011:
5975:Brownian motion
5960:
5931:
5854:
5847:
5817:
5799:
5790:Pythagoras tree
5647:
5640:
5636:Self-similarity
5580:Characteristics
5575:
5570:
5513:Cantor Function
5501:
5478:
5459:
5397:
5347:10.2307/2690689
5282:
5227:
5208:
5203:
5146:
5142:
5126:
5125:
5081:
5077:
5062:10.2307/2159830
5046:
5042:
5034:
5030:
5021:
5020:
5016:
5008:
5004:
4996:
4992:
4988:
4970:
4933:
4904:
4900:
4899:
4895:
4894:
4881:
4844:
4842:
4834:
4830:
4823:
4799:
4792:
4788:
4779:
4775:
4773:
4770:
4769:
4764:. Subsequently
4748:
4744:
4730:
4716:
4713:
4712:
4694:
4665:
4629:
4625:
4619:
4615:
4609:
4598:
4576:
4572:
4570:
4567:
4566:
4557:
4519:
4515:
4509:
4505:
4499:
4488:
4476:
4473:
4472:
4466:
4464:Generalizations
4433:
4416:
4413:
4412:
4381:
4377:
4356:
4352:
4350:
4347:
4346:
4321:
4318:
4317:
4277:
4273:
4264:
4260:
4254:
4250:
4248:
4245:
4244:
4243:and generally,
4227:
4223:
4217:
4213:
4207:
4203:
4194:
4190:
4188:
4185:
4184:
4163:
4159:
4157:
4154:
4153:
4136:
4132:
4130:
4127:
4126:
4085:
4081:
4079:
4076:
4075:
4058:
4054:
4045:
4041:
4035:
4031:
4025:
4021:
4010:
4007:
4006:
3981:
3977:
3972:
3961:
3958:
3957:
3933:
3929:
3923:
3919:
3913:
3909:
3903:
3899:
3893:
3889:
3868:
3864:
3858:
3854:
3848:
3844:
3838:
3834:
3828:
3824:
3822:
3819:
3818:
3799:
3796:
3795:
3764:
3761:
3760:
3740:
3736:
3715:
3711:
3709:
3706:
3705:
3682:
3678:
3657:
3653:
3651:
3648:
3647:
3624:
3620:
3599:
3595:
3593:
3590:
3589:
3556:
3554:
3536:
3532:
3530:
3527:
3526:
3499:
3497:
3479:
3475:
3473:
3470:
3469:
3446:
3442:
3421:
3417:
3415:
3412:
3411:
3384:
3366:
3362:
3360:
3357:
3356:
3335:
3317:
3313:
3311:
3308:
3307:
3222:
3219:
3218:
3202:
3199:
3198:
3161:
3158:
3157:
3114:
3111:
3110:
3064:
3062:
3039:
3037:
3033:
3028:
3025:
3024:
2988:
2986:
2969:
2965:
2960:
2957:
2956:
2898:
2895:
2894:
2863:
2860:
2859:
2852:
2850:Self-similarity
2800:
2796:
2779:
2776:
2775:
2733:
2710:
2707:
2706:
2685:
2681:
2679:
2676:
2675:
2651:
2648:
2647:
2631:
2628:
2627:
2612:
2588:
2573:
2569:
2551:
2547:
2542:
2518:
2498:
2494:
2486:
2471:
2467:
2447:
2423:
2417:
2414:
2413:
2370:
2349:
2345:
2327:
2323:
2318:
2294:
2279:
2271:
2256:
2252:
2228:
2224:
2219:
2195:
2189:
2186:
2185:
2180:
2170:
2157:
2148:
2130:
2122:
2113:
2107:
2088:
2082:
2063:
2055:
2046:
2040:
2023:
2010:
1995:
1982:
1972:
1955:
1950:
1916:
1912:
1894:
1890:
1875:
1864:
1858:
1855:
1854:
1828:
1824:
1815:
1811:
1802:
1791:
1785:
1782:
1781:
1770:
1763:
1718:
1682:
1679:
1678:
1647:
1644:
1643:
1539:
1536:
1535:
1527:of the 1/2-1/2
1507:
1500:
1494:
1488:
1481:
1474:
1468:
1462:
1425:
1422:
1421:
1396:
1393:
1392:
1377:
1355:
1354:
1328:
1325:
1324:
1287:
1284:
1283:
1266:
1265:
1263:
1260:
1259:
1243:
1240:
1239:
1208:
1205:
1204:
1173:
1170:
1169:
1167:
1163:
1138:
1136:
1133:
1132:
1115:
1114:
1112:
1109:
1108:
1084:
1083:
1074:
1073:
1047:
1024:
1023:
1003:
996:
995:
968:
964:
950:
941:
940:
929:
925:
918:
914:
910:
908:
902:
891:
879:
868:
864:
858:
854:
852:
846:
835:
824:
823:
806:
803:
802:
747:
744:
743:
722:
721:
719:
716:
715:
688:
670:
662:
659:
658:
635:
633:
630:
629:
606:
604:
601:
600:
575:
557:
549:
546:
545:
522:
520:
517:
516:
493:
491:
488:
487:
462:
444:
436:
433:
432:
409:
407:
404:
403:
380:
378:
375:
374:
342:
339:
338:
316:
313:
312:
293:
290:
289:
260:
257:
256:
228:
225:
224:
208:
205:
204:
152:
149:
148:
137:
125:Lebesgue (1904)
38:Cantor function
19:
12:
11:
5:
6206:
6196:
6195:
6193:De Rham curves
6190:
6185:
6180:
6178:Measure theory
6175:
6158:
6157:
6155:
6154:
6149:
6144:
6136:
6128:
6120:
6115:
6110:
6109:
6108:
6095:
6093:
6089:
6088:
6086:
6085:
6080:
6075:
6070:
6065:
6060:
6055:
6053:Helge von Koch
6050:
6045:
6040:
6035:
6030:
6025:
6019:
6017:
6013:
6012:
6010:
6009:
6004:
5999:
5994:
5989:
5988:
5987:
5985:Brownian motor
5982:
5971:
5969:
5962:
5961:
5959:
5958:
5956:Pickover stalk
5953:
5948:
5942:
5940:
5933:
5932:
5930:
5929:
5924:
5919:
5914:
5912:Newton fractal
5909:
5904:
5903:
5902:
5895:Mandelbrot set
5892:
5887:
5886:
5885:
5880:
5878:Newton fractal
5875:
5865:
5859:
5857:
5849:
5848:
5846:
5845:
5844:
5843:
5833:
5831:Fractal canopy
5827:
5825:
5819:
5818:
5816:
5815:
5809:
5807:
5801:
5800:
5798:
5797:
5792:
5787:
5782:
5777:
5775:Vicsek fractal
5772:
5767:
5762:
5757:
5756:
5755:
5750:
5745:
5740:
5735:
5730:
5725:
5720:
5715:
5714:
5713:
5703:
5693:
5691:Fibonacci word
5688:
5683:
5678:
5673:
5668:
5666:Koch snowflake
5663:
5658:
5652:
5650:
5642:
5641:
5639:
5638:
5633:
5628:
5627:
5626:
5621:
5616:
5611:
5606:
5605:
5604:
5594:
5583:
5581:
5577:
5576:
5569:
5568:
5561:
5554:
5546:
5540:
5539:
5520:
5510:
5500:
5499:External links
5497:
5496:
5495:
5482:
5477:978-0471249771
5476:
5463:
5457:
5444:
5409:
5395:
5380:
5367:
5330:
5293:
5280:
5272:Curious curves
5267:
5231:
5225:
5207:
5204:
5202:
5201:
5156:(2): 295–311.
5140:
5091:(1): 167–174.
5075:
5056:(1): 105–108.
5040:
5028:
5014:
5012:, p. 252.
5002:
5000:, Section 4.6.
4989:
4987:
4984:
4983:
4982:
4976:
4969:
4966:
4936:
4931:
4927:
4924:
4921:
4918:
4915:
4912:
4907:
4903:
4898:
4893:
4889:
4878:
4874:
4871:
4868:
4865:
4862:
4859:
4856:
4853:
4850:
4847:
4837:
4833:
4829:
4826:
4822:
4818:
4815:
4812:
4809:
4805:
4802:
4798:
4795:
4791:
4787:
4782:
4778:
4751:
4747:
4743:
4740:
4737:
4733:
4729:
4726:
4723:
4720:
4690:
4663:
4649:
4648:
4637:
4632:
4628:
4622:
4618:
4612:
4607:
4604:
4601:
4597:
4593:
4590:
4587:
4584:
4579:
4575:
4553:
4539:
4538:
4525:
4522:
4518:
4512:
4508:
4502:
4497:
4494:
4491:
4487:
4483:
4480:
4465:
4462:
4443:
4440:
4436:
4432:
4429:
4426:
4423:
4420:
4406:de Rham curves
4398:
4397:
4384:
4380:
4376:
4373:
4370:
4367:
4364:
4359:
4355:
4331:
4328:
4325:
4283:
4280:
4276:
4272:
4267:
4263:
4257:
4253:
4230:
4226:
4220:
4216:
4210:
4206:
4202:
4197:
4193:
4166:
4162:
4139:
4135:
4111:
4108:
4105:
4102:
4099:
4096:
4093:
4088:
4084:
4061:
4057:
4053:
4048:
4044:
4038:
4034:
4028:
4024:
4020:
4017:
4014:
3984:
3980:
3975:
3971:
3968:
3965:
3950:
3949:
3936:
3932:
3926:
3922:
3916:
3912:
3906:
3902:
3896:
3892:
3888:
3885:
3882:
3879:
3876:
3871:
3867:
3861:
3857:
3851:
3847:
3841:
3837:
3831:
3827:
3803:
3783:
3780:
3777:
3774:
3771:
3768:
3757:
3756:
3743:
3739:
3735:
3732:
3729:
3726:
3723:
3718:
3714:
3701:and likewise,
3699:
3698:
3685:
3681:
3677:
3674:
3671:
3668:
3665:
3660:
3656:
3641:
3640:
3627:
3623:
3619:
3616:
3613:
3610:
3607:
3602:
3598:
3583:
3582:
3569:
3565:
3562:
3559:
3553:
3550:
3547:
3544:
3539:
3535:
3512:
3508:
3505:
3502:
3496:
3493:
3490:
3487:
3482:
3478:
3463:
3462:
3449:
3445:
3441:
3438:
3435:
3432:
3429:
3424:
3420:
3405:
3404:
3391:
3388:
3383:
3380:
3377:
3374:
3369:
3365:
3342:
3339:
3334:
3331:
3328:
3325:
3320:
3316:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3206:
3195:
3194:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3151:
3150:
3139:
3136:
3133:
3130:
3127:
3124:
3121:
3118:
3100:
3099:
3086:
3082:
3079:
3076:
3073:
3070:
3067:
3061:
3057:
3052:
3048:
3045:
3042:
3036:
3032:
3018:
3017:
3004:
3000:
2997:
2994:
2991:
2985:
2981:
2976:
2973:
2968:
2964:
2950:
2949:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2879:
2876:
2873:
2870:
2867:
2851:
2848:
2847:
2846:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2803:
2799:
2795:
2792:
2789:
2786:
2783:
2752:
2749:
2746:
2743:
2740:
2736:
2732:
2729:
2726:
2723:
2720:
2717:
2714:
2688:
2684:
2655:
2635:
2611:
2610:Fractal volume
2608:
2607:
2606:
2595:
2591:
2587:
2584:
2581:
2576:
2572:
2568:
2565:
2562:
2559:
2554:
2550:
2545:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2517:
2510:
2507:
2504:
2501:
2497:
2493:
2489:
2485:
2482:
2479:
2474:
2470:
2466:
2463:
2460:
2457:
2454:
2450:
2444:
2441:
2438:
2435:
2432:
2429:
2426:
2422:
2399:
2398:
2387:
2384:
2381:
2377:
2373:
2369:
2366:
2363:
2358:
2355:
2352:
2348:
2344:
2341:
2338:
2335:
2330:
2326:
2321:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2293:
2286:
2283:
2278:
2274:
2270:
2267:
2264:
2259:
2255:
2251:
2248:
2245:
2242:
2237:
2234:
2231:
2227:
2222:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2194:
2175:
2166:
2153:
2144:
2118:
2102:
2077:
2051:
2035:
2028:) as follows:
2019:
2005:
1980:
1968:
1954:
1951:
1949:
1946:
1933:
1930:
1927:
1924:
1919:
1915:
1911:
1908:
1905:
1902:
1897:
1893:
1889:
1886:
1883:
1878:
1873:
1870:
1867:
1863:
1842:
1839:
1836:
1831:
1827:
1823:
1818:
1814:
1810:
1805:
1800:
1797:
1794:
1790:
1768:
1761:
1717:
1714:
1701:
1698:
1695:
1692:
1689:
1686:
1666:
1663:
1660:
1657:
1654:
1651:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1516:subset of the
1505:
1498:
1492:
1486:
1479:
1472:
1466:
1460:
1448:of exponent
1433:{\textstyle x}
1429:
1409:
1406:
1403:
1400:
1376:
1373:
1358:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1269:
1247:
1227:
1224:
1221:
1218:
1215:
1212:
1192:
1189:
1186:
1183:
1180:
1177:
1165:
1161:
1146:
1143:
1118:
1101:
1100:
1087:
1082:
1077:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1046:
1043:
1040:
1037:
1034:
1027:
1022:
1019:
1015:
1012:
1009:
1006:
1002:
998:
997:
994:
991:
988:
985:
982:
979:
976:
971:
967:
959:
956:
953:
944:
939:
932:
928:
921:
917:
913:
905:
900:
897:
894:
890:
886:
883:
880:
878:
871:
867:
861:
857:
849:
844:
841:
838:
834:
830:
829:
827:
822:
819:
816:
813:
810:
787:
784:
781:
778:
775:
772:
769:
766:
763:
760:
757:
754:
751:
725:
712:
711:
696:
693:
687:
684:
678:
675:
669:
666:
643:
640:
614:
611:
598:
583:
580:
574:
571:
565:
562:
556:
553:
530:
527:
501:
498:
485:
470:
467:
461:
458:
452:
449:
443:
440:
417:
414:
388:
385:
368:
367:
355:
352:
349:
346:
335:
332:
320:
309:
297:
273:
270:
267:
264:
244:
241:
238:
235:
232:
212:
192:
189:
186:
183:
180:
177:
174:
171:
168:
165:
162:
159:
156:
136:
133:
109:counterexample
54:counterexample
17:
9:
6:
4:
3:
2:
6205:
6194:
6191:
6189:
6186:
6184:
6181:
6179:
6176:
6174:
6171:
6170:
6168:
6153:
6150:
6148:
6145:
6142:
6141:
6137:
6134:
6133:
6129:
6126:
6125:
6121:
6119:
6116:
6114:
6111:
6107:
6104:
6103:
6101:
6097:
6096:
6094:
6090:
6084:
6081:
6079:
6076:
6074:
6071:
6069:
6066:
6064:
6061:
6059:
6056:
6054:
6051:
6049:
6046:
6044:
6041:
6039:
6036:
6034:
6031:
6029:
6026:
6024:
6021:
6020:
6018:
6014:
6008:
6005:
6003:
6000:
5998:
5995:
5993:
5990:
5986:
5983:
5981:
5980:Brownian tree
5978:
5977:
5976:
5973:
5972:
5970:
5967:
5963:
5957:
5954:
5952:
5949:
5947:
5944:
5943:
5941:
5938:
5934:
5928:
5925:
5923:
5920:
5918:
5915:
5913:
5910:
5908:
5907:Multibrot set
5905:
5901:
5898:
5897:
5896:
5893:
5891:
5888:
5884:
5883:Douady rabbit
5881:
5879:
5876:
5874:
5871:
5870:
5869:
5866:
5864:
5861:
5860:
5858:
5856:
5850:
5842:
5839:
5838:
5837:
5834:
5832:
5829:
5828:
5826:
5824:
5820:
5814:
5811:
5810:
5808:
5806:
5802:
5796:
5793:
5791:
5788:
5786:
5783:
5781:
5778:
5776:
5773:
5771:
5768:
5766:
5763:
5761:
5758:
5754:
5753:Z-order curve
5751:
5749:
5746:
5744:
5741:
5739:
5736:
5734:
5731:
5729:
5726:
5724:
5723:Hilbert curve
5721:
5719:
5716:
5712:
5709:
5708:
5707:
5706:De Rham curve
5704:
5702:
5699:
5698:
5697:
5694:
5692:
5689:
5687:
5684:
5682:
5679:
5677:
5674:
5672:
5671:Menger sponge
5669:
5667:
5664:
5662:
5659:
5657:
5656:Barnsley fern
5654:
5653:
5651:
5649:
5643:
5637:
5634:
5632:
5629:
5625:
5622:
5620:
5617:
5615:
5612:
5610:
5607:
5603:
5600:
5599:
5598:
5595:
5593:
5590:
5589:
5588:
5585:
5584:
5582:
5578:
5574:
5567:
5562:
5560:
5555:
5553:
5548:
5547:
5544:
5535:
5534:
5529:
5526:
5521:
5518:
5514:
5511:
5509:
5507:
5503:
5502:
5492:
5488:
5483:
5479:
5473:
5469:
5464:
5460:
5454:
5450:
5445:
5441:
5437:
5432:
5427:
5423:
5419:
5415:
5410:
5406:
5402:
5398:
5392:
5388:
5387:
5381:
5377:
5373:
5368:
5364:
5360:
5356:
5352:
5348:
5344:
5340:
5336:
5331:
5327:
5323:
5319:
5315:
5311:
5307:
5303:
5299:
5294:
5291:
5287:
5283:
5277:
5273:
5268:
5263:
5259:
5254:
5249:
5245:
5241:
5237:
5232:
5228:
5222:
5218:
5214:
5210:
5209:
5197:
5193:
5189:
5185:
5181:
5177:
5173:
5169:
5164:
5159:
5155:
5151:
5144:
5136:
5130:
5122:
5118:
5114:
5110:
5106:
5102:
5098:
5094:
5090:
5086:
5079:
5071:
5067:
5063:
5059:
5055:
5051:
5044:
5038:, p. 28.
5037:
5032:
5024:
5018:
5011:
5006:
4999:
4994:
4990:
4980:
4977:
4975:
4972:
4971:
4965:
4962:
4958:
4954:
4952:
4934:
4929:
4922:
4916:
4913:
4910:
4905:
4901:
4896:
4891:
4887:
4876:
4866:
4863:
4860:
4857:
4854:
4845:
4835:
4831:
4824:
4816:
4810:
4803:
4800:
4796:
4793:
4789:
4785:
4780:
4776:
4767:
4749:
4741:
4738:
4735:
4731:
4727:
4724:
4721:
4710:
4704:
4702:
4698:
4693:
4689:
4685:
4681:
4677:
4674: =
4673:
4669:
4662:
4658:
4654:
4635:
4630:
4626:
4620:
4616:
4605:
4602:
4599:
4595:
4591:
4585:
4577:
4573:
4565:
4564:
4563:
4561:
4556:
4552:
4548:
4544:
4523:
4520:
4516:
4510:
4506:
4495:
4492:
4489:
4485:
4481:
4478:
4471:
4470:
4469:
4461:
4459:
4454:
4441:
4430:
4427:
4421:
4418:
4411:
4410:modular group
4407:
4403:
4382:
4378:
4374:
4371:
4368:
4365:
4362:
4357:
4353:
4345:
4344:
4343:
4329:
4326:
4323:
4315:
4311:
4310:concatenation
4307:
4303:
4299:
4281:
4278:
4274:
4270:
4265:
4261:
4255:
4251:
4228:
4224:
4218:
4214:
4208:
4204:
4200:
4195:
4191:
4182:
4164:
4160:
4137:
4133:
4123:
4109:
4103:
4100:
4097:
4091:
4086:
4082:
4059:
4055:
4051:
4046:
4042:
4036:
4032:
4026:
4022:
4018:
4015:
4012:
4004:
4000:
3982:
3978:
3973:
3969:
3966:
3963:
3955:
3934:
3930:
3924:
3920:
3914:
3910:
3904:
3900:
3894:
3890:
3886:
3883:
3880:
3877:
3874:
3869:
3865:
3859:
3855:
3849:
3845:
3839:
3835:
3829:
3825:
3817:
3816:
3815:
3801:
3781:
3778:
3775:
3772:
3769:
3766:
3741:
3737:
3733:
3730:
3727:
3724:
3721:
3716:
3712:
3704:
3703:
3702:
3683:
3679:
3675:
3672:
3669:
3666:
3663:
3658:
3654:
3646:
3645:
3644:
3625:
3621:
3617:
3614:
3611:
3608:
3605:
3600:
3596:
3588:
3587:
3586:
3567:
3563:
3560:
3557:
3551:
3545:
3537:
3533:
3510:
3506:
3503:
3500:
3494:
3488:
3480:
3476:
3468:
3467:
3466:
3447:
3443:
3439:
3436:
3433:
3430:
3427:
3422:
3418:
3410:
3409:
3408:
3389:
3386:
3381:
3375:
3367:
3363:
3340:
3337:
3332:
3326:
3318:
3314:
3306:
3305:
3304:
3287:
3281:
3278:
3275:
3272:
3263:
3257:
3251:
3248:
3242:
3233:
3230:
3227:
3204:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3156:
3155:
3154:
3137:
3134:
3131:
3128:
3122:
3116:
3109:
3108:
3107:
3105:
3104:dyadic monoid
3084:
3077:
3071:
3068:
3065:
3059:
3055:
3050:
3046:
3043:
3040:
3034:
3030:
3023:
3022:
3021:
3002:
2995:
2989:
2983:
2979:
2974:
2971:
2966:
2962:
2955:
2954:
2953:
2933:
2930:
2927:
2921:
2918:
2915:
2912:
2906:
2900:
2893:
2892:
2891:
2877:
2874:
2871:
2868:
2865:
2857:
2833:
2824:
2821:
2818:
2812:
2809:
2801:
2797:
2793:
2787:
2781:
2774:
2773:
2772:
2770:
2766:
2747:
2741:
2738:
2734:
2727:
2721:
2718:
2715:
2712:
2704:
2686:
2682:
2673:
2669:
2653:
2633:
2625:
2621:
2617:
2593:
2582:
2574:
2570:
2566:
2560:
2552:
2548:
2534:
2531:
2528:
2522:
2519:
2508:
2505:
2502:
2499:
2495:
2491:
2480:
2472:
2468:
2464:
2458:
2452:
2439:
2436:
2433:
2427:
2424:
2412:
2411:
2410:
2408:
2404:
2385:
2382:
2379:
2375:
2364:
2356:
2353:
2350:
2346:
2342:
2336:
2328:
2324:
2310:
2307:
2304:
2298:
2295:
2284:
2281:
2276:
2265:
2257:
2253:
2249:
2243:
2235:
2232:
2229:
2225:
2211:
2208:
2205:
2199:
2196:
2184:
2183:
2182:
2178:
2174:
2169:
2165:
2161:
2156:
2152:
2147:
2143:
2138:
2134:
2129:, when
2126:
2121:
2117:
2111:
2105:
2101:
2096:
2092:
2086:
2080:
2076:
2071:
2067:
2062:, when
2059:
2054:
2050:
2044:
2038:
2034:
2029:
2027:
2022:
2018:
2014:
2008:
2004:
1998:
1992:
1990:
1986:
1979:
1974:
1971:
1967:
1959:
1945:
1931:
1928:
1917:
1913:
1906:
1903:
1895:
1891:
1884:
1876:
1871:
1868:
1865:
1840:
1837:
1829:
1825:
1821:
1816:
1812:
1803:
1798:
1795:
1792:
1779:
1776: ≤
1775:
1771:
1764:
1757:
1752:
1750:
1746:
1742:
1738:
1734:
1730:
1727:
1723:
1713:
1699:
1696:
1690:
1684:
1664:
1661:
1655:
1649:
1641:
1637:
1632:
1630:
1625:
1623:
1618:
1616:
1615:Vitali (1905)
1612:
1608:
1604:
1599:
1597:
1593:
1571:
1568:
1565:
1556:
1553:
1547:
1541:
1533:
1530:
1526:
1521:
1519:
1515:
1511:
1504:
1497:
1491:
1485:
1482:022222..., 0.
1478:
1471:
1465:
1459:
1455:
1451:
1447:
1443:
1427:
1404:
1398:
1390:
1386:
1382:
1372:
1351:
1345:
1342:
1339:
1333:
1330:
1310:
1307:
1301:
1295:
1292:
1289:
1245:
1225:
1222:
1216:
1210:
1190:
1187:
1181:
1175:
1144:
1141:
1106:
1080:
1070:
1064:
1061:
1058:
1052:
1049:
1044:
1038:
1032:
1020:
1017:
1013:
1010:
1007:
1004:
992:
986:
983:
980:
974:
969:
965:
937:
930:
926:
919:
915:
911:
898:
895:
892:
888:
884:
881:
876:
869:
865:
859:
855:
842:
839:
836:
832:
825:
820:
814:
808:
801:
800:
799:
782:
779:
776:
764:
761:
758:
752:
749:
741:
694:
691:
685:
676:
673:
664:
641:
638:
612:
609:
599:
581:
578:
572:
563:
560:
551:
528:
525:
499:
496:
486:
468:
465:
459:
450:
447:
438:
415:
412:
386:
383:
373:
372:
371:
370:For example:
350:
344:
336:
333:
318:
310:
295:
287:
286:
285:
268:
262:
239:
236:
233:
210:
187:
184:
181:
169:
166:
163:
157:
154:
141:
132:
130:
129:Vitali (1905)
126:
122:
118:
114:
110:
106:
102:
98:
94:
90:
86:
82:
78:
74:
69:
67:
66:monotonically
63:
59:
55:
51:
47:
43:
39:
35:
28:
27:unit interval
23:
16:
6188:Georg Cantor
6152:Chaos theory
6147:Kaleidoscope
6138:
6130:
6122:
6048:Gaston Julia
6028:Georg Cantor
5853:Escape-time
5785:Gosper curve
5733:Lévy C curve
5718:Dragon curve
5597:Box-counting
5531:
5505:
5490:
5486:
5467:
5448:
5421:
5417:
5385:
5375:
5371:
5338:
5334:
5301:
5297:
5271:
5243:
5239:
5216:
5153:
5149:
5143:
5129:cite journal
5088:
5084:
5078:
5053:
5049:
5043:
5031:
5017:
5005:
4998:Vestrup 2003
4993:
4955:
4705:
4700:
4696:
4691:
4687:
4683:
4679:
4675:
4671:
4667:
4660:
4656:
4652:
4650:
4554:
4550:
4546:
4540:
4467:
4455:
4399:
4313:
4305:
4301:
4297:
4124:
4002:
3998:
3997:for integer
3951:
3758:
3700:
3642:
3584:
3464:
3406:
3196:
3152:
3101:
3019:
2951:
2853:
2768:
2764:
2671:
2619:
2613:
2406:
2402:
2400:
2176:
2172:
2167:
2163:
2159:
2154:
2150:
2145:
2141:
2139:
2132:
2124:
2119:
2115:
2109:
2103:
2099:
2097:
2093:≤ 2/3
2090:
2084:
2078:
2074:
2072:
2068:≤ 1/3
2065:
2057:
2052:
2048:
2042:
2036:
2032:
2030:
2025:
2020:
2016:
2012:
2006:
2002:
1996:
1993:
1988:
1984:
1977:
1975:
1969:
1965:
1963:
1777:
1773:
1766:
1759:
1755:
1753:
1748:
1744:
1736:
1732:
1720:Because the
1719:
1633:
1626:
1619:
1609:that is not
1600:
1531:
1522:
1502:
1495:
1489:
1483:
1476:
1469:
1463:
1457:
1449:
1378:
1164:= 0.02222...
1104:
1102:
713:
369:
146:
96:
92:
88:
84:
80:
76:
72:
70:
37:
31:
15:
6143:(1987 book)
6135:(1986 book)
6127:(1982 book)
6113:Fractal art
6033:Bill Gosper
5997:Lévy flight
5743:Peano curve
5738:Moore curve
5624:Topological
5609:Correlation
5493:: 1021–1034
4402:binary tree
2047:1/2 ×
1514:uncountable
255:and obtain
115:claimed by
34:mathematics
6167:Categories
5951:Orbit trap
5946:Buddhabrot
5939:techniques
5927:Mandelbulb
5728:Koch curve
5661:Cantor set
5206:References
2856:symmetries
2616:Cantor set
2127:− 2)
1729:Cantor set
1518:Cantor set
1510:derivative
1381:continuity
1375:Properties
740:Cantor set
135:Definition
95:, and the
58:derivative
48:, but not
46:continuous
6058:Paul Lévy
5937:Rendering
5922:Mandelbox
5868:Julia set
5780:Hexaflake
5711:Minkowski
5631:Recursion
5614:Hausdorff
5533:MathWorld
5440:0001-5962
5405:976406106
5355:0025-570X
5318:0723-0869
5262:0001-5962
5215:(2013) .
5188:1469-8064
5163:1301.1286
5121:122381614
5113:1469-8064
5036:Bass 2013
4923:μ
4917:
4911:
4846:μ
4828:→
4786:
4739:
4725:
4659:= 2
4611:∞
4596:∑
4521:−
4501:∞
4486:∑
4379:γ
4375:∘
4363:∘
4354:γ
4327:∈
4324:γ
4092:∈
4052:⋯
3887:∘
3875:∘
3802:∘
3734:∘
3722:∘
3676:∘
3664:∘
3618:∘
3606:∘
3440:∘
3428:∘
3279:−
3231:∘
3205:∘
3179:∘
3167:∘
3135:−
2931:−
2919:−
2875:≤
2869:≤
2813:∩
2742:
2722:
2654:…
2634:…
2567:−
2523:∈
2500:−
2492:≤
2465:−
2428:∈
2383:≥
2354:−
2343:−
2299:∈
2277:≤
2250:−
2200:∈
2123:(3
2112:) =
2045:) =
1999:≥ 0
1904:−
1862:∑
1841:δ
1822:−
1789:∑
1557:μ
1352:∖
1334:∈
1308:≤
1293:≤
1071:∖
1053:∈
1021:∈
1008:≤
975:∈
938:∈
904:∞
889:∑
848:∞
833:∑
771:→
176:→
6173:Fractals
5968:fractals
5855:fractals
5823:L-system
5765:T-square
5573:Fractals
5196:56402751
4968:See also
4804:′
4766:Falconer
4304:, where
1596:atomless
288:Express
44:that is
42:function
5917:Tricorn
5770:n-flake
5619:Packing
5602:Higuchi
5592:Assouad
5363:2690689
5326:2195181
5290:2681574
5168:Bibcode
5093:Bibcode
5070:2159830
4541:be the
2668:fractal
1780:) with
1724:of the
1385:measure
738:is the
117:Harnack
103: (
62:measure
6016:People
5966:Random
5873:Filled
5841:H tree
5760:String
5648:system
5474:
5455:
5438:
5403:
5393:
5361:
5353:
5324:
5316:
5288:
5278:
5260:
5223:
5194:
5186:
5119:
5111:
5068:
4543:dyadic
4181:monoid
2858:. For
2131:2/3 ≤
2089:1/3 ≤
1238:, and
1105:unique
962:
948:
203:, let
91:, the
87:, the
83:, the
75:, the
68:grow.
60:, and
36:, the
6092:Other
5374:[
5359:JSTOR
5192:S2CID
5158:arXiv
5117:S2CID
5066:JSTOR
4986:Notes
4074:with
1160:= 0.1
657:, so
544:, so
431:, so
5472:ISBN
5453:ISBN
5436:ISSN
5401:OCLC
5391:ISBN
5351:ISSN
5314:ISSN
5276:ISBN
5258:ISSN
5221:ISBN
5184:ISSN
5135:link
5109:ISSN
4914:supp
4651:For
4468:Let
4152:and
4001:and
3525:and
3355:and
3020:and
2098:Let
2073:Let
2064:0 ≤
2031:Let
1987:) =
1976:Let
1853:and
1838:<
1677:and
1383:and
1203:and
127:and
105:1884
5426:doi
5343:doi
5306:doi
5248:doi
5176:doi
5154:156
5101:doi
5089:136
5058:doi
5054:119
4959:'s
4902:dim
4821:lim
4777:dim
4736:log
4722:log
4664:1/3
4196:010
2739:log
2719:log
2516:max
2421:max
2401:If
2292:max
2193:max
2135:≤ 1
1638:.
1501:...
1475:...
1001:sup
677:243
674:200
613:243
610:200
32:In
6169::
6102:"
5530:.
5491:40
5489:,
5434:.
5420:.
5399:.
5357:.
5349:.
5339:67
5337:.
5322:MR
5320:.
5312:.
5302:24
5300:.
5286:MR
5284:,
5256:.
5242:.
5190:.
5182:.
5174:.
5166:.
5152:.
5131:}}
5127:{{
5115:.
5099:.
5087:.
5064:.
5052:.
4953:.
4306:AB
4300:,
4019:0.
2386:1.
2179:+1
2137:.
2106:+1
2095:;
2081:+1
2070:;
2056:(3
2039:+1
2009:+1
1991:.
1944:.
1751:.
1624:.
1391:,
1371:.
131:.
123:,
99:.
79:,
6098:"
5565:e
5558:t
5551:v
5536:.
5519:.
5480:.
5461:.
5442:.
5428::
5422:5
5407:.
5365:.
5345::
5328:.
5308::
5264:.
5250::
5244:4
5229:.
5198:.
5178::
5170::
5160::
5137:)
5123:.
5103::
5095::
5072:.
5060::
5025:.
4935:2
4930:)
4926:)
4920:(
4906:H
4897:(
4892:=
4888:}
4877:h
4873:)
4870:]
4867:h
4864:+
4861:x
4858:,
4855:x
4852:[
4849:(
4836:+
4832:0
4825:h
4817:=
4814:)
4811:x
4808:(
4801:f
4797::
4794:x
4790:{
4781:H
4750:2
4746:)
4742:3
4732:/
4728:2
4719:(
4701:z
4697:y
4695:(
4692:z
4688:C
4684:z
4680:x
4678:(
4676:y
4672:y
4668:y
4666:(
4661:C
4657:x
4653:z
4636:.
4631:k
4627:z
4621:k
4617:b
4606:1
4603:=
4600:k
4592:=
4589:)
4586:y
4583:(
4578:z
4574:C
4555:k
4551:b
4547:y
4524:k
4517:2
4511:k
4507:b
4496:1
4493:=
4490:k
4482:=
4479:y
4442:.
4439:)
4435:Z
4431:,
4428:2
4425:(
4422:L
4419:S
4383:C
4372:c
4369:=
4366:c
4358:D
4330:M
4314:M
4302:B
4298:A
4282:B
4279:A
4275:g
4271:=
4266:B
4262:g
4256:A
4252:g
4229:0
4225:g
4219:1
4215:g
4209:0
4205:g
4201:=
4192:g
4165:1
4161:g
4138:0
4134:g
4110:.
4107:}
4104:1
4101:,
4098:0
4095:{
4087:k
4083:b
4060:m
4056:b
4047:3
4043:b
4037:2
4033:b
4027:1
4023:b
4016:=
4013:y
4003:m
3999:n
3983:m
3979:2
3974:/
3970:n
3967:=
3964:y
3935:C
3931:R
3925:C
3921:L
3915:C
3911:L
3905:C
3901:R
3895:C
3891:L
3884:c
3881:=
3878:c
3870:D
3866:R
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3856:L
3850:D
3846:L
3840:D
3836:R
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3826:L
3782:.
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3773:L
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3742:C
3738:R
3731:r
3728:=
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3684:D
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3626:C
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3612:=
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3382:=
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3333:=
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3243:x
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3237:)
3234:c
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3123:x
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3060:=
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3051:3
3047:2
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3041:x
3035:(
3031:c
3003:2
2999:)
2996:x
2993:(
2990:c
2984:=
2980:)
2975:3
2972:x
2967:(
2963:c
2937:)
2934:x
2928:1
2925:(
2922:c
2916:1
2913:=
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2907:x
2904:(
2901:c
2878:1
2872:x
2866:0
2834:.
2831:)
2828:)
2825:x
2822:,
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2807:(
2802:D
2798:H
2794:=
2791:)
2788:x
2785:(
2782:f
2769:D
2765:C
2751:)
2748:3
2745:(
2735:/
2731:)
2728:2
2725:(
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2687:D
2683:H
2672:D
2620:C
2594:.
2590:|
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2583:x
2580:(
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2561:x
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2459:x
2456:(
2453:f
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2380:n
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2368:)
2365:x
2362:(
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2337:x
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2329:n
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2308:,
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2302:[
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2244:x
2241:(
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2212:1
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2206:0
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2041:(
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2024:(
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2017:f
2013:x
2011:(
2007:n
2003:f
1997:n
1989:x
1985:x
1983:(
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1932:1
1929:=
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1923:)
1918:k
1914:x
1910:(
1907:c
1901:)
1896:k
1892:y
1888:(
1885:c
1882:(
1877:M
1872:1
1869:=
1866:k
1835:)
1830:k
1826:x
1817:k
1813:y
1809:(
1804:M
1799:1
1796:=
1793:k
1778:M
1774:k
1769:k
1767:y
1765:,
1762:k
1760:x
1756:δ
1749:ε
1745:δ
1737:δ
1733:ε
1700:1
1697:=
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1691:1
1688:(
1685:f
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1662:=
1659:)
1656:0
1653:(
1650:f
1578:)
1575:]
1572:x
1569:,
1566:0
1563:[
1560:(
1554:=
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1548:x
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1542:c
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1506:n
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1493:2
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1302:x
1299:(
1296:c
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1191:0
1188:=
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877:,
870:n
866:2
860:n
856:a
843:1
840:=
837:n
826:{
821:=
818:)
815:x
812:(
809:c
786:]
783:1
780:,
777:0
774:[
768:]
765:1
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756:[
753::
750:c
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710:.
695:4
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686:=
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668:(
665:c
642:4
639:3
597:.
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579:1
573:=
570:)
564:5
561:1
555:(
552:c
529:4
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497:1
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469:3
466:1
460:=
457:)
451:4
448:1
442:(
439:c
416:3
413:1
387:4
384:1
366:.
354:)
351:x
348:(
345:c
319:x
296:x
272:)
269:x
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231:[
211:x
191:]
188:1
185:,
182:0
179:[
173:]
170:1
167:,
164:0
161:[
158::
155:c
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