31:
2374:
sequence of functions with bounded
Lipschitz constant, then it has a convergent subsequence. By the result of the previous paragraph, the limit function is also Lipschitz, with the same bound for the Lipschitz constant. In particular the set of all real-valued Lipschitz functions on a compact
1309:
3698:
1642:
3248:
3098:
2890:
879:
2335:
Lipschitz constants, however. In fact, the space of all
Lipschitz functions on a compact metric space is a subalgebra of the Banach space of continuous functions, and thus dense in it, an elementary consequence of the
1105:
2629:
436:
892:
passing through a point on the graph of the function forms a circular cone, and a function is
Lipschitz if and only if the graph of the function everywhere lies completely outside of this cone (see figure).
3450:
1071:
3375:
648:
2158:
2282:
2477:
2440:
1759:≤1 gives an example of a function that is differentiable on a compact set while not locally Lipschitz because its derivative function is not bounded. See also the first property below.
2327:. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the
2094:
1526:
1425:
185:
2041:
34:
For a
Lipschitz continuous function, there exists a double cone (white) whose origin can be moved along the graph so that the whole graph always stays outside the double cone
2795:
2782:
2750:
3134:
2714:
1855:. In particular, any continuously differentiable function is locally Lipschitz, as continuous functions are locally bounded so its gradient is locally bounded as well.
1501:
143:
1668:
2218:
2198:
2178:
2961:
884:
For real-valued functions of several real variables, this holds if and only if the absolute value of the slopes of all secant lines are bounded by
754:
3649:
Prospects in
Mathematics: Invited Talks on the Occasion of the 250th Anniversary of Princeton University, March 17-21, 1996, Princeton University
1304:{\displaystyle {\frac {1}{K}}d_{X}(x_{1},x_{2})\leq d_{Y}(f(x_{1}),f(x_{2}))\leq Kd_{X}(x_{1},x_{2})\quad {\text{ for all }}x_{1},x_{2}\in X,}
3540:
3507:
3128:
could have a very large
Lipschitz constant but a moderately sized, or even negative, one-sided Lipschitz constant. For example, the function
2672:. Such a structure allows one to define locally Lipschitz maps between such manifolds, similarly to how one defines smooth maps between
2531:
68:
is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the
319:
3694:
2911:: a PL structure gives rise to a unique Lipschitz structure. While Lipschitz manifolds are closely related to topological manifolds,
3656:
3589:
3564:
3480:
2341:
3380:
3788:
978:
90:
3706:
3644:
3327:
543:
2479:) is Lipschitz continuous as well, with the same Lipschitz constant, provided it assumes a finite value at least at a point.
2337:
82:). For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous.
3750:
Donchev, Tzanko; Farkhi, Elza (1998). "Stability and Euler
Approximation of One-sided Lipschitz Differential Inclusions".
3736:
2360:
1511:
on a vector space is
Lipschitz continuous with respect to the associated metric, with the Lipschitz constant equal to 1.
3731:
3307:
1504:
1446:
function is
Lipschitz continuous because its derivative, the cosine function, is bounded above by 1 in absolute value.
117:
2099:
72:
of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the
909:
102:
2904:
2223:
2445:
2408:
1637:{\displaystyle f(x)\;=\;{\begin{cases}x^{2}\sin(1/x)&{\text{if }}x\neq 0\\0&{\text{if }}x=0\end{cases}}}
2058:
3783:
2912:
1961:
1432:
1378:
3726:
1435:
and the absolute value of the derivative is bounded above by 1. See the first property listed below under "
61:
3297:
2384:
1871:
1716:
196:
158:
94:
86:
78:
39:
2908:
2661:
2348:
2002:
1964:
extends the differentiability result to
Lipschitz mappings between Euclidean spaces: a Lipschitz map
1772:
1708:
1517:
Lipschitz continuous functions that are everywhere differentiable but not continuously differentiable
509:
255:
202:
147:
3243:{\displaystyle {\begin{cases}F:\mathbf {R} ^{2}\to \mathbf {R} ,\\F(x,y)=-50(y-\cos(x))\end{cases}}}
3143:
1552:
3286:
2352:
2316:
1883:
1356:
onto its image. A bilipschitz function is the same thing as an injective Lipschitz function whose
960:
501:
98:
65:
2755:
2723:
2331:
of continuous functions. This result does not hold for sequences in which the functions may have
1707:
approaches 0 since its derivative becomes infinite. However, it is uniformly continuous, and both
523:
is called Lipschitz continuous if there exists a positive real constant K such that, for all real
2519:
1852:
1508:
1349:
57:
943:. In spaces that are not locally compact, this is a necessary but not a sufficient condition.
3652:
3626:
3585:
3560:
3536:
3524:
3503:
3497:
3476:
3470:
2687:
2371:
1985:
1875:
1784:
1461:
128:
3759:
3618:
2665:
2044:
1879:
1848:
1503:
defined on the reals is Lipschitz continuous with the Lipschitz constant equal to 1, by the
1357:
53:
3714:
3711:
3532:
3292:
2932:
2673:
939:
is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of
932:
30:
2294:} is a sequence of Lipschitz continuous mappings between two metric spaces, and that all
1647:
2668:
whose transition maps are bilipschitz; this is possible because bilipschitz maps form a
3302:
2356:
2203:
2183:
2163:
727:
69:
3777:
3622:
3093:{\displaystyle (x_{1}-x_{2})^{T}(F(x_{1})-F(x_{2}))\leq C\Vert x_{1}-x_{2}\Vert ^{2}}
2885:{\displaystyle \psi ^{-1}\circ f\circ \phi :U\cap (f\circ \phi )^{-1}(\psi (V))\to V}
1353:
50:
3673:
2328:
1929:
is absolutely continuous and thus differentiable almost everywhere, and satisfies |
216:
17:
2502:
is a Lipschitz continuous function, there always exist Lipschitz continuous maps
1427:
defined for all real numbers is Lipschitz continuous with the Lipschitz constant
2669:
658:
109:
2355:. More generally, a set of functions with bounded Lipschitz constant forms an
726:. Otherwise, one can equivalently define a function to be Lipschitz continuous
3763:
3324:, and any real number 0<ε<1, there exists a (1+ε)-bi-Lipschitz function
1988:
3261:= 0. An example which is one-sided Lipschitz but not Lipschitz continuous is
2892:
is locally Lipschitz. This definition does not rely on defining a metric on
874:{\displaystyle {\frac {d_{Y}(f(x_{1}),f(x_{2}))}{d_{X}(x_{1},x_{2})}}\leq K.}
487:
3703:
Miniconferences on harmonic analysis and operator algebras (Canberra, 1987)
3630:
3607:"'Dilatation' and 'dilation': trends in use on both sides of the Atlantic"
3606:
47:
108:
We have the following chain of strict inclusions for functions over a
2792:
are open sets in the corresponding Euclidean spaces, the composition
1452:
Lipschitz continuous functions that are not everywhere differentiable
3496:
Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2001).
1725:
Differentiable functions that are not (locally) Lipschitz continuous
93:
which guarantees the existence and uniqueness of the solution to an
2624:{\displaystyle {\tilde {f}}(x):=\inf _{u\in U}\{f(u)+k\,d(x,u)\},}
29:
1806:
Lipschitz continuous. This function becomes arbitrarily steep as
1676:
Continuous functions that are not (globally) Lipschitz continuous
1369:
Lipschitz continuous functions that are everywhere differentiable
431:{\displaystyle d_{Y}(f(x_{1}),f(x_{2}))\leq Kd_{X}(x_{1},x_{2}).}
1810:
approaches infinity. It is however locally Lipschitz continuous.
1703:
Lipschitz continuous. This function becomes infinitely steep as
1644:, whose derivative exists but has an essential discontinuity at
1443:
3531:, Springer undergraduate mathematics series, Berlin, New York:
2323:
is also Lipschitz, with Lipschitz constant bounded by the same
1765:
Analytic functions that are not (globally) Lipschitz continuous
3647:(1999). "Quantitative Homotopy Theory". In Rossi, Hugo (ed.).
3605:
Mahroo, Omar A; Shalchi, Zaid; Hammond, Christopher J (2014).
3445:{\displaystyle d=\lceil 15(\ln |X|)/\varepsilon ^{2}\rceil .}
2344:, because every polynomial is locally Lipschitz continuous).
3236:
1630:
1088:. Sometimes a Hölder condition of order α is also called a
2915:
allows one to do analysis, yielding various applications.
1878:, that is, differentiable at every point outside a set of
1066:{\displaystyle d_{Y}(f(x),f(y))\leq Md_{X}(x,y)^{\alpha }}
1953:
is Lipschitz continuous with Lipschitz constant at most
499:
maps a metric space to itself, the function is called a
3370:{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{d},}
712:
In general, the inequality is (trivially) satisfied if
643:{\displaystyle |f(x_{1})-f(x_{2})|\leq K|x_{1}-x_{2}|.}
89:, Lipschitz continuity is the central condition of the
3580:
Burago, Dmitri; Burago, Yuri; Ivanov, Sergei (2001).
3383:
3330:
3289: – Function reducing distance between all points
3137:
2964:
2798:
2758:
2726:
2690:
2534:
2448:
2411:
2226:
2206:
2186:
2166:
2102:
2061:
2005:
1650:
1529:
1464:
1381:
1108:
981:
757:
546:
322:
161:
131:
2720:if and only if for every pair of coordinate charts
3444:
3369:
3242:
3092:
2884:
2776:
2744:
2708:
2623:
2471:
2434:
2276:
2212:
2192:
2172:
2152:
2088:
2035:
1662:
1636:
1495:
1419:
1303:
1065:
873:
642:
430:
179:
137:
3699:"Applications of analysis on Lipschitz manifolds"
2903:This structure is intermediate between that of a
97:. A special type of Lipschitz continuity, called
2560:
2450:
2413:
1886:in magnitude by the Lipschitz constant, and for
1783:globally Lipschitz continuous, despite being an
2398:For a family of Lipschitz continuous functions
2153:{\displaystyle \|Df\|_{W^{1,\infty }(U)}\leq K}
3555:Benyamini, Yoav; Lindenstrauss, Joram (2000).
3651:. American Mathematical Society. p. 46.
3559:. American Mathematical Society. p. 11.
1910:) is equal to the integral of the derivative
8:
3436:
3390:
3081:
3054:
2615:
2575:
2237:
2227:
2113:
2103:
2024:
2006:
461:. The smallest constant is sometimes called
3689:
3687:
2277:{\displaystyle \|Df\|_{W^{1,\infty }(U)}=K}
927:is Lipschitz continuous. Equivalently, if
3475:. Vol. 231. Birkhäuser. p. 142.
2472:{\displaystyle \inf _{\alpha }f_{\alpha }}
2435:{\displaystyle \sup _{\alpha }f_{\alpha }}
1546:
1542:
3430:
3421:
3413:
3405:
3382:
3358:
3354:
3353:
3343:
3339:
3338:
3329:
3167:
3158:
3153:
3138:
3136:
3084:
3074:
3061:
3036:
3014:
2995:
2985:
2972:
2963:
2849:
2803:
2797:
2757:
2725:
2689:
2684:are Lipschitz manifolds, then a function
2596:
2563:
2536:
2535:
2533:
2463:
2453:
2447:
2426:
2416:
2410:
2245:
2240:
2225:
2205:
2185:
2165:
2121:
2116:
2101:
2080:
2076:
2075:
2060:
2004:
1649:
1613:
1590:
1577:
1559:
1547:
1528:
1488:
1480:
1463:
1431: = 1, because it is everywhere
1403:
1397:
1380:
1286:
1273:
1264:
1254:
1241:
1228:
1206:
1184:
1165:
1149:
1136:
1123:
1109:
1107:
1057:
1035:
986:
980:
850:
837:
824:
806:
784:
765:
758:
756:
632:
626:
613:
604:
593:
584:
562:
547:
545:
416:
403:
390:
368:
346:
327:
321:
160:
130:
3752:SIAM Journal on Control and Optimization
3257:= 50 and a one-sided Lipschitz constant
2514:and have the same Lipschitz constant as
2301:have Lipschitz constant bounded by some
3557:Geometric Nonlinear Functional Analysis
3461:
2089:{\displaystyle f:U\to \mathbb {R} ^{m}}
112:non-trivial interval of the real line:
76:of the function (and is related to the
2160:holds for the best Lipschitz constant
1823:An everywhere differentiable function
1420:{\displaystyle f(x)={\sqrt {x^{2}+5}}}
1719:on (both of which imply the former).
7:
1847:)|) if and only if it has a bounded
1090:uniform Lipschitz condition of order
2405:with common constant, the function
2055:For a differentiable Lipschitz map
2947:) is a closed, convex set for all
2387:convex subset of the Banach space
2347:Every Lipschitz continuous map is
2252:
2128:
1995:is the best Lipschitz constant of
180:{\displaystyle 0<\alpha \leq 1}
25:
3675:Continuity and Uniform Continuity
3584:. American Mathematical Society.
3124:It is possible that the function
2379:having Lipschitz constant ≤
2342:Weierstrass approximation theorem
1851:; one direction follows from the
27:Strong form of uniform continuity
3623:10.1136/bjophthalmol-2014-304986
3611:British Journal of Ophthalmology
3168:
3154:
2486:is a subset of the metric space
1874:and therefore is differentiable
1802:with domain all real numbers is
291:if there exists a real constant
2338:Stone–Weierstrass theorem
2036:{\displaystyle \|Df(x)\|\leq K}
1715:for α ≤ 1/2 and also
1436:
1263:
3707:Australian National University
3502:. Prentice-Hall. p. 623.
3418:
3414:
3406:
3396:
3349:
3230:
3227:
3221:
3206:
3194:
3182:
3164:
3045:
3042:
3029:
3020:
3007:
3001:
2992:
2965:
2876:
2873:
2870:
2864:
2858:
2846:
2833:
2768:
2736:
2700:
2612:
2600:
2587:
2581:
2553:
2547:
2541:
2522:). An extension is provided by
2263:
2257:
2139:
2133:
2071:
2021:
2015:
1835:is Lipschitz continuous (with
1585:
1571:
1539:
1533:
1489:
1481:
1474:
1468:
1391:
1385:
1260:
1234:
1215:
1212:
1199:
1190:
1177:
1171:
1155:
1129:
1054:
1041:
1022:
1019:
1013:
1004:
998:
992:
856:
830:
815:
812:
799:
790:
777:
771:
633:
605:
594:
590:
577:
568:
555:
548:
422:
396:
377:
374:
361:
352:
339:
333:
1:
2639:is a Lipschitz constant for
1775:becomes arbitrarily steep as
888:. The set of lines of slope
485:= 1 the function is called a
463:the (best) Lipschitz constant
79:modulus of uniform continuity
3523:Searcóid, Mícheál Ó (2006),
2777:{\displaystyle \psi :V\to N}
2745:{\displaystyle \phi :U\to M}
1344:to mean there exists such a
898:locally Lipschitz continuous
3732:Encyclopedia of Mathematics
3582:A Course in Metric Geometry
3308:Johnson-Lindenstrauss lemma
1505:reverse triangle inequality
1348:. A bilipschitz mapping is
968:if there exists a constant
946:More generally, a function
457:may also be referred to as
118:Continuously differentiable
64:. Intuitively, a Lipschitz
3805:
2955:is one-sided Lipschitz if
103:Banach fixed-point theorem
3764:10.1137/S0363012995293694
2905:piecewise-linear manifold
1882:zero. Its derivative is
664:with the standard metric
3499:Elementary Real Analysis
2709:{\displaystyle f:M\to N}
2340:(or as a consequence of
1496:{\displaystyle f(x)=|x|}
730:there exists a constant
3789:Structures on manifolds
3727:"Topology of manifolds"
3253:has Lipschitz constant
2312:converges to a mapping
2220:is convex then in fact
1839: = sup |
734:≥ 0 such that, for all
295:≥ 0 such that, for all
138:{\displaystyle \alpha }
91:Picard–Lindelöf theorem
3469:Sohrab, H. H. (2003).
3446:
3371:
3314:≥0, any finite subset
3244:
3094:
2886:
2778:
2746:
2710:
2625:
2473:
2436:
2278:
2214:
2194:
2174:
2154:
2090:
2037:
1779:→ ∞, and therefore is
1739:(0) = 0 and
1664:
1638:
1497:
1421:
1305:
1067:
875:
644:
432:
181:
139:
87:differential equations
56:, is a strong form of
35:
3525:"Lipschitz Functions"
3447:
3372:
3298:Modulus of continuity
3245:
3095:
2933:upper semi-continuous
2887:
2779:
2747:
2711:
2626:
2474:
2437:
2361:Arzelà–Ascoli theorem
2279:
2215:
2195:
2175:
2155:
2091:
2038:
1872:absolutely continuous
1858:A Lipschitz function
1717:absolutely continuous
1665:
1639:
1498:
1422:
1306:
1068:
964:of order α > 0 on
896:A function is called
876:
645:
433:
271:is the metric on set
197:absolutely continuous
182:
140:
95:initial value problem
40:mathematical analysis
33:
3381:
3328:
3135:
2962:
2913:Rademacher's theorem
2909:topological manifold
2796:
2756:
2724:
2688:
2664:is defined using an
2662:topological manifold
2532:
2446:
2409:
2349:uniformly continuous
2224:
2204:
2184:
2164:
2100:
2059:
2003:
1962:Rademacher's theorem
1914:′ on the interval .
1773:exponential function
1648:
1527:
1507:. More generally, a
1462:
1379:
1106:
979:
972:≥ 0 such that
755:
544:
510:real-valued function
447:a Lipschitz constant
320:
289:Lipschitz continuous
203:uniformly continuous
192:Lipschitz continuous
159:
129:
123:Lipschitz continuous
44:Lipschitz continuity
3709:. pp. 269–283.
3695:Rosenberg, Jonathan
3472:Basic Real Analysis
3287:Contraction mapping
2919:One-sided Lipschitz
2658:Lipschitz structure
2652:Lipschitz manifolds
1884:essentially bounded
1663:{\displaystyle x=0}
1360:is also Lipschitz.
1352:, and is in fact a
1266: for all
935:metric space, then
445:is referred to as
66:continuous function
18:Lipschitz condition
3442:
3367:
3310:– For any integer
3240:
3235:
3090:
2882:
2774:
2742:
2706:
2621:
2574:
2520:Kirszbraun theorem
2469:
2458:
2432:
2421:
2274:
2210:
2190:
2170:
2150:
2086:
2033:
1980:is an open set in
1853:mean value theorem
1660:
1634:
1629:
1493:
1417:
1301:
1095:For a real number
1063:
871:
640:
428:
177:
135:
110:closed and bounded
74:Lipschitz constant
58:uniform continuity
36:
3672:Robbin, Joel W.,
3542:978-1-84628-369-7
3509:978-0-13-019075-8
2718:locally Lipschitz
2559:
2544:
2449:
2412:
2372:uniformly bounded
2363:implies that if {
2213:{\displaystyle U}
2193:{\displaystyle f}
2173:{\displaystyle K}
1986:almost everywhere
1894:, the difference
1876:almost everywhere
1785:analytic function
1709:Hölder continuous
1616:
1593:
1415:
1267:
1117:
956:Hölder continuous
860:
508:In particular, a
449:for the function
148:Hölder continuous
101:, is used in the
85:In the theory of
16:(Redirected from
3796:
3768:
3767:
3747:
3741:
3740:
3723:
3717:
3710:
3691:
3682:
3681:
3680:
3669:
3663:
3662:
3641:
3635:
3634:
3602:
3596:
3595:
3577:
3571:
3570:
3552:
3546:
3545:
3520:
3514:
3513:
3493:
3487:
3486:
3466:
3451:
3449:
3448:
3443:
3435:
3434:
3425:
3417:
3409:
3376:
3374:
3373:
3368:
3363:
3362:
3357:
3348:
3347:
3342:
3249:
3247:
3246:
3241:
3239:
3238:
3171:
3163:
3162:
3157:
3099:
3097:
3096:
3091:
3089:
3088:
3079:
3078:
3066:
3065:
3041:
3040:
3019:
3018:
3000:
2999:
2990:
2989:
2977:
2976:
2899:
2895:
2891:
2889:
2888:
2883:
2857:
2856:
2811:
2810:
2791:
2787:
2783:
2781:
2780:
2775:
2751:
2749:
2748:
2743:
2715:
2713:
2712:
2707:
2683:
2679:
2674:smooth manifolds
2630:
2628:
2627:
2622:
2573:
2546:
2545:
2537:
2478:
2476:
2475:
2470:
2468:
2467:
2457:
2441:
2439:
2438:
2433:
2431:
2430:
2420:
2283:
2281:
2280:
2275:
2267:
2266:
2256:
2255:
2219:
2217:
2216:
2211:
2200:. If the domain
2199:
2197:
2196:
2191:
2179:
2177:
2176:
2171:
2159:
2157:
2156:
2151:
2143:
2142:
2132:
2131:
2095:
2093:
2092:
2087:
2085:
2084:
2079:
2045:total derivative
2042:
2040:
2039:
2034:
1960:More generally,
1880:Lebesgue measure
1849:first derivative
1698:
1697:
1669:
1667:
1666:
1661:
1643:
1641:
1640:
1635:
1633:
1632:
1617:
1614:
1594:
1591:
1581:
1564:
1563:
1502:
1500:
1499:
1494:
1492:
1484:
1426:
1424:
1423:
1418:
1416:
1408:
1407:
1398:
1358:inverse function
1310:
1308:
1307:
1302:
1291:
1290:
1278:
1277:
1268:
1265:
1259:
1258:
1246:
1245:
1233:
1232:
1211:
1210:
1189:
1188:
1170:
1169:
1154:
1153:
1141:
1140:
1128:
1127:
1118:
1110:
1072:
1070:
1069:
1064:
1062:
1061:
1040:
1039:
991:
990:
961:Hölder condition
958:or to satisfy a
880:
878:
877:
872:
861:
859:
855:
854:
842:
841:
829:
828:
818:
811:
810:
789:
788:
770:
769:
759:
649:
647:
646:
641:
636:
631:
630:
618:
617:
608:
597:
589:
588:
567:
566:
551:
437:
435:
434:
429:
421:
420:
408:
407:
395:
394:
373:
372:
351:
350:
332:
331:
186:
184:
183:
178:
144:
142:
141:
136:
54:Rudolf Lipschitz
21:
3804:
3803:
3799:
3798:
3797:
3795:
3794:
3793:
3774:
3773:
3772:
3771:
3749:
3748:
3744:
3725:
3724:
3720:
3693:
3692:
3685:
3678:
3671:
3670:
3666:
3659:
3645:Gromov, Mikhael
3643:
3642:
3638:
3604:
3603:
3599:
3592:
3579:
3578:
3574:
3567:
3554:
3553:
3549:
3543:
3533:Springer-Verlag
3522:
3521:
3517:
3510:
3495:
3494:
3490:
3483:
3468:
3467:
3463:
3458:
3426:
3379:
3378:
3352:
3337:
3326:
3325:
3293:Dini continuity
3283:
3234:
3233:
3176:
3175:
3152:
3139:
3133:
3132:
3120:
3113:
3080:
3070:
3057:
3032:
3010:
2991:
2981:
2968:
2960:
2959:
2921:
2897:
2893:
2845:
2799:
2794:
2793:
2789:
2785:
2754:
2753:
2722:
2721:
2686:
2685:
2681:
2677:
2666:atlas of charts
2654:
2530:
2529:
2459:
2444:
2443:
2422:
2407:
2406:
2404:
2385:locally compact
2368:
2310:
2299:
2292:
2241:
2236:
2222:
2221:
2202:
2201:
2182:
2181:
2162:
2161:
2117:
2112:
2098:
2097:
2096:the inequality
2074:
2057:
2056:
2001:
2000:
1991:. Moreover, if
1941:for almost all
1917:Conversely, if
1820:
1813:
1762:
1722:
1699:defined on is
1693:
1691:
1673:
1646:
1645:
1628:
1627:
1611:
1605:
1604:
1588:
1555:
1548:
1525:
1524:
1514:
1460:
1459:
1449:
1399:
1377:
1376:
1366:
1282:
1269:
1250:
1237:
1224:
1202:
1180:
1161:
1145:
1132:
1119:
1104:
1103:
1053:
1031:
982:
977:
976:
933:locally compact
908:there exists a
846:
833:
820:
819:
802:
780:
761:
760:
753:
752:
747:
740:
725:
718:
705:is a subset of
699:
692:
685:
678:
672:
622:
609:
580:
558:
542:
541:
536:
529:
412:
399:
386:
364:
342:
323:
318:
317:
308:
301:
270:
253:
244:
231:
213:
187:. We also have
157:
156:
127:
126:
28:
23:
22:
15:
12:
11:
5:
3802:
3800:
3792:
3791:
3786:
3784:Lipschitz maps
3776:
3775:
3770:
3769:
3758:(2): 780–796.
3742:
3718:
3683:
3664:
3657:
3636:
3617:(6): 845–846.
3597:
3590:
3572:
3565:
3547:
3541:
3515:
3508:
3488:
3481:
3460:
3459:
3457:
3454:
3453:
3452:
3441:
3438:
3433:
3429:
3424:
3420:
3416:
3412:
3408:
3404:
3401:
3398:
3395:
3392:
3389:
3386:
3366:
3361:
3356:
3351:
3346:
3341:
3336:
3333:
3305:
3303:Quasi-isometry
3300:
3295:
3290:
3282:
3279:
3251:
3250:
3237:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3177:
3174:
3170:
3166:
3161:
3156:
3151:
3148:
3145:
3144:
3142:
3118:
3111:
3101:
3100:
3087:
3083:
3077:
3073:
3069:
3064:
3060:
3056:
3053:
3050:
3047:
3044:
3039:
3035:
3031:
3028:
3025:
3022:
3017:
3013:
3009:
3006:
3003:
2998:
2994:
2988:
2984:
2980:
2975:
2971:
2967:
2920:
2917:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2855:
2852:
2848:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2809:
2806:
2802:
2773:
2770:
2767:
2764:
2761:
2741:
2738:
2735:
2732:
2729:
2705:
2702:
2699:
2696:
2693:
2653:
2650:
2649:
2648:
2633:
2632:
2631:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2572:
2569:
2566:
2562:
2558:
2555:
2552:
2549:
2543:
2540:
2524:
2523:
2480:
2466:
2462:
2456:
2452:
2429:
2425:
2419:
2415:
2402:
2396:
2366:
2357:equicontinuous
2345:
2308:
2297:
2290:
2287:Suppose that {
2285:
2273:
2270:
2265:
2262:
2259:
2254:
2251:
2248:
2244:
2239:
2235:
2232:
2229:
2209:
2189:
2169:
2149:
2146:
2141:
2138:
2135:
2130:
2127:
2124:
2120:
2115:
2111:
2108:
2105:
2083:
2078:
2073:
2070:
2067:
2064:
2053:
2052:
2051:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
1989:differentiable
1958:
1902:) −
1856:
1819:
1816:
1815:
1814:
1812:
1811:
1798:) =
1788:
1768:
1766:
1763:
1761:
1760:
1747:) =
1728:
1726:
1723:
1721:
1720:
1690:) =
1679:
1677:
1674:
1672:
1671:
1659:
1656:
1653:
1631:
1626:
1623:
1620:
1612:
1610:
1607:
1606:
1603:
1600:
1597:
1589:
1587:
1584:
1580:
1576:
1573:
1570:
1567:
1562:
1558:
1554:
1553:
1551:
1545:
1541:
1538:
1535:
1532:
1520:
1518:
1515:
1513:
1512:
1491:
1487:
1483:
1479:
1476:
1473:
1470:
1467:
1455:
1453:
1450:
1448:
1447:
1442:Likewise, the
1440:
1433:differentiable
1414:
1411:
1406:
1402:
1396:
1393:
1390:
1387:
1384:
1372:
1370:
1365:
1362:
1325:(also written
1312:
1311:
1300:
1297:
1294:
1289:
1285:
1281:
1276:
1272:
1262:
1257:
1253:
1249:
1244:
1240:
1236:
1231:
1227:
1223:
1220:
1217:
1214:
1209:
1205:
1201:
1198:
1195:
1192:
1187:
1183:
1179:
1176:
1173:
1168:
1164:
1160:
1157:
1152:
1148:
1144:
1139:
1135:
1131:
1126:
1122:
1116:
1113:
1099:≥ 1, if
1074:
1073:
1060:
1056:
1052:
1049:
1046:
1043:
1038:
1034:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
989:
985:
954:is said to be
923:restricted to
882:
881:
870:
867:
864:
858:
853:
849:
845:
840:
836:
832:
827:
823:
817:
814:
809:
805:
801:
798:
795:
792:
787:
783:
779:
776:
773:
768:
764:
745:
738:
728:if and only if
723:
716:
697:
690:
683:
676:
668:
657:is the set of
653:In this case,
651:
650:
639:
635:
629:
625:
621:
616:
612:
607:
603:
600:
596:
592:
587:
583:
579:
576:
573:
570:
565:
561:
557:
554:
550:
534:
527:
439:
438:
427:
424:
419:
415:
411:
406:
402:
398:
393:
389:
385:
382:
379:
376:
371:
367:
363:
360:
357:
354:
349:
345:
341:
338:
335:
330:
326:
306:
299:
266:
249:
240:
227:
212:
209:
208:
207:
176:
173:
170:
167:
164:
153:
152:
134:
70:absolute value
46:, named after
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3801:
3790:
3787:
3785:
3782:
3781:
3779:
3765:
3761:
3757:
3753:
3746:
3743:
3738:
3734:
3733:
3728:
3722:
3719:
3716:
3713:
3708:
3704:
3700:
3696:
3690:
3688:
3684:
3677:
3676:
3668:
3665:
3660:
3658:0-8218-0975-X
3654:
3650:
3646:
3640:
3637:
3632:
3628:
3624:
3620:
3616:
3612:
3608:
3601:
3598:
3593:
3591:0-8218-2129-6
3587:
3583:
3576:
3573:
3568:
3566:0-8218-0835-4
3562:
3558:
3551:
3548:
3544:
3538:
3534:
3530:
3529:Metric Spaces
3526:
3519:
3516:
3511:
3505:
3501:
3500:
3492:
3489:
3484:
3482:0-8176-4211-0
3478:
3474:
3473:
3465:
3462:
3455:
3439:
3431:
3427:
3422:
3410:
3402:
3399:
3393:
3387:
3384:
3364:
3359:
3344:
3334:
3331:
3323:
3322:
3317:
3313:
3309:
3306:
3304:
3301:
3299:
3296:
3294:
3291:
3288:
3285:
3284:
3280:
3278:
3276:
3272:
3268:
3264:
3260:
3256:
3224:
3218:
3215:
3212:
3209:
3203:
3200:
3197:
3191:
3188:
3185:
3179:
3172:
3159:
3149:
3146:
3140:
3131:
3130:
3129:
3127:
3122:
3117:
3110:
3106:
3085:
3075:
3071:
3067:
3062:
3058:
3051:
3048:
3037:
3033:
3026:
3023:
3015:
3011:
3004:
2996:
2986:
2982:
2978:
2973:
2969:
2958:
2957:
2956:
2954:
2950:
2946:
2942:
2938:
2934:
2930:
2926:
2918:
2916:
2914:
2910:
2906:
2901:
2879:
2867:
2861:
2853:
2850:
2842:
2839:
2836:
2830:
2827:
2824:
2821:
2818:
2815:
2812:
2807:
2804:
2800:
2771:
2765:
2762:
2759:
2739:
2733:
2730:
2727:
2719:
2703:
2697:
2694:
2691:
2675:
2671:
2667:
2663:
2659:
2651:
2646:
2642:
2638:
2634:
2618:
2609:
2606:
2603:
2597:
2593:
2590:
2584:
2578:
2570:
2567:
2564:
2556:
2550:
2538:
2528:
2527:
2526:
2525:
2521:
2517:
2513:
2510:which extend
2509:
2505:
2501:
2497:
2493:
2489:
2485:
2481:
2464:
2460:
2454:
2427:
2423:
2417:
2401:
2397:
2394:
2390:
2386:
2383: is a
2382:
2378:
2375:metric space
2373:
2369:
2362:
2358:
2354:
2350:
2346:
2343:
2339:
2334:
2330:
2326:
2322:
2318:
2315:
2311:
2304:
2300:
2293:
2286:
2271:
2268:
2260:
2249:
2246:
2242:
2233:
2230:
2207:
2187:
2167:
2147:
2144:
2136:
2125:
2122:
2118:
2109:
2106:
2081:
2068:
2065:
2062:
2054:
2049:
2046:
2043:whenever the
2030:
2027:
2018:
2012:
2009:
1998:
1994:
1990:
1987:
1983:
1979:
1975:
1972: →
1971:
1968: :
1967:
1963:
1959:
1956:
1952:
1948:
1944:
1940:
1936:
1932:
1928:
1924:
1920:
1916:
1915:
1913:
1909:
1905:
1901:
1897:
1893:
1889:
1885:
1881:
1877:
1873:
1869:
1866: →
1865:
1862: :
1861:
1857:
1854:
1850:
1846:
1842:
1838:
1834:
1831: →
1830:
1827: :
1826:
1822:
1821:
1817:
1809:
1805:
1801:
1797:
1793:
1790:The function
1789:
1786:
1782:
1778:
1774:
1770:
1769:
1767:
1764:
1758:
1754:
1750:
1746:
1742:
1738:
1734:
1731:The function
1730:
1729:
1727:
1724:
1718:
1714:
1710:
1706:
1702:
1696:
1689:
1685:
1682:The function
1681:
1680:
1678:
1675:
1657:
1654:
1651:
1624:
1621:
1618:
1608:
1601:
1598:
1595:
1582:
1578:
1574:
1568:
1565:
1560:
1556:
1549:
1543:
1536:
1530:
1523:The function
1522:
1521:
1519:
1516:
1510:
1506:
1485:
1477:
1471:
1465:
1458:The function
1457:
1456:
1454:
1451:
1445:
1441:
1438:
1434:
1430:
1412:
1409:
1404:
1400:
1394:
1388:
1382:
1375:The function
1374:
1373:
1371:
1368:
1367:
1363:
1361:
1359:
1355:
1354:homeomorphism
1351:
1347:
1343:
1339:
1335:
1331:
1330:-bi-Lipschitz
1329:
1324:
1322:
1317:
1298:
1295:
1292:
1287:
1283:
1279:
1274:
1270:
1255:
1251:
1247:
1242:
1238:
1229:
1225:
1221:
1218:
1207:
1203:
1196:
1193:
1185:
1181:
1174:
1166:
1162:
1158:
1150:
1146:
1142:
1137:
1133:
1124:
1120:
1114:
1111:
1102:
1101:
1100:
1098:
1093:
1091:
1087:
1083:
1079:
1058:
1050:
1047:
1044:
1036:
1032:
1028:
1025:
1016:
1010:
1007:
1001:
995:
987:
983:
975:
974:
973:
971:
967:
963:
962:
957:
953:
949:
944:
942:
938:
934:
930:
926:
922:
918:
914:
911:
907:
903:
900:if for every
899:
894:
891:
887:
868:
865:
862:
851:
847:
843:
838:
834:
825:
821:
807:
803:
796:
793:
785:
781:
774:
766:
762:
751:
750:
749:
744:
737:
733:
729:
722:
715:
710:
708:
704:
700:
693:
686:
679:
671:
667:
663:
660:
656:
637:
627:
623:
619:
614:
610:
601:
598:
585:
581:
574:
571:
563:
559:
552:
540:
539:
538:
533:
526:
522:
518:
514:
511:
506:
504:
503:
498:
494:
491:, and if 0 ≤
490:
489:
484:
480:
476:
472:
468:
464:
460:
456:
452:
448:
444:
425:
417:
413:
409:
404:
400:
391:
387:
383:
380:
369:
365:
358:
355:
347:
343:
336:
328:
324:
316:
315:
314:
312:
305:
298:
294:
290:
286:
282:
278:
275:, a function
274:
269:
265:
261:
257:
252:
248:
243:
239:
235:
230:
226:
222:
218:
217:metric spaces
210:
205:
204:
199:
198:
193:
190:
189:
188:
174:
171:
168:
165:
162:
150:
149:
132:
124:
120:
119:
115:
114:
113:
111:
106:
104:
100:
96:
92:
88:
83:
81:
80:
75:
71:
67:
63:
59:
55:
52:
51:mathematician
49:
45:
41:
32:
19:
3755:
3751:
3745:
3730:
3721:
3705:. Canberra:
3702:
3674:
3667:
3648:
3639:
3614:
3610:
3600:
3581:
3575:
3556:
3550:
3528:
3518:
3498:
3491:
3471:
3464:
3320:
3319:
3315:
3311:
3274:
3270:
3266:
3262:
3258:
3254:
3252:
3125:
3123:
3115:
3108:
3107:and for all
3104:
3102:
2952:
2948:
2944:
2940:
2936:
2935:function of
2928:
2924:
2922:
2902:
2717:
2657:
2655:
2644:
2640:
2636:
2515:
2511:
2507:
2503:
2499:
2495:
2491:
2487:
2483:
2399:
2392:
2388:
2380:
2376:
2364:
2351:, and hence
2332:
2329:Banach space
2324:
2320:
2313:
2306:
2302:
2295:
2288:
2047:
1996:
1992:
1981:
1977:
1973:
1969:
1965:
1954:
1950:
1946:
1942:
1938:
1934:
1930:
1926:
1922:
1918:
1911:
1907:
1903:
1899:
1895:
1891:
1887:
1867:
1863:
1859:
1844:
1840:
1836:
1832:
1828:
1824:
1807:
1803:
1799:
1795:
1791:
1780:
1776:
1756:
1752:
1748:
1744:
1740:
1736:
1732:
1712:
1704:
1700:
1694:
1687:
1683:
1428:
1345:
1342:bi-Lipschitz
1341:
1337:
1333:
1327:
1326:
1323:-bilipschitz
1320:
1319:
1315:
1313:
1096:
1094:
1089:
1085:
1081:
1077:
1075:
969:
965:
959:
955:
951:
947:
945:
940:
936:
928:
924:
920:
916:
912:
910:neighborhood
905:
901:
897:
895:
889:
885:
883:
742:
735:
731:
720:
713:
711:
706:
702:
695:
688:
681:
674:
669:
665:
661:
659:real numbers
654:
652:
531:
524:
520:
516:
512:
507:
500:
496:
492:
486:
482:
478:
474:
470:
466:
462:
458:
454:
450:
446:
442:
440:
310:
303:
296:
292:
288:
284:
280:
276:
272:
267:
263:
259:
254:denotes the
250:
246:
241:
237:
233:
228:
224:
220:
214:
201:
195:
191:
154:
145:
122:
116:
107:
84:
77:
73:
43:
37:
2939:, and that
2670:pseudogroup
1890: <
1755:) for 0<
1735:defined by
1338:bilipschitz
950:defined on
502:contraction
495:< 1 and
459:K-Lipschitz
258:on the set
211:Definitions
99:contraction
3778:Categories
3456:References
2518:(see also
2359:set. The
2353:continuous
1937:)| ≤
1818:Properties
1437:Properties
1332:). We say
1318:is called
1092:α > 0.
919:such that
475:dilatation
441:Any such
287:is called
215:Given two
3737:EMS Press
3437:⌉
3428:ε
3403:
3391:⌈
3350:→
3219:
3213:−
3201:−
3165:→
3103:for some
3082:‖
3068:−
3055:‖
3049:≤
3024:−
2979:−
2877:→
2862:ψ
2851:−
2843:ϕ
2840:∘
2831:∩
2822:ϕ
2819:∘
2813:∘
2805:−
2801:ψ
2769:→
2760:ψ
2737:→
2728:ϕ
2701:→
2568:∈
2542:~
2465:α
2455:α
2428:α
2418:α
2333:unbounded
2317:uniformly
2253:∞
2238:‖
2228:‖
2145:≤
2129:∞
2114:‖
2104:‖
2072:→
2028:≤
2025:‖
2007:‖
1711:of class
1599:≠
1569:
1350:injective
1293:∈
1219:≤
1159:≤
1059:α
1026:≤
863:≤
620:−
599:≤
572:−
488:short map
381:≤
245:), where
172:≤
169:α
133:α
62:functions
3697:(1988).
3631:24568871
3281:See also
2931:) be an
2784:, where
2506: →
2498: →
2494: :
1976:, where
1925: →
1921: :
1615:if
1592:if
1364:Examples
1076:for all
515: :
471:dilation
279: :
200:⊂
194:⊂
125:⊂
121:⊂
3739:, 2001
3273:, with
2951:. Then
2370:} is a
2319:, then
2050:exists.
1999:, then
1949:, then
1692:√
701:|, and
469:or the
232:) and (
3715:954004
3655:
3629:
3588:
3563:
3539:
3506:
3479:
3377:where
2907:and a
2635:where
2305:. If
1751:sin(1/
256:metric
155:where
48:German
3679:(PDF)
3277:= 0.
2676:: if
2660:on a
2442:(and
1984:, is
1314:then
931:is a
687:) = |
481:. If
3653:ISBN
3627:PMID
3586:ISBN
3561:ISBN
3537:ISBN
3504:ISBN
3477:ISBN
3269:) =
3114:and
2923:Let
2788:and
2752:and
2680:and
2490:and
1771:The
1509:norm
1444:sine
1080:and
748:,
530:and
453:and
302:and
262:and
166:<
60:for
3760:doi
3619:doi
3216:cos
2896:or
2716:is
2643:on
2561:inf
2482:If
2451:inf
2414:sup
2180:of
1945:in
1870:is
1804:not
1781:not
1701:not
1566:sin
1340:or
1336:is
1084:in
915:of
904:in
477:of
473:or
465:of
309:in
38:In
3780::
3756:36
3754:.
3735:,
3729:,
3712:MR
3701:.
3686:^
3625:.
3615:98
3613:.
3609:.
3535:,
3527:,
3400:ln
3394:15
3204:50
3121:.
2900:.
2656:A
2557::=
2395:).
2048:Df
1931:f′
1843:′(
1439:".
741:≠
719:=
709:.
694:−
680:,
537:,
519:→
505:.
313:,
283:→
236:,
223:,
105:.
42:,
3766:.
3762::
3661:.
3633:.
3621::
3594:.
3569:.
3512:.
3485:.
3440:.
3432:2
3423:/
3419:)
3415:|
3411:X
3407:|
3397:(
3388:=
3385:d
3365:,
3360:d
3355:R
3345:n
3340:R
3335::
3332:f
3321:R
3318:⊆
3316:X
3312:n
3275:C
3271:e
3267:x
3265:(
3263:F
3259:C
3255:K
3231:)
3228:)
3225:x
3222:(
3210:y
3207:(
3198:=
3195:)
3192:y
3189:,
3186:x
3183:(
3180:F
3173:,
3169:R
3160:2
3155:R
3150::
3147:F
3141:{
3126:F
3119:2
3116:x
3112:1
3109:x
3105:C
3086:2
3076:2
3072:x
3063:1
3059:x
3052:C
3046:)
3043:)
3038:2
3034:x
3030:(
3027:F
3021:)
3016:1
3012:x
3008:(
3005:F
3002:(
2997:T
2993:)
2987:2
2983:x
2974:1
2970:x
2966:(
2953:F
2949:x
2945:x
2943:(
2941:F
2937:x
2929:x
2927:(
2925:F
2898:N
2894:M
2880:V
2874:)
2871:)
2868:V
2865:(
2859:(
2854:1
2847:)
2837:f
2834:(
2828:U
2825::
2816:f
2808:1
2790:V
2786:U
2772:N
2766:V
2763::
2740:M
2734:U
2731::
2704:N
2698:M
2695::
2692:f
2682:N
2678:M
2647:.
2645:U
2641:f
2637:k
2619:,
2616:}
2613:)
2610:u
2607:,
2604:x
2601:(
2598:d
2594:k
2591:+
2588:)
2585:u
2582:(
2579:f
2576:{
2571:U
2565:u
2554:)
2551:x
2548:(
2539:f
2516:f
2512:f
2508:R
2504:M
2500:R
2496:U
2492:f
2488:M
2484:U
2461:f
2424:f
2403:α
2400:f
2393:X
2391:(
2389:C
2381:K
2377:X
2367:n
2365:f
2325:K
2321:f
2314:f
2309:n
2307:f
2303:K
2298:n
2296:f
2291:n
2289:f
2284:.
2272:K
2269:=
2264:)
2261:U
2258:(
2250:,
2247:1
2243:W
2234:f
2231:D
2208:U
2188:f
2168:K
2148:K
2140:)
2137:U
2134:(
2126:,
2123:1
2119:W
2110:f
2107:D
2082:m
2077:R
2069:U
2066::
2063:f
2031:K
2022:)
2019:x
2016:(
2013:f
2010:D
1997:f
1993:K
1982:R
1978:U
1974:R
1970:U
1966:f
1957:.
1955:K
1951:f
1947:I
1943:x
1939:K
1935:x
1933:(
1927:R
1923:I
1919:f
1912:g
1908:a
1906:(
1904:g
1900:b
1898:(
1896:g
1892:b
1888:a
1868:R
1864:R
1860:g
1845:x
1841:g
1837:K
1833:R
1829:R
1825:g
1808:x
1800:x
1796:x
1794:(
1792:f
1787:.
1777:x
1757:x
1753:x
1749:x
1745:x
1743:(
1741:f
1737:f
1733:f
1713:C
1705:x
1695:x
1688:x
1686:(
1684:f
1670:.
1658:0
1655:=
1652:x
1625:0
1622:=
1619:x
1609:0
1602:0
1596:x
1586:)
1583:x
1579:/
1575:1
1572:(
1561:2
1557:x
1550:{
1544:=
1540:)
1537:x
1534:(
1531:f
1490:|
1486:x
1482:|
1478:=
1475:)
1472:x
1469:(
1466:f
1429:K
1413:5
1410:+
1405:2
1401:x
1395:=
1392:)
1389:x
1386:(
1383:f
1346:K
1334:f
1328:K
1321:K
1316:f
1299:,
1296:X
1288:2
1284:x
1280:,
1275:1
1271:x
1261:)
1256:2
1252:x
1248:,
1243:1
1239:x
1235:(
1230:X
1226:d
1222:K
1216:)
1213:)
1208:2
1204:x
1200:(
1197:f
1194:,
1191:)
1186:1
1182:x
1178:(
1175:f
1172:(
1167:Y
1163:d
1156:)
1151:2
1147:x
1143:,
1138:1
1134:x
1130:(
1125:X
1121:d
1115:K
1112:1
1097:K
1086:X
1082:y
1078:x
1055:)
1051:y
1048:,
1045:x
1042:(
1037:X
1033:d
1029:M
1023:)
1020:)
1017:y
1014:(
1011:f
1008:,
1005:)
1002:x
999:(
996:f
993:(
988:Y
984:d
970:M
966:X
952:X
948:f
941:X
937:f
929:X
925:U
921:f
917:x
913:U
906:X
902:x
890:K
886:K
869:.
866:K
857:)
852:2
848:x
844:,
839:1
835:x
831:(
826:X
822:d
816:)
813:)
808:2
804:x
800:(
797:f
794:,
791:)
786:1
782:x
778:(
775:f
772:(
767:Y
763:d
746:2
743:x
739:1
736:x
732:K
724:2
721:x
717:1
714:x
707:R
703:X
698:2
696:y
691:1
689:y
684:2
682:y
677:1
675:y
673:(
670:Y
666:d
662:R
655:Y
638:.
634:|
628:2
624:x
615:1
611:x
606:|
602:K
595:|
591:)
586:2
582:x
578:(
575:f
569:)
564:1
560:x
556:(
553:f
549:|
535:2
532:x
528:1
525:x
521:R
517:R
513:f
497:f
493:K
483:K
479:f
467:f
455:f
451:f
443:K
426:.
423:)
418:2
414:x
410:,
405:1
401:x
397:(
392:X
388:d
384:K
378:)
375:)
370:2
366:x
362:(
359:f
356:,
353:)
348:1
344:x
340:(
337:f
334:(
329:Y
325:d
311:X
307:2
304:x
300:1
297:x
293:K
285:Y
281:X
277:f
273:Y
268:Y
264:d
260:X
251:X
247:d
242:Y
238:d
234:Y
229:X
225:d
221:X
219:(
206:.
175:1
163:0
151:,
146:-
20:)
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