828:
1642:
2271:
is a concave function. In the case of thermodynamic entropy, without phase transition, entropy as a function of extensive variables is strictly concave. If the system can undergo phase transition, and if it is allowed to split into two subsystems of different phase
831:
A cubic function is concave (left half) when its first derivative (red) is monotonically decreasing i.e. its second derivative (orange) is negative, and convex (right half) when its first derivative is monotonically increasing i.e. its second derivative is
1406:
2276:, e.g. boiling), the entropy-maximal parameters of the subsystems will result in a combined entropy precisely on the straight line between the two phases. This means that the "Effective Entropy" of a system with phase transition is the
40:
is one for which the value at any convex combination of elements in the domain is greater than or equal to the convex combination of the values at the endpoints. Equivalently, a concave function is any function for which the
1141:
412:
300:
1355:
1693:
in the interior of the domain of a function, the function must be concave; as a partial converse, if the derivative of a strictly concave function is zero at some point, then that point is a local maximum.
1913:
518:
3734:
1027:
817:
1843:
1401:
728:
1800:
453:
2022:
2165:
1963:
1995:
1761:
1194:
196:
2072:
721:
680:
619:
479:
2195:
751:
639:
578:
558:
538:
158:
138:
102:
3727:
2219:
2111:
1061:
1217:
3586:
1637:{\displaystyle f(a)+f(b)=f\left((a+b){\frac {a}{a+b}}\right)+f\left((a+b){\frac {b}{a+b}}\right)\geq {\frac {a}{a+b}}f(a+b)+{\frac {b}{a+b}}f(a+b)=f(a+b)}
3720:
3574:
2506:
2487:
315:
204:
3696:
3581:
2458:. Heil, Christopher, 1960-, Weir, Maurice D.,, Thomas, George B. Jr. (George Brinton), 1914-2006. (Fourteenth ed.). . p. 203.
3564:
3559:
3924:
3569:
3554:
2668:
2856:
3824:
3549:
2280:
of entropy without phase separation; therefore, the entropy of a system including phase separation will be non-strictly concave.
3916:
3166:
2920:
2637:
2618:
2577:
2552:
2463:
2342:
953:
3929:
2718:
3664:
3523:
2372:
17:
487:
3078:
2994:
2303:
3949:
3866:
3659:
3591:
3216:
3071:
3039:
2798:
3292:
3269:
2984:
1848:
3382:
3320:
3115:
2989:
2661:
2868:
2846:
2268:
1032:
760:
3969:
3691:
4168:
3676:
3442:
3056:
2878:
2202:
1362:
3934:
3061:
2831:
1766:
420:
3974:
3964:
3480:
3427:
2888:
2568:
Callen, Herbert B.; Callen, Herbert B. (1985). "8.1: Intrinsic
Stability of Thermodynamic Systems".
2000:
4163:
3954:
3939:
3781:
3596:
3367:
2915:
2654:
2230:
2119:
1925:
862:
2598:
4024:
4001:
3895:
3819:
3362:
3034:
1968:
1721:
1167:
891:
843:
163:
42:
4142:
4103:
4019:
3944:
3871:
3856:
3809:
3490:
3372:
3193:
3141:
2947:
2925:
2793:
2298:
2033:
1805:
850:
105:
82:
4088:
4080:
4076:
4072:
4068:
4064:
2542:
685:
644:
583:
3876:
3616:
3475:
3387:
3044:
2979:
2952:
2942:
2863:
2851:
2836:
2808:
2308:
458:
3881:
3747:
3712:
3432:
3051:
2898:
2245:
947:
754:
8:
3814:
3804:
3799:
3452:
3377:
3264:
3221:
2972:
2957:
2788:
2776:
2763:
2723:
2703:
2253:
2249:
2170:
4043:
3761:
3541:
3516:
3347:
3300:
3241:
3206:
3201:
3181:
3176:
3171:
3136:
3083:
3066:
2967:
2841:
2826:
2771:
2738:
2523:
2481:
2264:
736:
624:
563:
543:
523:
143:
123:
87:
2084:
827:
3861:
3681:
3505:
3437:
3259:
3236:
3110:
3103:
3006:
2821:
2713:
2633:
2614:
2573:
2548:
2469:
2459:
2434:
2378:
2368:
2338:
1682:
of two concave functions, i.e. the set of concave functions on a given domain form a
1679:
45:
is convex. The class of concave functions is in a sense the opposite of the class of
4014:
3959:
3845:
3639:
3422:
3335:
3315:
3246:
3156:
3098:
3090:
3024:
2937:
2693:
2606:
2527:
2515:
2424:
2273:
2234:
2226:
918:
880:
872:
2252:
are usually assumed to be concave over some or all of their domains, resulting in
4034:
4005:
3979:
3900:
3885:
3794:
3766:
3743:
3701:
3686:
3470:
3325:
3305:
3274:
3251:
3231:
3125:
2781:
2728:
2501:
2313:
2293:
2277:
2028:
1672:
876:
46:
865:
on that interval, that is, a concave function has a non-increasing (decreasing)
4121:
4029:
3890:
3789:
3611:
3510:
3357:
3310:
3211:
3014:
2610:
2260:
1701:
1664:
1042:
899:
3029:
4157:
3905:
3485:
3340:
3226:
2930:
2905:
2473:
2438:
2238:
2074:
is both concave and convex, but neither strictly-concave nor strictly-convex.
1697:
1690:
1161:
2382:
4126:
3495:
3465:
3330:
2893:
2504:; Thomas, J. A. (1988). "Determinant inequalities via information theory".
946:
is concave and differentiable, then it is bounded above by its first-order
910:
906:
113:
2453:
2337:. Mathematical and Computational Biology Series. Chapman & Hall/ CRC.
4116:
4111:
3995:
2743:
2685:
2362:
2198:
1136:{\displaystyle f\left({\frac {x+y}{2}}\right)\geq {\frac {f(x)+f(y)}{2}}}
407:{\displaystyle f((1-\alpha )x+\alpha y)>(1-\alpha )f(x)+\alpha f(y)\,}
33:
295:{\displaystyle f((1-\alpha )x+\alpha y)\geq (1-\alpha )f(x)+\alpha f(y)}
4039:
4009:
3771:
3460:
3392:
3146:
3019:
2883:
2873:
2816:
1350:{\displaystyle f(tx)=f(tx+(1-t)\cdot 0)\geq tf(x)+(1-t)f(0)\geq tf(x).}
854:
109:
2429:
2412:
3654:
3402:
3397:
2708:
1919:
1683:
2519:
4048:
3649:
3151:
2677:
3829:
3500:
2753:
1678:
The sum of two concave functions is itself concave and so is the
50:
3669:
2733:
925:
is strictly concave, but the converse is not true, as shown by
2748:
2601:. In Durlauf, Steven N.; Blume, Lawrence E (eds.).
866:
727:
3742:
2411:
Gradshteyn, I. S.; Ryzhik, I. M.; Hays, D. F. (1976-07-01).
2646:
2078:
1802:
are concave on their domains, as their second derivatives
2605:(Second ed.). Palgrave Macmillan. pp. 815â816.
2570:
Thermodynamics and an introduction to thermostatistics
2220:
computation of radiowave attenuation in the atmosphere
1851:
1708:
concave function will have at most one global maximum.
520:, this second definition merely states that for every
2173:
2122:
2087:
2036:
2003:
1971:
1928:
1808:
1769:
1724:
1409:
1365:
1220:
1170:
1064:
956:
763:
739:
688:
647:
627:
586:
566:
546:
526:
490:
461:
423:
318:
207:
166:
146:
126:
90:
2544:
Mathematics for
Economists: An Introductory Textbook
2410:
2572:(2nd ed.). New York: Wiley. pp. 203â206.
2189:
2159:
2105:
2066:
2016:
1989:
1957:
1907:
1837:
1794:
1755:
1636:
1395:
1349:
1188:
1135:
1021:
811:
745:
715:
674:
633:
613:
572:
552:
532:
512:
473:
447:
406:
294:
190:
152:
132:
96:
836:
4155:
2507:SIAM Journal on Matrix Analysis and Applications
2367:(3rd ed.). New York: Norton. p. 489.
2360:
2332:
641:is above the straight line joining the points
513:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
3728:
2662:
2630:Engineering Optimization: Theory and Practice
2603:The New Palgrave Dictionary of Economics
2547:. Oxford University Press. pp. 363â364.
875:where concavity changes (between concave and
2567:
2540:
2335:Optimal Control Applied to Biological Models
806:
779:
2500:
1649:
3735:
3721:
2669:
2655:
2541:Pemberton, Malcolm; Rau, Nicholas (2015).
2486:: CS1 maint: location missing publisher (
2451:
2413:"Table of Integrals, Series, and Products"
1908:{\textstyle g''(x)=-{\frac {1}{4x^{3/2}}}}
757:if the upper contour sets of the function
3697:Regiomontanus' angle maximization problem
2494:
2428:
1045:it is midpoint concave, that is, for any
506:
498:
403:
3540:
2596:
826:
3825:Locally convex topological vector space
3045:Differentiating under the integral sign
14:
4156:
3716:
2921:Inverse functions and differentiation
2650:
2590:
2395:
812:{\displaystyle S(a)=\{x:f(x)\geq a\}}
2632:. John Wiley and Sons. p. 779.
2356:
2354:
2333:Lenhart, S.; Workman, J. T. (2007).
2081:function is concave on the interval
2627:
1022:{\displaystyle f(y)\leq f(x)+f'(x)}
24:
2719:Free variables and bound variables
2024:is a strictly decreasing function.
1981:
1396:{\displaystyle a,b\in [0,\infty )}
1387:
1180:
25:
4180:
3524:The Method of Mechanical Theorems
2417:Journal of Lubrication Technology
2351:
3079:Partial fractions in integration
2995:Stochastic differential equation
2304:Logarithmically concave function
1795:{\displaystyle g(x)={\sqrt {x}}}
1700:of a concave function is also a
726:
448:{\displaystyle \alpha \in (0,1)}
3930:Ekeland's variational principle
3217:Jacobian matrix and determinant
3072:Tangent half-angle substitution
3040:Fundamental theorem of calculus
2212:
3293:Arithmetico-geometric sequence
2985:Ordinary differential equation
2561:
2534:
2445:
2404:
2389:
2326:
2183:
2175:
2153:
2145:
2132:
2126:
2100:
2088:
2046:
2040:
2017:{\displaystyle {\frac {1}{x}}}
1984:
1972:
1938:
1932:
1866:
1860:
1823:
1817:
1779:
1773:
1734:
1728:
1631:
1619:
1610:
1598:
1571:
1559:
1509:
1497:
1460:
1448:
1434:
1428:
1419:
1413:
1390:
1378:
1341:
1335:
1323:
1317:
1311:
1299:
1293:
1287:
1275:
1266:
1254:
1242:
1233:
1224:
1183:
1171:
1124:
1118:
1109:
1103:
1016:
1004:
1001:
995:
981:
975:
966:
960:
837:Functions of a single variable
797:
791:
773:
767:
710:
707:
701:
689:
669:
666:
660:
648:
608:
605:
599:
587:
502:
442:
430:
400:
394:
382:
376:
370:
358:
352:
337:
325:
322:
289:
283:
271:
265:
259:
247:
241:
226:
214:
211:
185:
173:
13:
1:
3116:Integro-differential equation
2990:Partial differential equation
2319:
2160:{\displaystyle f(B)=\log |B|}
1958:{\displaystyle f(x)=\log {x}}
1663:is concave over a convex set
822:
76:
49:. A concave function is also
27:Negative of a convex function
2676:
2452:Hass, Joel (13 March 2017).
2241:decision makers are concave.
1033:Lebesgue measurable function
849:is (strictly) concave on an
160:in the interval and for any
7:
3950:HermiteâHadamard inequality
3270:Generalized Stokes' theorem
3057:Integration by substitution
2287:
2203:nonnegative-definite matrix
1990:{\displaystyle (0,\infty )}
1756:{\displaystyle f(x)=-x^{2}}
1712:
1189:{\displaystyle [0,\infty )}
191:{\displaystyle \alpha \in }
10:
4185:
2799:(Δ, Ύ)-definition of limit
2628:Rao, Singiresu S. (2009).
2611:10.1057/9780230226203.1375
2222:involve concave functions.
4135:
4102:
4057:
3988:
3914:
3838:
3780:
3754:
3692:Proof that 22/7 exceeds Ï
3629:
3607:
3533:
3481:Gottfried Wilhelm Leibniz
3451:
3428:e (mathematical constant)
3413:
3285:
3192:
3124:
3005:
2807:
2762:
2684:
2067:{\displaystyle f(x)=ax+b}
1965:is concave on its domain
1838:{\displaystyle f''(x)=-2}
909:(or, informally, if the "
4136:Applications and related
3940:Fenchel-Young inequality
3443:Stirling's approximation
2916:Implicit differentiation
2864:Rules of differentiation
2597:Crouzeix, J.-P. (2008).
2231:choice under uncertainty
863:monotonically decreasing
716:{\displaystyle (y,f(y))}
675:{\displaystyle (x,f(x))}
614:{\displaystyle (z,f(z))}
3896:Legendre transformation
3820:Legendre transformation
3677:EulerâMaclaurin formula
3582:trigonometric functions
3035:Constant of integration
2361:Varian, Hal R. (1992).
913:" is non-positive). If
844:differentiable function
474:{\displaystyle x\neq y}
108:(or, more generally, a
4143:Convexity in economics
4077:(lower) ideally convex
3935:FenchelâMoreau theorem
3925:Carathéodory's theorem
3646:Differential geometry
3491:Infinitesimal calculus
3194:Multivariable calculus
3142:Directional derivative
2948:Second derivative test
2926:Logarithmic derivative
2899:General Leibniz's rule
2794:Order of approximation
2396:Rudin, Walter (1976).
2364:Microeconomic analysis
2191:
2161:
2107:
2068:
2018:
1991:
1959:
1909:
1839:
1796:
1757:
1638:
1397:
1351:
1190:
1137:
1023:
833:
813:
747:
717:
676:
635:
615:
574:
554:
534:
514:
475:
449:
408:
296:
192:
154:
134:
98:
4065:Convex series related
3965:ShapleyâFolkman lemma
3565:logarithmic functions
3560:exponential functions
3476:Generality of algebra
3354:Tests of convergence
2980:Differential equation
2964:Further applications
2953:Extreme value theorem
2943:First derivative test
2837:Differential operator
2809:Differential calculus
2309:Quasiconcave function
2192:
2162:
2108:
2069:
2019:
1992:
1960:
1910:
1840:
1797:
1758:
1639:
1398:
1352:
1191:
1138:
1024:
830:
814:
748:
718:
677:
636:
616:
575:
555:
535:
515:
476:
450:
409:
305:A function is called
297:
193:
155:
135:
99:
3955:KreinâMilman theorem
3748:variational analysis
3630:Miscellaneous topics
3570:hyperbolic functions
3555:irrational functions
3433:Exponential function
3286:Sequences and series
3052:Integration by parts
2250:production functions
2246:microeconomic theory
2218:Rays bending in the
2171:
2120:
2085:
2034:
2001:
1997:, as its derivative
1969:
1926:
1915:are always negative.
1849:
1806:
1767:
1722:
1407:
1363:
1218:
1168:
1062:
954:
948:Taylor approximation
761:
737:
686:
645:
625:
584:
564:
544:
524:
488:
459:
421:
316:
205:
164:
144:
124:
88:
3945:Jensen's inequality
3815:Lagrange multiplier
3805:Convex optimization
3800:Convex metric space
3617:List of derivatives
3453:History of calculus
3368:Cauchy condensation
3265:Exterior derivative
3222:Lagrange multiplier
2958:Maximum and minimum
2789:Limit of a sequence
2777:Limit of a function
2724:Graph of a function
2704:Continuous function
2299:Jensen's inequality
2254:diminishing returns
2190:{\displaystyle |B|}
853:if and only if its
4169:Types of functions
4073:(cs, bcs)-complete
4044:Algebraic interior
3762:Convex combination
3550:rational functions
3517:Method of Fluxions
3363:Alternating series
3260:Differential forms
3242:Partial derivative
3202:Divergence theorem
3084:Quadratic integral
2852:Leibniz's notation
2842:Mean value theorem
2827:Partial derivative
2772:Indeterminate form
2591:Further References
2265:Information Theory
2187:
2157:
2103:
2064:
2014:
1987:
1955:
1905:
1835:
1792:
1753:
1634:
1393:
1347:
1186:
1133:
1019:
834:
809:
743:
713:
672:
631:
611:
570:
550:
530:
510:
471:
445:
404:
292:
188:
150:
130:
94:
4151:
4150:
3710:
3709:
3636:Complex calculus
3625:
3624:
3506:Law of Continuity
3438:Natural logarithm
3423:Bernoulli numbers
3414:Special functions
3373:Direct comparison
3237:Multiple integral
3111:Integral equation
3007:Integral calculus
2938:Stationary points
2912:Other techniques
2857:Newton's notation
2822:Second derivative
2714:Finite difference
2639:978-0-470-18352-6
2620:978-0-333-78676-5
2599:"Quasi-concavity"
2579:978-0-471-86256-7
2554:978-1-78499-148-7
2465:978-0-13-443898-6
2430:10.1115/1.3452897
2344:978-1-58488-640-2
2256:to input factors.
2012:
1903:
1790:
1680:pointwise minimum
1593:
1554:
1528:
1479:
1131:
1088:
881:inflection points
819:are convex sets.
746:{\displaystyle f}
634:{\displaystyle f}
573:{\displaystyle y}
553:{\displaystyle x}
540:strictly between
533:{\displaystyle z}
153:{\displaystyle y}
133:{\displaystyle x}
97:{\displaystyle f}
55:concave downwards
18:Concave functions
16:(Redirected from
4176:
4069:(cs, lcs)-closed
4015:Effective domain
3970:RobinsonâUrsescu
3846:Convex conjugate
3737:
3730:
3723:
3714:
3713:
3640:Contour integral
3538:
3537:
3388:Limit comparison
3297:Types of series
3256:Advanced topics
3247:Surface integral
3091:Trapezoidal rule
3030:Basic properties
3025:Riemann integral
2973:Taylor's theorem
2699:Concave function
2694:Binomial theorem
2671:
2664:
2657:
2648:
2647:
2643:
2624:
2584:
2583:
2565:
2559:
2558:
2538:
2532:
2531:
2502:Cover, Thomas M.
2498:
2492:
2491:
2485:
2477:
2455:Thomas' calculus
2449:
2443:
2442:
2432:
2408:
2402:
2401:
2393:
2387:
2386:
2358:
2349:
2348:
2330:
2274:phase separation
2235:cardinal utility
2227:expected utility
2196:
2194:
2193:
2188:
2186:
2178:
2166:
2164:
2163:
2158:
2156:
2148:
2112:
2110:
2109:
2106:{\displaystyle }
2104:
2073:
2071:
2070:
2065:
2023:
2021:
2020:
2015:
2013:
2005:
1996:
1994:
1993:
1988:
1964:
1962:
1961:
1956:
1954:
1914:
1912:
1911:
1906:
1904:
1902:
1901:
1900:
1896:
1876:
1859:
1844:
1842:
1841:
1836:
1816:
1801:
1799:
1798:
1793:
1791:
1786:
1762:
1760:
1759:
1754:
1752:
1751:
1670:
1662:
1643:
1641:
1640:
1635:
1594:
1592:
1578:
1555:
1553:
1539:
1534:
1530:
1529:
1527:
1513:
1485:
1481:
1480:
1478:
1464:
1402:
1400:
1399:
1394:
1356:
1354:
1353:
1348:
1213:
1206:
1202:
1195:
1193:
1192:
1187:
1159:
1155:
1149:is concave, and
1148:
1142:
1140:
1139:
1134:
1132:
1127:
1098:
1093:
1089:
1084:
1073:
1058:
1052:
1048:
1040:
1028:
1026:
1025:
1020:
994:
945:
938:
924:
916:
915:f ′′
904:
903:f ′′
897:
889:
860:
848:
818:
816:
815:
810:
752:
750:
749:
744:
730:
722:
720:
719:
714:
681:
679:
678:
673:
640:
638:
637:
632:
621:on the graph of
620:
618:
617:
612:
579:
577:
576:
571:
559:
557:
556:
551:
539:
537:
536:
531:
519:
517:
516:
511:
509:
501:
480:
478:
477:
472:
454:
452:
451:
446:
413:
411:
410:
405:
307:strictly concave
301:
299:
298:
293:
197:
195:
194:
189:
159:
157:
156:
151:
139:
137:
136:
131:
116:) is said to be
103:
101:
100:
95:
47:convex functions
38:concave function
21:
4184:
4183:
4179:
4178:
4177:
4175:
4174:
4173:
4164:Convex analysis
4154:
4153:
4152:
4147:
4131:
4098:
4053:
3984:
3910:
3901:Semi-continuity
3886:Convex function
3867:Logarithmically
3834:
3795:Convex geometry
3776:
3767:Convex function
3750:
3744:Convex analysis
3741:
3711:
3706:
3702:Steinmetz solid
3687:Integration Bee
3621:
3603:
3529:
3471:Colin Maclaurin
3447:
3415:
3409:
3281:
3275:Tensor calculus
3252:Volume integral
3188:
3163:Basic theorems
3126:Vector calculus
3120:
3001:
2968:Newton's method
2803:
2782:One-sided limit
2758:
2739:Rolle's theorem
2729:Linear function
2680:
2675:
2640:
2621:
2593:
2588:
2587:
2580:
2566:
2562:
2555:
2539:
2535:
2520:10.1137/0609033
2499:
2495:
2479:
2478:
2466:
2450:
2446:
2409:
2405:
2394:
2390:
2375:
2359:
2352:
2345:
2331:
2327:
2322:
2314:Concavification
2294:Concave polygon
2290:
2284:
2278:convex envelope
2215:
2182:
2174:
2172:
2169:
2168:
2152:
2144:
2121:
2118:
2117:
2086:
2083:
2082:
2035:
2032:
2031:
2029:affine function
2004:
2002:
1999:
1998:
1970:
1967:
1966:
1950:
1927:
1924:
1923:
1892:
1888:
1884:
1880:
1875:
1852:
1850:
1847:
1846:
1809:
1807:
1804:
1803:
1785:
1768:
1765:
1764:
1747:
1743:
1723:
1720:
1719:
1715:
1673:convex function
1668:
1660:
1656:
1582:
1577:
1543:
1538:
1517:
1512:
1496:
1492:
1468:
1463:
1447:
1443:
1408:
1405:
1404:
1364:
1361:
1360:
1219:
1216:
1215:
1208:
1204:
1203:is concave and
1200:
1169:
1166:
1165:
1157:
1150:
1146:
1099:
1097:
1074:
1072:
1068:
1063:
1060:
1059:
1054:
1050:
1046:
1036:
1035:on an interval
987:
955:
952:
951:
943:
926:
922:
914:
902:
895:
887:
858:
846:
839:
825:
762:
759:
758:
738:
735:
734:
687:
684:
683:
646:
643:
642:
626:
623:
622:
585:
582:
581:
565:
562:
561:
545:
542:
541:
525:
522:
521:
505:
497:
489:
486:
485:
484:For a function
460:
457:
456:
422:
419:
418:
317:
314:
313:
206:
203:
202:
165:
162:
161:
145:
142:
141:
125:
122:
121:
89:
86:
85:
79:
28:
23:
22:
15:
12:
11:
5:
4182:
4172:
4171:
4166:
4149:
4148:
4146:
4145:
4139:
4137:
4133:
4132:
4130:
4129:
4124:
4122:Strong duality
4119:
4114:
4108:
4106:
4100:
4099:
4097:
4096:
4061:
4059:
4055:
4054:
4052:
4051:
4046:
4037:
4032:
4030:John ellipsoid
4027:
4022:
4017:
4012:
3998:
3992:
3990:
3986:
3985:
3983:
3982:
3977:
3972:
3967:
3962:
3957:
3952:
3947:
3942:
3937:
3932:
3927:
3921:
3919:
3917:results (list)
3912:
3911:
3909:
3908:
3903:
3898:
3893:
3891:Invex function
3888:
3879:
3874:
3869:
3864:
3859:
3853:
3848:
3842:
3840:
3836:
3835:
3833:
3832:
3827:
3822:
3817:
3812:
3807:
3802:
3797:
3792:
3790:Choquet theory
3786:
3784:
3778:
3777:
3775:
3774:
3769:
3764:
3758:
3756:
3755:Basic concepts
3752:
3751:
3740:
3739:
3732:
3725:
3717:
3708:
3707:
3705:
3704:
3699:
3694:
3689:
3684:
3682:Gabriel's horn
3679:
3674:
3673:
3672:
3667:
3662:
3657:
3652:
3644:
3643:
3642:
3633:
3631:
3627:
3626:
3623:
3622:
3620:
3619:
3614:
3612:List of limits
3608:
3605:
3604:
3602:
3601:
3600:
3599:
3594:
3589:
3579:
3578:
3577:
3567:
3562:
3557:
3552:
3546:
3544:
3535:
3531:
3530:
3528:
3527:
3520:
3513:
3511:Leonhard Euler
3508:
3503:
3498:
3493:
3488:
3483:
3478:
3473:
3468:
3463:
3457:
3455:
3449:
3448:
3446:
3445:
3440:
3435:
3430:
3425:
3419:
3417:
3411:
3410:
3408:
3407:
3406:
3405:
3400:
3395:
3390:
3385:
3380:
3375:
3370:
3365:
3360:
3352:
3351:
3350:
3345:
3344:
3343:
3338:
3328:
3323:
3318:
3313:
3308:
3303:
3295:
3289:
3287:
3283:
3282:
3280:
3279:
3278:
3277:
3272:
3267:
3262:
3254:
3249:
3244:
3239:
3234:
3229:
3224:
3219:
3214:
3212:Hessian matrix
3209:
3204:
3198:
3196:
3190:
3189:
3187:
3186:
3185:
3184:
3179:
3174:
3169:
3167:Line integrals
3161:
3160:
3159:
3154:
3149:
3144:
3139:
3130:
3128:
3122:
3121:
3119:
3118:
3113:
3108:
3107:
3106:
3101:
3093:
3088:
3087:
3086:
3076:
3075:
3074:
3069:
3064:
3054:
3049:
3048:
3047:
3037:
3032:
3027:
3022:
3017:
3015:Antiderivative
3011:
3009:
3003:
3002:
3000:
2999:
2998:
2997:
2992:
2987:
2977:
2976:
2975:
2970:
2962:
2961:
2960:
2955:
2950:
2945:
2935:
2934:
2933:
2928:
2923:
2918:
2910:
2909:
2908:
2903:
2902:
2901:
2891:
2886:
2881:
2876:
2871:
2861:
2860:
2859:
2854:
2844:
2839:
2834:
2829:
2824:
2819:
2813:
2811:
2805:
2804:
2802:
2801:
2796:
2791:
2786:
2785:
2784:
2774:
2768:
2766:
2760:
2759:
2757:
2756:
2751:
2746:
2741:
2736:
2731:
2726:
2721:
2716:
2711:
2706:
2701:
2696:
2690:
2688:
2682:
2681:
2674:
2673:
2666:
2659:
2651:
2645:
2644:
2638:
2625:
2619:
2592:
2589:
2586:
2585:
2578:
2560:
2553:
2533:
2514:(3): 384â392.
2493:
2464:
2444:
2403:
2400:. p. 101.
2388:
2373:
2350:
2343:
2324:
2323:
2321:
2318:
2317:
2316:
2311:
2306:
2301:
2296:
2289:
2286:
2282:
2281:
2261:Thermodynamics
2257:
2242:
2223:
2214:
2211:
2210:
2209:
2185:
2181:
2177:
2155:
2151:
2147:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2114:
2102:
2099:
2096:
2093:
2090:
2075:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2042:
2039:
2025:
2011:
2008:
1986:
1983:
1980:
1977:
1974:
1953:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1916:
1899:
1895:
1891:
1887:
1883:
1879:
1874:
1871:
1868:
1865:
1862:
1858:
1855:
1834:
1831:
1828:
1825:
1822:
1819:
1815:
1812:
1789:
1784:
1781:
1778:
1775:
1772:
1750:
1746:
1742:
1739:
1736:
1733:
1730:
1727:
1718:The functions
1714:
1711:
1710:
1709:
1702:global maximum
1694:
1689:Near a strict
1687:
1676:
1665:if and only if
1655:
1648:
1647:
1646:
1645:
1644:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1591:
1588:
1585:
1581:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1552:
1549:
1546:
1542:
1537:
1533:
1526:
1523:
1520:
1516:
1511:
1508:
1505:
1502:
1499:
1495:
1491:
1488:
1484:
1477:
1474:
1471:
1467:
1462:
1459:
1456:
1453:
1450:
1446:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1357:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1185:
1182:
1179:
1176:
1173:
1145:If a function
1143:
1130:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1096:
1092:
1087:
1083:
1080:
1077:
1071:
1067:
1043:if and only if
1029:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
993:
990:
986:
983:
980:
977:
974:
971:
968:
965:
962:
959:
940:
900:if and only if
892:differentiable
884:
870:
861:is (strictly)
838:
835:
824:
821:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
778:
775:
772:
769:
766:
742:
712:
709:
706:
703:
700:
697:
694:
691:
671:
668:
665:
662:
659:
656:
653:
650:
630:
610:
607:
604:
601:
598:
595:
592:
589:
569:
549:
529:
508:
504:
500:
496:
493:
470:
467:
464:
444:
441:
438:
435:
432:
429:
426:
415:
414:
402:
399:
396:
393:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
360:
357:
354:
351:
348:
345:
342:
339:
336:
333:
330:
327:
324:
321:
303:
302:
291:
288:
285:
282:
279:
276:
273:
270:
267:
264:
261:
258:
255:
252:
249:
246:
243:
240:
237:
234:
231:
228:
225:
222:
219:
216:
213:
210:
187:
184:
181:
178:
175:
172:
169:
149:
129:
93:
81:A real-valued
78:
75:
63:convex upwards
26:
9:
6:
4:
3:
2:
4181:
4170:
4167:
4165:
4162:
4161:
4159:
4144:
4141:
4140:
4138:
4134:
4128:
4125:
4123:
4120:
4118:
4115:
4113:
4110:
4109:
4107:
4105:
4101:
4094:
4092:
4086:
4084:
4078:
4074:
4070:
4066:
4063:
4062:
4060:
4056:
4050:
4047:
4045:
4041:
4038:
4036:
4033:
4031:
4028:
4026:
4023:
4021:
4018:
4016:
4013:
4011:
4007:
4003:
3999:
3997:
3994:
3993:
3991:
3987:
3981:
3978:
3976:
3973:
3971:
3968:
3966:
3963:
3961:
3960:Mazur's lemma
3958:
3956:
3953:
3951:
3948:
3946:
3943:
3941:
3938:
3936:
3933:
3931:
3928:
3926:
3923:
3922:
3920:
3918:
3913:
3907:
3906:Subderivative
3904:
3902:
3899:
3897:
3894:
3892:
3889:
3887:
3883:
3880:
3878:
3875:
3873:
3870:
3868:
3865:
3863:
3860:
3858:
3854:
3852:
3849:
3847:
3844:
3843:
3841:
3837:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3811:
3808:
3806:
3803:
3801:
3798:
3796:
3793:
3791:
3788:
3787:
3785:
3783:
3782:Topics (list)
3779:
3773:
3770:
3768:
3765:
3763:
3760:
3759:
3757:
3753:
3749:
3745:
3738:
3733:
3731:
3726:
3724:
3719:
3718:
3715:
3703:
3700:
3698:
3695:
3693:
3690:
3688:
3685:
3683:
3680:
3678:
3675:
3671:
3668:
3666:
3663:
3661:
3658:
3656:
3653:
3651:
3648:
3647:
3645:
3641:
3638:
3637:
3635:
3634:
3632:
3628:
3618:
3615:
3613:
3610:
3609:
3606:
3598:
3595:
3593:
3590:
3588:
3585:
3584:
3583:
3580:
3576:
3573:
3572:
3571:
3568:
3566:
3563:
3561:
3558:
3556:
3553:
3551:
3548:
3547:
3545:
3543:
3539:
3536:
3532:
3526:
3525:
3521:
3519:
3518:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3494:
3492:
3489:
3487:
3486:Infinitesimal
3484:
3482:
3479:
3477:
3474:
3472:
3469:
3467:
3464:
3462:
3459:
3458:
3456:
3454:
3450:
3444:
3441:
3439:
3436:
3434:
3431:
3429:
3426:
3424:
3421:
3420:
3418:
3412:
3404:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3384:
3381:
3379:
3376:
3374:
3371:
3369:
3366:
3364:
3361:
3359:
3356:
3355:
3353:
3349:
3346:
3342:
3339:
3337:
3334:
3333:
3332:
3329:
3327:
3324:
3322:
3319:
3317:
3314:
3312:
3309:
3307:
3304:
3302:
3299:
3298:
3296:
3294:
3291:
3290:
3288:
3284:
3276:
3273:
3271:
3268:
3266:
3263:
3261:
3258:
3257:
3255:
3253:
3250:
3248:
3245:
3243:
3240:
3238:
3235:
3233:
3230:
3228:
3227:Line integral
3225:
3223:
3220:
3218:
3215:
3213:
3210:
3208:
3205:
3203:
3200:
3199:
3197:
3195:
3191:
3183:
3180:
3178:
3175:
3173:
3170:
3168:
3165:
3164:
3162:
3158:
3155:
3153:
3150:
3148:
3145:
3143:
3140:
3138:
3135:
3134:
3132:
3131:
3129:
3127:
3123:
3117:
3114:
3112:
3109:
3105:
3102:
3100:
3099:Washer method
3097:
3096:
3094:
3092:
3089:
3085:
3082:
3081:
3080:
3077:
3073:
3070:
3068:
3065:
3063:
3062:trigonometric
3060:
3059:
3058:
3055:
3053:
3050:
3046:
3043:
3042:
3041:
3038:
3036:
3033:
3031:
3028:
3026:
3023:
3021:
3018:
3016:
3013:
3012:
3010:
3008:
3004:
2996:
2993:
2991:
2988:
2986:
2983:
2982:
2981:
2978:
2974:
2971:
2969:
2966:
2965:
2963:
2959:
2956:
2954:
2951:
2949:
2946:
2944:
2941:
2940:
2939:
2936:
2932:
2931:Related rates
2929:
2927:
2924:
2922:
2919:
2917:
2914:
2913:
2911:
2907:
2904:
2900:
2897:
2896:
2895:
2892:
2890:
2887:
2885:
2882:
2880:
2877:
2875:
2872:
2870:
2867:
2866:
2865:
2862:
2858:
2855:
2853:
2850:
2849:
2848:
2845:
2843:
2840:
2838:
2835:
2833:
2830:
2828:
2825:
2823:
2820:
2818:
2815:
2814:
2812:
2810:
2806:
2800:
2797:
2795:
2792:
2790:
2787:
2783:
2780:
2779:
2778:
2775:
2773:
2770:
2769:
2767:
2765:
2761:
2755:
2752:
2750:
2747:
2745:
2742:
2740:
2737:
2735:
2732:
2730:
2727:
2725:
2722:
2720:
2717:
2715:
2712:
2710:
2707:
2705:
2702:
2700:
2697:
2695:
2692:
2691:
2689:
2687:
2683:
2679:
2672:
2667:
2665:
2660:
2658:
2653:
2652:
2649:
2641:
2635:
2631:
2626:
2622:
2616:
2612:
2608:
2604:
2600:
2595:
2594:
2581:
2575:
2571:
2564:
2556:
2550:
2546:
2545:
2537:
2529:
2525:
2521:
2517:
2513:
2509:
2508:
2503:
2497:
2489:
2483:
2475:
2471:
2467:
2461:
2457:
2456:
2448:
2440:
2436:
2431:
2426:
2422:
2418:
2414:
2407:
2399:
2392:
2384:
2380:
2376:
2374:0-393-95735-7
2370:
2366:
2365:
2357:
2355:
2346:
2340:
2336:
2329:
2325:
2315:
2312:
2310:
2307:
2305:
2302:
2300:
2297:
2295:
2292:
2291:
2285:
2279:
2275:
2270:
2266:
2262:
2258:
2255:
2251:
2247:
2243:
2240:
2237:functions of
2236:
2232:
2228:
2224:
2221:
2217:
2216:
2208:, is concave.
2207:
2204:
2200:
2179:
2149:
2141:
2138:
2135:
2129:
2123:
2116:The function
2115:
2097:
2094:
2091:
2080:
2076:
2061:
2058:
2055:
2052:
2049:
2043:
2037:
2030:
2026:
2009:
2006:
1978:
1975:
1951:
1947:
1944:
1941:
1935:
1929:
1921:
1917:
1897:
1893:
1889:
1885:
1881:
1877:
1872:
1869:
1863:
1856:
1853:
1832:
1829:
1826:
1820:
1813:
1810:
1787:
1782:
1776:
1770:
1748:
1744:
1740:
1737:
1731:
1725:
1717:
1716:
1707:
1703:
1699:
1698:local maximum
1695:
1692:
1691:local maximum
1688:
1685:
1681:
1677:
1675:over the set.
1674:
1667:the function
1666:
1658:
1657:
1653:
1650:Functions of
1628:
1625:
1622:
1616:
1613:
1607:
1604:
1601:
1595:
1589:
1586:
1583:
1579:
1574:
1568:
1565:
1562:
1556:
1550:
1547:
1544:
1540:
1535:
1531:
1524:
1521:
1518:
1514:
1506:
1503:
1500:
1493:
1489:
1486:
1482:
1475:
1472:
1469:
1465:
1457:
1454:
1451:
1444:
1440:
1437:
1431:
1425:
1422:
1416:
1410:
1384:
1381:
1375:
1372:
1369:
1366:
1358:
1344:
1338:
1332:
1329:
1326:
1320:
1314:
1308:
1305:
1302:
1296:
1290:
1284:
1281:
1278:
1272:
1269:
1263:
1260:
1257:
1251:
1248:
1245:
1239:
1236:
1230:
1227:
1221:
1211:
1198:
1197:
1177:
1174:
1163:
1153:
1144:
1128:
1121:
1115:
1112:
1106:
1100:
1094:
1090:
1085:
1081:
1078:
1075:
1069:
1065:
1057:
1044:
1039:
1034:
1030:
1013:
1010:
1007:
998:
991:
988:
984:
978:
972:
969:
963:
957:
949:
941:
937:
933:
929:
920:
912:
908:
901:
893:
885:
882:
878:
874:
871:
868:
864:
856:
852:
845:
841:
840:
829:
820:
803:
800:
794:
788:
785:
782:
776:
770:
764:
756:
740:
731:
729:
724:
704:
698:
695:
692:
663:
657:
654:
651:
628:
602:
596:
593:
590:
567:
547:
527:
494:
491:
482:
468:
465:
462:
439:
436:
433:
427:
424:
397:
391:
388:
385:
379:
373:
367:
364:
361:
355:
349:
346:
343:
340:
334:
331:
328:
319:
312:
311:
310:
308:
286:
280:
277:
274:
268:
262:
256:
253:
250:
244:
238:
235:
232:
229:
223:
220:
217:
208:
201:
200:
199:
182:
179:
176:
170:
167:
147:
127:
119:
115:
111:
107:
91:
84:
74:
72:
68:
64:
60:
56:
52:
48:
44:
39:
35:
30:
19:
4127:Weak duality
4090:
4082:
4002:Orthogonally
3850:
3597:Secant cubed
3522:
3515:
3496:Isaac Newton
3466:Brook Taylor
3133:Derivatives
3104:Shell method
2832:Differential
2698:
2629:
2602:
2569:
2563:
2543:
2536:
2511:
2505:
2496:
2454:
2447:
2420:
2416:
2406:
2397:
2391:
2363:
2334:
2328:
2283:
2213:Applications
2205:
1705:
1651:
1209:
1151:
1055:
1037:
935:
931:
927:
911:acceleration
907:non-positive
755:quasiconcave
732:
725:
580:, the point
483:
416:
306:
304:
120:if, for any
117:
114:vector space
80:
71:upper convex
70:
66:
62:
59:concave down
58:
54:
51:synonymously
37:
31:
29:
4117:Duality gap
4112:Dual system
3996:Convex hull
3665:of surfaces
3416:and numbers
3378:Dirichlet's
3348:Telescoping
3301:Alternating
2889:L'HĂŽpital's
2686:Precalculus
2239:risk averse
2229:theory for
2199:determinant
1659:A function
1162:subadditive
1041:is concave
934:) = −
898:is concave
733:A function
34:mathematics
4158:Categories
4040:Radial set
4010:Convex set
3772:Convex set
3461:Adequality
3147:Divergence
3020:Arc length
2817:Derivative
2423:(3): 479.
2320:References
1207:, letting
855:derivative
823:Properties
110:convex set
77:Definition
67:convex cap
4025:Hypograph
3660:of curves
3655:Curvature
3542:Integrals
3336:Maclaurin
3316:Geometric
3207:Geometric
3157:Laplacian
2869:linearity
2709:Factorial
2482:cite book
2474:965446428
2439:0022-2305
2142:
2098:π
1982:∞
1948:
1922:function
1920:logarithm
1873:−
1830:−
1741:−
1684:semifield
1654:variables
1536:≥
1388:∞
1376:∈
1327:≥
1306:−
1279:≥
1270:⋅
1261:−
1205:1 â„ t â„ 0
1196:. Proof:
1181:∞
1095:≥
1011:−
970:≤
890:is twice-
859:f ′
857:function
801:≥
503:→
466:≠
428:∈
425:α
389:α
368:α
365:−
347:α
335:α
332:−
278:α
257:α
254:−
245:≥
236:α
224:α
221:−
171:∈
168:α
43:hypograph
4049:Zonotope
4020:Epigraph
3650:Manifold
3383:Integral
3326:Infinite
3321:Harmonic
3306:Binomial
3152:Gradient
3095:Volumes
2906:Quotient
2847:Notation
2678:Calculus
2398:Analysis
2383:24847759
2288:See also
2167:, where
1857:″
1814:″
1713:Examples
1706:strictly
1214:we have
992:′
919:negative
851:interval
832:positive
417:for any
106:interval
83:function
4104:Duality
4006:Pseudo-
3980:Ursescu
3877:Pseudo-
3851:Concave
3830:Simplex
3810:Duality
3587:inverse
3575:inverse
3501:Fluxion
3311:Fourier
3177:Stokes'
3172:Green's
2894:Product
2754:Tangent
2528:5491763
2269:Entropy
2197:is the
1156:, then
1154:(0) â„ 0
894:, then
118:concave
53:called
4087:, and
4058:Series
3975:Simons
3882:Quasi-
3872:Proper
3857:Closed
3670:Tensor
3592:Secant
3358:Abel's
3341:Taylor
3232:Matrix
3182:Gauss'
2764:Limits
2744:Secant
2734:Radian
2636:
2617:
2576:
2551:
2526:
2472:
2462:
2437:
2381:
2371:
2341:
1199:Since
879:) are
877:convex
873:Points
104:on an
3915:Main
3534:Lists
3393:Ratio
3331:Power
3067:Euler
2884:Chain
2874:Power
2749:Slope
2524:S2CID
2201:of a
1671:is a
921:then
867:slope
69:, or
4035:Lens
3989:Sets
3839:Maps
3746:and
3403:Term
3398:Root
3137:Curl
2634:ISBN
2615:ISBN
2574:ISBN
2549:ISBN
2488:link
2470:OCLC
2460:ISBN
2435:ISSN
2379:OCLC
2369:ISBN
2339:ISBN
2263:and
2079:sine
2077:The
2027:Any
1918:The
1845:and
1763:and
1704:. A
1696:Any
1359:For
1049:and
682:and
560:and
455:and
356:>
140:and
36:, a
4089:(Hw
2879:Sum
2607:doi
2516:doi
2425:doi
2259:In
2244:In
2225:In
2139:log
1945:log
1212:= 0
1164:on
1160:is
1053:in
942:If
917:is
905:is
886:If
753:is
309:if
112:in
32:In
4160::
4081:(H
4079:,
4075:,
4071:,
4008:)
4004:,
3884:)
3862:K-
2613:.
2522:.
2510:.
2484:}}
2480:{{
2468:.
2433:.
2421:98
2419:.
2415:.
2377:.
2353:^
2267:,
2248:,
2233:,
1669:âf
1403::
1031:A
950::
842:A
723:.
481:.
198:,
73:.
65:,
61:,
57:,
4095:)
4093:)
4091:x
4085:)
4083:x
4067:(
4042:/
4000:(
3855:(
3736:e
3729:t
3722:v
2670:e
2663:t
2656:v
2642:.
2623:.
2609::
2582:.
2557:.
2530:.
2518::
2512:9
2490:)
2476:.
2441:.
2427::
2385:.
2347:.
2272:(
2206:B
2184:|
2180:B
2176:|
2154:|
2150:B
2146:|
2136:=
2133:)
2130:B
2127:(
2124:f
2113:.
2101:]
2095:,
2092:0
2089:[
2062:b
2059:+
2056:x
2053:a
2050:=
2047:)
2044:x
2041:(
2038:f
2010:x
2007:1
1985:)
1979:,
1976:0
1973:(
1952:x
1942:=
1939:)
1936:x
1933:(
1930:f
1898:2
1894:/
1890:3
1886:x
1882:4
1878:1
1870:=
1867:)
1864:x
1861:(
1854:g
1833:2
1827:=
1824:)
1821:x
1818:(
1811:f
1788:x
1783:=
1780:)
1777:x
1774:(
1771:g
1749:2
1745:x
1738:=
1735:)
1732:x
1729:(
1726:f
1686:.
1661:f
1652:n
1632:)
1629:b
1626:+
1623:a
1620:(
1617:f
1614:=
1611:)
1608:b
1605:+
1602:a
1599:(
1596:f
1590:b
1587:+
1584:a
1580:b
1575:+
1572:)
1569:b
1566:+
1563:a
1560:(
1557:f
1551:b
1548:+
1545:a
1541:a
1532:)
1525:b
1522:+
1519:a
1515:b
1510:)
1507:b
1504:+
1501:a
1498:(
1494:(
1490:f
1487:+
1483:)
1476:b
1473:+
1470:a
1466:a
1461:)
1458:b
1455:+
1452:a
1449:(
1445:(
1441:f
1438:=
1435:)
1432:b
1429:(
1426:f
1423:+
1420:)
1417:a
1414:(
1411:f
1391:)
1385:,
1382:0
1379:[
1373:b
1370:,
1367:a
1345:.
1342:)
1339:x
1336:(
1333:f
1330:t
1324:)
1321:0
1318:(
1315:f
1312:)
1309:t
1303:1
1300:(
1297:+
1294:)
1291:x
1288:(
1285:f
1282:t
1276:)
1273:0
1267:)
1264:t
1258:1
1255:(
1252:+
1249:x
1246:t
1243:(
1240:f
1237:=
1234:)
1231:x
1228:t
1225:(
1222:f
1210:y
1201:f
1184:)
1178:,
1175:0
1172:[
1158:f
1152:f
1147:f
1129:2
1125:)
1122:y
1119:(
1116:f
1113:+
1110:)
1107:x
1104:(
1101:f
1091:)
1086:2
1082:y
1079:+
1076:x
1070:(
1066:f
1056:C
1051:y
1047:x
1038:C
1017:]
1014:x
1008:y
1005:[
1002:)
999:x
996:(
989:f
985:+
982:)
979:x
976:(
973:f
967:)
964:y
961:(
958:f
944:f
939:.
936:x
932:x
930:(
928:f
923:f
896:f
888:f
883:.
869:.
847:f
807:}
804:a
798:)
795:x
792:(
789:f
786::
783:x
780:{
777:=
774:)
771:a
768:(
765:S
741:f
711:)
708:)
705:y
702:(
699:f
696:,
693:y
690:(
670:)
667:)
664:x
661:(
658:f
655:,
652:x
649:(
629:f
609:)
606:)
603:z
600:(
597:f
594:,
591:z
588:(
568:y
548:x
528:z
507:R
499:R
495::
492:f
469:y
463:x
443:)
440:1
437:,
434:0
431:(
401:)
398:y
395:(
392:f
386:+
383:)
380:x
377:(
374:f
371:)
362:1
359:(
353:)
350:y
344:+
341:x
338:)
329:1
326:(
323:(
320:f
290:)
287:y
284:(
281:f
275:+
272:)
269:x
266:(
263:f
260:)
251:1
248:(
242:)
239:y
233:+
230:x
227:)
218:1
215:(
212:(
209:f
186:]
183:1
180:,
177:0
174:[
148:y
128:x
92:f
20:)
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