4155:
bundle to "become" the sum of the prices of the two separate items. Thus proving that it is not a sufficient condition for a natural monopoly; since the unit of exchange may not be the actual cost of an item. This situation is familiar to everyone in the political arena where some minority asserts that the loss of some particular freedom at some particular level of government means that many governments are better; whereas the majority assert that there is some other correct unit of cost.
4117:. Entropy appears always as a subadditive quantity in all of its formulations, meaning the entropy of a supersystem or a set union of random variables is always less or equal than the sum of the entropies of its individual components. Additionally, entropy in physics satisfies several more strict inequalities such as the Strong Subadditivity of Entropy in classical statistical mechanics and its
1641:
4154:
Except in the case of complementary goods, the price of goods (as a function of quantity) must be subadditive. Otherwise, if the sum of the cost of two items is cheaper than the cost of the bundle of two of them together, then nobody would ever buy the bundle, effectively causing the price of the
4367:
4171:. The economic intuition behind risk measure subadditivity is that a portfolio risk exposure should, at worst, simply equal the sum of the risk exposures of the individual positions that compose the portfolio. The lack of subadditivity is one of the main critiques of
1351:
4625:
4011:
4228:
2754:
1742:
654:
3466:
Besides, analogues of Fekete's lemma have been proved for subadditive real maps (with additional assumptions) from finite subsets of an amenable group , and further, of a cancellative left-amenable semigroup.
4511:
1042:
299:
1636:{\textstyle {\begin{aligned}a_{n_{2}}&\leq a_{n_{1}}+a_{m}(n_{2}-n_{1})/m\\a_{n_{3}}&\leq a_{n_{2}}+a_{m}(n_{3}-n_{2})/m\leq a_{n_{1}}+a_{m}(n_{3}-n_{1})/m\\\cdots &\cdots \end{aligned}}}
899:
842:
1210:
3164:
2890:
599:
198:
3589:
1973:
35:
always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly
5241:
1356:
3382:
2211:
1122:
782:
3646:
4468:
3535:
352:
3871:
4141:. It implies that production from only one firm is socially less expensive (in terms of average costs) than production of a fraction of the original quantity by an equal number of firms.
2615:
3239:
1807:
3302:
2053:
1911:
694:
418:
3820:
3749:
3454:
4918:
2378:
2248:
2554:
5169:
5136:
2136:
246:
1341:
1264:
949:
481:
4676:
975:
88:
1849:
4712:
3065:
5268:
4498:
3906:
2992:
2478:
5075:
4953:
3672:
2652:
2414:
717:
5011:
2779:
2090:
4394:
3402:
2917:
2645:
1149:
4982:
4836:
4069:
4040:
5195:
4223:
5103:
5031:
4856:
4807:
4783:
4759:
4418:
4197:
3911:
1308:
1288:
1062:
513:
5811:
N.H. Bingham, A.J. Ostaszewski. "Generic subadditive functions." Proceedings of
American Mathematical Society, vol. 136, no. 12 (2008), pp. 4257–4266.
3459:
There are also results that allow one to deduce the rate of convergence to the limit whose existence is stated in Fekete's lemma if some kind of both
1648:
251:
4362:{\displaystyle {\text{VaR}}_{p}\equiv z_{p}\sigma _{\Delta V}=z_{p}{\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}+2\rho _{xy}\sigma _{x}\sigma _{y}}}}
114:
607:
980:
4179:
of risk factors. The
Gaussian VaR ensures subadditivity: for example, the Gaussian VaR of a two unitary long positions portfolio
4118:
847:
790:
5641:
1154:
364:
5316:
Fekete, M. (1923). "Über die
Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten".
3070:
2784:
549:
428:. This is a special case of subadditive function, if a sequence is interpreted as a function on the set of natural numbers.
3540:
5580:
5494:
1932:
5200:
2255:
3307:
2141:
1070:
730:
4620:{\displaystyle {\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}+2\rho _{xy}\sigma _{x}\sigma _{y}}}\leq \sigma _{x}+\sigma _{y}}
4134:
3594:
5795:
5786:
5390:
4423:
3491:
309:
4397:
3830:
2559:
4501:
5842:
3172:
1750:
5847:
5433:
3248:
1987:
1857:
662:
5660:"Bigger Is Not Always Safer: A Critical Analysis of the Subadditivity Assumption for Coherent Risk Measures"
3754:
3689:
5852:
4861:
3407:
2620:
Though we don't have continuous variables, we can still cover enough integers to complete the proof. Let
2219:
1216:
4225:
is, assuming that the mean portfolio value variation is zero and the VaR is defined as a negative loss,
2483:
5531:
Gromov, Misha (1999). "Topological
Invariants of Dynamical Systems and Spaces of Holomorphic Maps: I".
5141:
5108:
5078:
2095:
431:
Note that while a concave sequence is subadditive, the converse is false. For example, randomly assign
216:
4762:
1313:
1222:
907:
434:
4630:
954:
61:
5447:
5385:
Michael J. Steele. "Probability theory and combinatorial optimization". SIAM, Philadelphia (1997).
1812:
359:
4681:
27:
is a property of a function that states, roughly, that evaluating the function for the sum of two
5279:
5246:
4473:
3876:
2922:
2419:
5481:
5442:
5299: – Property of geometry, also used to generalize the notion of "distance" in metric spaces
5036:
4923:
3651:
2385:
56:
1310:. This sequence, continued for long enough, would be forced by subadditivity to dip below the
699:
5576:"An analogue of Fekete's lemma for subadditive functions on cancellative amenable semigroups"
4990:
4164:
2997:
108:
5351:
de Bruijn, N.G.; Erdös, P. (1952). "Some linear and some quadratic recursion formulas. II".
5282: – Difference in properties of one mole of substance in a mixture vs. an ideal solution
2058:
4372:
3387:
2895:
2623:
1981:
Continue the proof as before, until we have just used the infinite pigeonhole principle.
1127:
657:
91:
32:
4958:
4812:
4045:
4016:
4006:{\displaystyle f(x)\geq \textstyle {\frac {y}{x+y}}f(0)+\textstyle {\frac {x}{x+y}}f(x+y)}
8:
5296:
5174:
4202:
4176:
4114:
4106:
3485:
602:
529:
5805:
2759:
5826:
5757:
5589:
5556:
5420:
5333:
5088:
5016:
4841:
4792:
4768:
4744:
4403:
4182:
4144:
4102:
2382:
If we were dealing with continuous variables, then we can use subadditivity to go from
1293:
1273:
1267:
1047:
36:
5619:
Hille 1948, Theorem 6.6.1. (Measurability is stipulated in Sect. 6.2 "Preliminaries".)
5364:
486:
5791:
5777:
5749:
5637:
5560:
5548:
5513:
5462:
5386:
5337:
4110:
5781:
5761:
5741:
5710:
5681:
5671:
5599:
5540:
5503:
5452:
5360:
5325:
5285:
4138:
3825:
1747:
The analogue of Fekete's lemma holds for superadditive sequences as well, that is:
2749:{\displaystyle \ln(2)>\ln(1.5)+\ln \left({\frac {1.5n_{k}+m}{1.5n_{k}}}\right)}
5729:
5714:
5290:
4739:
4168:
4079:
5629:
5485:
5424:
4955:
is subadditive, and hence Fekete's lemma can be used to estimate the growth of
4727:
4723:
3067:
touch in the middle. Thus, by repeating this process, we cover the entirety of
533:
5604:
5575:
5544:
5836:
5753:
5552:
5517:
5466:
4172:
3460:
98:
28:
5745:
5574:
Ceccherini-Silberstein, Tullio; Krieger, Fabrice; Coornaert, Michel (2014).
4678:
and, in particular, it equals the sum of the individual risk exposures when
4809:. In combinatorics on words, a common problem is to determine the number
4148:
207:
44:
40:
5404:
1854:
There are extensions of Fekete's lemma that do not require the inequality
1737:{\displaystyle \limsup _{k}a_{n_{k}}/n_{k}\leq a_{m}/m<s^{*}+\epsilon }
5801:
5676:
5659:
210:
203:
20:
5686:
5822:
5508:
5489:
5457:
5428:
5329:
4130:
5730:"Improved bounds on the average length of longest common subsequences"
5573:
4786:
528:
A useful result pertaining to subadditive sequences is the following
649:{\displaystyle \displaystyle \lim _{n\to \infty }{\frac {a_{n}}{n}}}
4714:
which is the case of no diversification effects on portfolio risk.
4505:
304:
101:
5701:
Shur, Arseny (2012). "Growth properties of power-free languages".
5594:
5406:
CBMS Lectures on
Probability Theory and Combinatorial Optimization
5490:"Entropy and isomorphism theorems for actions of amenable groups"
4098:
1809:(The limit then may be positive infinity: consider the sequence
1037:{\displaystyle {\frac {a_{n_{k}}}{n_{k}}}>s^{*}+\epsilon }
4722:
Subadditivity occurs in the thermodynamic properties of non-
3682:
is a subadditive function, and if 0 is in its domain, then
4129:
Subadditivity is an essential property of some particular
3908:
is also subadditive. To see this, one first observes that
294:{\displaystyle {\sqrt {x+y}}\leq {\sqrt {x}}+{\sqrt {y}}.}
5821:
This article incorporates material from subadditivity on
5270:, however, is only known to be between 0.788 and 0.827.
894:{\displaystyle \limsup _{n}{\frac {a_{n}}{n}}\leq s^{*}}
837:{\displaystyle \liminf _{n}{\frac {a_{n}}{n}}\geq s^{*}}
2919:, it's easy to see (draw a picture) that the intervals
1205:{\displaystyle {\frac {a_{m}}{m}}<s^{*}+\epsilon /2}
4627:
Thus the
Gaussian VaR is subadditive for any value of
3964:
3930:
3686:(0) ≥ 0. To see this, take the inequality at the top.
3410:
3159:{\displaystyle (n_{k}+m\mathbb {Z} )\cap (\ln n_{k}+)}
2885:{\displaystyle (n_{k}+m\mathbb {Z} )\cap (\ln n_{k}+)}
1935:
1354:
594:{\displaystyle {\left\{a_{n}\right\}}_{n=1}^{\infty }}
193:{\displaystyle \forall x,y\in A,f(x+y)\leq f(x)+f(y).}
5249:
5203:
5177:
5144:
5111:
5091:
5039:
5019:
4993:
4961:
4926:
4864:
4844:
4815:
4795:
4771:
4747:
4684:
4633:
4514:
4476:
4426:
4406:
4375:
4231:
4205:
4185:
4048:
4019:
3914:
3879:
3833:
3757:
3692:
3654:
3597:
3584:{\displaystyle \lim _{t\to \infty }{\frac {f(t)}{t}}}
3543:
3494:
3390:
3310:
3251:
3175:
3073:
3000:
2925:
2898:
2787:
2762:
2655:
2626:
2562:
2486:
2422:
2388:
2258:
2222:
2144:
2098:
2061:
1990:
1968:{\textstyle {\frac {1}{2}}\leq {\frac {m}{n}}\leq 2.}
1860:
1815:
1753:
1651:
1316:
1296:
1276:
1225:
1157:
1130:
1073:
1050:
983:
957:
910:
850:
793:
733:
702:
665:
611:
610:
552:
489:
437:
367:
312:
254:
219:
117:
64:
4504:
between the two individual positions returns. Since
4163:
Subadditivity is one of the desirable properties of
515:; then the sequence is subadditive but not concave.
5236:{\displaystyle {\frac {1}{k}}<\gamma _{k}\leq 1}
4470:are the individual positions returns variances and
5808:". American Mathematical Society, New York (1948).
5419:
5262:
5235:
5189:
5163:
5130:
5097:
5069:
5025:
5005:
4976:
4947:
4912:
4850:
4830:
4801:
4777:
4753:
4706:
4670:
4619:
4492:
4462:
4412:
4388:
4361:
4217:
4191:
4063:
4034:
4005:
3900:
3865:
3814:
3743:
3666:
3640:
3583:
3529:
3448:
3396:
3377:{\displaystyle a_{n+m}\leq a_{n}+a_{m}+\phi (n+m)}
3376:
3296:
3233:
3158:
3059:
2986:
2911:
2884:
2773:
2748:
2639:
2609:
2548:
2472:
2408:
2372:
2242:
2206:{\displaystyle a_{3m}\leq a_{2m}+a_{m}\leq 3a_{m}}
2205:
2130:
2084:
2047:
1967:
1905:
1843:
1801:
1736:
1635:
1335:
1302:
1282:
1258:
1204:
1143:
1117:{\displaystyle s^{*}:=\inf _{n}{\frac {a_{n}}{n}}}
1116:
1056:
1036:
969:
943:
893:
836:
777:{\displaystyle s^{*}:=\inf _{n}{\frac {a_{n}}{n}}}
776:
711:
688:
648:
593:
507:
475:
412:
346:
293:
240:
192:
82:
5402:
3404:is an increasing function such that the integral
5834:
5827:Creative Commons Attribution/Share-Alike License
3641:{\displaystyle \inf _{t>0}{\frac {f(t)}{t}}.}
3599:
3545:
1653:
1088:
852:
795:
748:
666:
613:
4463:{\displaystyle \sigma _{x}^{2},\sigma _{y}^{2}}
5350:
4726:and mixtures like the excess molar volume and
4175:models which do not rely on the assumption of
4088:
3530:{\displaystyle f:(0,\infty )\to \mathbb {R} ,}
2556:, and so on, which covers the entire interval
347:{\displaystyle \left\{a_{n}\right\}_{n\geq 1}}
5480:
3866:{\displaystyle f:[0,\infty )\to \mathbb {R} }
111:under addition, with the following property:
5790:, vol. 1. Springer-Verlag, New York (1976).
4013:. Then looking at the sum of this bound for
2610:{\displaystyle a_{n_{k}}+[\ln 1.5,+\infty )}
47:are special cases of subadditive functions.
5533:Mathematical Physics, Analysis and Geometry
2781:be the smallest number in the intersection
5657:
5651:
4078:The negative of a subadditive function is
3241:are forced down as in the previous proof.
1346:In more detail, by subadditivity, we have
5685:
5675:
5628:
5603:
5593:
5507:
5456:
5446:
5138:, such that the expected length grows as
4733:
3859:
3520:
3439:
3234:{\displaystyle a_{n_{k}},a_{n_{k+1}},...}
3094:
2808:
2250:, we can use subadditivity on them if
2236:
5634:Handbook of Analysis and its Foundations
4858:words in a factorial language. Clearly
1802:{\displaystyle a_{n+m}\geq a_{n}+a_{m}.}
904:If not, then there exists a subsequence
5376:.) See also Steele 1997, Theorem 1.9.2.
3297:{\displaystyle a_{n+m}\leq a_{n}+a_{m}}
2048:{\displaystyle a_{m},a_{2m},a_{3m},...}
1906:{\displaystyle a_{n+m}\leq a_{n}+a_{m}}
1266:, whose indices all belong to the same
689:{\displaystyle \inf {\frac {a_{n}}{n}}}
413:{\displaystyle a_{n+m}\leq a_{n}+a_{m}}
16:Property of some mathematical functions
5835:
5727:
5530:
5315:
3815:{\displaystyle f(0)\geq f(0+y)-f(y)=0}
1290:, and so they advance by multiples of
5700:
5694:
5353:Nederl. Akad. Wetensch. Proc. Ser. A
5033:uniformly at random on the alphabet
4113:in a generalized formulation due to
3744:{\displaystyle f(x)\geq f(x+y)-f(y)}
3449:{\textstyle \int \phi (t)t^{-2}\,dt}
5806:Functional analysis and semi-groups
4913:{\displaystyle A(m+n)\leq A(m)A(n)}
2373:{\displaystyle \ln(s+t)\in =\ln s+}
2243:{\displaystyle s,t\in \mathbb {N} }
13:
4262:
4135:necessary and sufficient condition
3849:
3658:
3555:
3510:
3147:
2601:
2549:{\displaystyle a_{n_{k}}+\ln 1.5+}
706:
623:
586:
220:
118:
14:
5864:
5815:
5787:Problems and Theorems in Analysis
5164:{\displaystyle \sim \gamma _{k}n}
5131:{\displaystyle \gamma _{k}\geq 0}
5105:, and thus there exists a number
4717:
2131:{\displaystyle a_{2m}\leq 2a_{m}}
1219:, there exists a sub-subsequence
241:{\displaystyle \forall x,y\geq 0}
4398:cumulative distribution function
5721:
5622:
5013:, sample two strings of length
4147:are represented by subadditive
3456:converges (near the infinity).
1343:slope line, a contradiction.
1336:{\displaystyle s^{*}+\epsilon }
1259:{\displaystyle (a_{n_{k}})_{k}}
1217:infinitary pigeonhole principle
944:{\displaystyle (a_{n_{k}})_{k}}
546:For every subadditive sequence
476:{\displaystyle a_{1},a_{2},...}
5825:, which is licensed under the
5728:Lueker, George S. (May 2009).
5613:
5581:Journal d'Analyse Mathématique
5567:
5524:
5495:Journal d'Analyse Mathématique
5474:
5413:
5396:
5379:
5344:
5309:
5293: – Property of a function
4971:
4965:
4942:
4936:
4907:
4901:
4895:
4889:
4880:
4868:
4825:
4819:
4671:{\displaystyle \rho _{xy}\in }
4665:
4650:
4058:
4052:
4029:
4023:
3998:
3986:
3958:
3952:
3924:
3918:
3889:
3883:
3855:
3852:
3840:
3803:
3797:
3788:
3776:
3767:
3761:
3738:
3732:
3723:
3711:
3702:
3696:
3626:
3620:
3572:
3566:
3552:
3516:
3513:
3501:
3463:and subadditivity is present.
3423:
3417:
3371:
3359:
3153:
3150:
3141:
3135:
3126:
3104:
3098:
3074:
3054:
3051:
3045:
3033:
3027:
3018:
2981:
2978:
2972:
2960:
2954:
2945:
2879:
2876:
2873:
2867:
2855:
2849:
2840:
2818:
2812:
2788:
2686:
2680:
2668:
2662:
2647:be large enough, such that
2604:
2583:
2543:
2519:
2467:
2443:
2367:
2343:
2325:
2322:
2313:
2301:
2292:
2283:
2277:
2265:
1604:
1578:
1534:
1508:
1439:
1413:
1247:
1226:
970:{\displaystyle \epsilon >0}
932:
911:
620:
502:
490:
213:as domain and codomain: since
184:
178:
169:
163:
154:
142:
83:{\displaystyle f\colon A\to B}
74:
50:
1:
5771:
5636:. San Diego: Academic Press.
5434:Israel Journal of Mathematics
5365:10.1016/S1385-7258(52)50021-0
5077:. The expected length of the
4396:is the inverse of the normal
1844:{\displaystyle a_{n}=\log n!}
518:
5715:10.1016/j.cosrev.2012.09.001
5429:"Mean topological dimension"
5171:. By checking the case with
4707:{\displaystyle \rho _{xy}=1}
4124:
4101:plays a fundamental role in
3470:
3304:may be weakened as follows:
523:
55:A subadditive function is a
7:
5273:
5263:{\displaystyle \gamma _{2}}
4089:Examples in various domains
4071:, will finally verify that
2216:By the assumption, for any
10:
5869:
5409:. University of Cambridge.
5403:Michael J. Steele (2011).
5243:. The exact value of even
5079:longest common subsequence
4502:linear correlation measure
4493:{\displaystyle \rho _{xy}}
4158:
4137:for the verification of a
4093:
3901:{\displaystyle f(0)\geq 0}
2987:{\displaystyle \ln n_{k}+}
2473:{\displaystyle a_{n_{k}}+}
541:Fekete's Subadditive Lemma
5605:10.1007/s11854-014-0027-4
5318:Mathematische Zeitschrift
5070:{\displaystyle 1,2,...,k}
4948:{\displaystyle \log A(n)}
4789:of that word are also in
3667:{\displaystyle -\infty .}
2409:{\displaystyle a_{n_{k}}}
844:. So it suffices to show
5303:
4199:at the confidence level
3245:Moreover, the condition
712:{\displaystyle -\infty }
5746:10.1145/1516512.1516519
5703:Computer Science Review
5658:Rau-Bredow, H. (2019).
5545:10.1023/A:1009841100168
5280:Apparent molar property
5006:{\displaystyle k\geq 1}
3591:exists and is equal to
3060:{\displaystyle \ln n'+}
2892:. By the assumption on
5264:
5237:
5191:
5165:
5132:
5099:
5085:-additive function of
5071:
5027:
5007:
4978:
4949:
4914:
4852:
4832:
4803:
4779:
4755:
4734:Combinatorics on words
4708:
4672:
4621:
4494:
4464:
4414:
4390:
4363:
4219:
4193:
4165:coherent risk measures
4133:. It is, generally, a
4065:
4036:
4007:
3902:
3867:
3816:
3745:
3668:
3642:
3585:
3531:
3450:
3398:
3378:
3298:
3235:
3160:
3061:
2988:
2913:
2886:
2775:
2750:
2641:
2611:
2550:
2474:
2410:
2374:
2244:
2207:
2132:
2086:
2085:{\displaystyle 2m/m=2}
2049:
1984:Consider the sequence
1969:
1907:
1845:
1803:
1738:
1637:
1337:
1304:
1284:
1260:
1206:
1145:
1118:
1058:
1038:
971:
945:
895:
838:
778:
713:
690:
650:
595:
509:
477:
414:
348:
295:
242:
194:
84:
5843:Mathematical analysis
5265:
5238:
5192:
5166:
5133:
5100:
5072:
5028:
5008:
4979:
4950:
4915:
4853:
4833:
4804:
4780:
4756:
4709:
4673:
4622:
4495:
4465:
4415:
4400:at probability level
4391:
4389:{\displaystyle z_{p}}
4364:
4220:
4194:
4066:
4037:
4008:
3903:
3868:
3817:
3746:
3669:
3643:
3586:
3532:
3488:subadditive function
3451:
3399:
3397:{\displaystyle \phi }
3379:
3299:
3236:
3161:
3062:
2989:
2914:
2912:{\displaystyle n_{k}}
2887:
2776:
2751:
2642:
2640:{\displaystyle n_{k}}
2612:
2551:
2475:
2411:
2375:
2245:
2208:
2138:. Similarly, we have
2133:
2087:
2050:
1970:
1908:
1846:
1804:
1739:
1638:
1338:
1305:
1285:
1261:
1207:
1146:
1144:{\displaystyle a_{m}}
1119:
1059:
1039:
972:
946:
896:
839:
779:
714:
696:. (The limit may be
691:
651:
596:
510:
478:
415:
349:
296:
243:
206:function, having the
195:
85:
5848:Sequences and series
5677:10.3390/risks7030091
5247:
5201:
5175:
5142:
5109:
5089:
5037:
5017:
4991:
4977:{\displaystyle A(n)}
4959:
4924:
4862:
4842:
4831:{\displaystyle A(n)}
4813:
4793:
4769:
4745:
4730:or excess enthalpy.
4682:
4631:
4512:
4508:is always positive,
4474:
4424:
4404:
4373:
4229:
4203:
4183:
4064:{\displaystyle f(y)}
4046:
4035:{\displaystyle f(x)}
4017:
3912:
3877:
3831:
3755:
3690:
3652:
3595:
3541:
3492:
3408:
3388:
3308:
3249:
3173:
3071:
2998:
2923:
2896:
2785:
2760:
2653:
2624:
2560:
2484:
2420:
2386:
2256:
2220:
2142:
2096:
2059:
1988:
1933:
1858:
1813:
1751:
1649:
1352:
1314:
1294:
1274:
1223:
1155:
1128:
1071:
1048:
981:
955:
908:
848:
791:
731:
700:
663:
608:
550:
487:
435:
365:
358:if it satisfies the
310:
252:
217:
115:
62:
5482:Ornstein, Donald S.
5421:Lindenstrauss, Elon
5297:Triangle inequality
5190:{\displaystyle n=1}
4549:
4531:
4459:
4441:
4317:
4299:
4218:{\displaystyle 1-p}
4107:statistical physics
3482: —
590:
544: —
5853:Types of functions
5734:Journal of the ACM
5509:10.1007/BF02790325
5458:10.1007/BF02810577
5371:Indagationes Math.
5330:10.1007/BF01504345
5260:
5233:
5187:
5161:
5128:
5095:
5067:
5023:
5003:
4974:
4945:
4910:
4848:
4828:
4799:
4775:
4761:is one where if a
4751:
4704:
4668:
4617:
4535:
4517:
4490:
4460:
4445:
4427:
4410:
4386:
4359:
4303:
4285:
4215:
4189:
4145:Economies of scale
4103:information theory
4061:
4032:
4003:
4002:
4001:
3898:
3863:
3812:
3741:
3664:
3648:(The limit may be
3638:
3613:
3581:
3559:
3527:
3480:
3446:
3394:
3374:
3294:
3231:
3156:
3057:
2984:
2909:
2882:
2774:{\displaystyle n'}
2771:
2746:
2637:
2607:
2546:
2470:
2406:
2370:
2240:
2203:
2128:
2082:
2045:
1979:
1965:
1903:
1841:
1799:
1734:
1661:
1633:
1631:
1333:
1300:
1280:
1256:
1202:
1141:
1124:, there exists an
1114:
1096:
1054:
1034:
967:
941:
891:
860:
834:
803:
774:
756:
725:
709:
686:
646:
645:
627:
591:
553:
542:
505:
473:
410:
344:
291:
238:
202:An example is the
190:
80:
5643:978-0-12-622760-4
5212:
5197:, we easily have
5098:{\displaystyle n}
5026:{\displaystyle n}
4851:{\displaystyle n}
4802:{\displaystyle L}
4778:{\displaystyle L}
4754:{\displaystyle L}
4589:
4413:{\displaystyle p}
4357:
4236:
4192:{\displaystyle V}
4111:quantum mechanics
3981:
3947:
3633:
3598:
3579:
3544:
3476:
2740:
1977:
1957:
1944:
1652:
1303:{\displaystyle m}
1283:{\displaystyle m}
1173:
1112:
1087:
1057:{\displaystyle k}
1013:
876:
851:
819:
794:
772:
747:
723:
684:
643:
612:
540:
286:
276:
266:
5860:
5766:
5765:
5725:
5719:
5718:
5709:(5–6): 187–208.
5698:
5692:
5691:
5689:
5679:
5655:
5649:
5647:
5626:
5620:
5617:
5611:
5609:
5607:
5597:
5571:
5565:
5564:
5528:
5522:
5521:
5511:
5478:
5472:
5470:
5460:
5450:
5417:
5411:
5410:
5400:
5394:
5383:
5377:
5368:
5348:
5342:
5341:
5313:
5286:Choquet integral
5269:
5267:
5266:
5261:
5259:
5258:
5242:
5240:
5239:
5234:
5226:
5225:
5213:
5205:
5196:
5194:
5193:
5188:
5170:
5168:
5167:
5162:
5157:
5156:
5137:
5135:
5134:
5129:
5121:
5120:
5104:
5102:
5101:
5096:
5076:
5074:
5073:
5068:
5032:
5030:
5029:
5024:
5012:
5010:
5009:
5004:
4983:
4981:
4980:
4975:
4954:
4952:
4951:
4946:
4919:
4917:
4916:
4911:
4857:
4855:
4854:
4849:
4837:
4835:
4834:
4829:
4808:
4806:
4805:
4800:
4784:
4782:
4781:
4776:
4760:
4758:
4757:
4752:
4713:
4711:
4710:
4705:
4697:
4696:
4677:
4675:
4674:
4669:
4646:
4645:
4626:
4624:
4623:
4618:
4616:
4615:
4603:
4602:
4590:
4588:
4587:
4578:
4577:
4568:
4567:
4548:
4543:
4530:
4525:
4516:
4499:
4497:
4496:
4491:
4489:
4488:
4469:
4467:
4466:
4461:
4458:
4453:
4440:
4435:
4419:
4417:
4416:
4411:
4395:
4393:
4392:
4387:
4385:
4384:
4368:
4366:
4365:
4360:
4358:
4356:
4355:
4346:
4345:
4336:
4335:
4316:
4311:
4298:
4293:
4284:
4282:
4281:
4269:
4268:
4256:
4255:
4243:
4242:
4237:
4234:
4224:
4222:
4221:
4216:
4198:
4196:
4195:
4190:
4139:natural monopoly
4109:, as well as in
4075:is subadditive.
4070:
4068:
4067:
4062:
4041:
4039:
4038:
4033:
4012:
4010:
4009:
4004:
3982:
3980:
3966:
3948:
3946:
3932:
3907:
3905:
3904:
3899:
3872:
3870:
3869:
3864:
3862:
3826:concave function
3821:
3819:
3818:
3813:
3750:
3748:
3747:
3742:
3673:
3671:
3670:
3665:
3647:
3645:
3644:
3639:
3634:
3629:
3615:
3612:
3590:
3588:
3587:
3582:
3580:
3575:
3561:
3558:
3536:
3534:
3533:
3528:
3523:
3483:
3455:
3453:
3452:
3447:
3438:
3437:
3403:
3401:
3400:
3395:
3383:
3381:
3380:
3375:
3352:
3351:
3339:
3338:
3326:
3325:
3303:
3301:
3300:
3295:
3293:
3292:
3280:
3279:
3267:
3266:
3240:
3238:
3237:
3232:
3218:
3217:
3216:
3215:
3192:
3191:
3190:
3189:
3165:
3163:
3162:
3157:
3122:
3121:
3097:
3086:
3085:
3066:
3064:
3063:
3058:
3014:
2993:
2991:
2990:
2985:
2941:
2940:
2918:
2916:
2915:
2910:
2908:
2907:
2891:
2889:
2888:
2883:
2836:
2835:
2811:
2800:
2799:
2780:
2778:
2777:
2772:
2770:
2755:
2753:
2752:
2747:
2745:
2741:
2739:
2738:
2737:
2724:
2717:
2716:
2703:
2646:
2644:
2643:
2638:
2636:
2635:
2616:
2614:
2613:
2608:
2579:
2578:
2577:
2576:
2555:
2553:
2552:
2547:
2503:
2502:
2501:
2500:
2479:
2477:
2476:
2471:
2439:
2438:
2437:
2436:
2415:
2413:
2412:
2407:
2405:
2404:
2403:
2402:
2379:
2377:
2376:
2371:
2249:
2247:
2246:
2241:
2239:
2212:
2210:
2209:
2204:
2202:
2201:
2186:
2185:
2173:
2172:
2157:
2156:
2137:
2135:
2134:
2129:
2127:
2126:
2111:
2110:
2091:
2089:
2088:
2083:
2072:
2054:
2052:
2051:
2046:
2032:
2031:
2016:
2015:
2000:
1999:
1974:
1972:
1971:
1966:
1958:
1950:
1945:
1937:
1913:to hold for all
1912:
1910:
1909:
1904:
1902:
1901:
1889:
1888:
1876:
1875:
1850:
1848:
1847:
1842:
1825:
1824:
1808:
1806:
1805:
1800:
1795:
1794:
1782:
1781:
1769:
1768:
1743:
1741:
1740:
1735:
1727:
1726:
1711:
1706:
1705:
1693:
1692:
1683:
1678:
1677:
1676:
1675:
1660:
1642:
1640:
1639:
1634:
1632:
1611:
1603:
1602:
1590:
1589:
1577:
1576:
1564:
1563:
1562:
1561:
1541:
1533:
1532:
1520:
1519:
1507:
1506:
1494:
1493:
1492:
1491:
1470:
1469:
1468:
1467:
1446:
1438:
1437:
1425:
1424:
1412:
1411:
1399:
1398:
1397:
1396:
1375:
1374:
1373:
1372:
1342:
1340:
1339:
1334:
1326:
1325:
1309:
1307:
1306:
1301:
1289:
1287:
1286:
1281:
1265:
1263:
1262:
1257:
1255:
1254:
1245:
1244:
1243:
1242:
1211:
1209:
1208:
1203:
1198:
1187:
1186:
1174:
1169:
1168:
1159:
1150:
1148:
1147:
1142:
1140:
1139:
1123:
1121:
1120:
1115:
1113:
1108:
1107:
1098:
1095:
1083:
1082:
1063:
1061:
1060:
1055:
1043:
1041:
1040:
1035:
1027:
1026:
1014:
1012:
1011:
1002:
1001:
1000:
999:
985:
976:
974:
973:
968:
950:
948:
947:
942:
940:
939:
930:
929:
928:
927:
900:
898:
897:
892:
890:
889:
877:
872:
871:
862:
859:
843:
841:
840:
835:
833:
832:
820:
815:
814:
805:
802:
783:
781:
780:
775:
773:
768:
767:
758:
755:
743:
742:
718:
716:
715:
710:
695:
693:
692:
687:
685:
680:
679:
670:
656:is equal to the
655:
653:
652:
647:
644:
639:
638:
629:
626:
600:
598:
597:
592:
589:
584:
573:
572:
568:
567:
545:
514:
512:
511:
508:{\displaystyle }
506:
482:
480:
479:
474:
460:
459:
447:
446:
419:
417:
416:
411:
409:
408:
396:
395:
383:
382:
353:
351:
350:
345:
343:
342:
331:
327:
326:
300:
298:
297:
292:
287:
282:
277:
272:
267:
256:
247:
245:
244:
239:
199:
197:
196:
191:
89:
87:
86:
81:
5868:
5867:
5863:
5862:
5861:
5859:
5858:
5857:
5833:
5832:
5818:
5774:
5769:
5726:
5722:
5699:
5695:
5656:
5652:
5644:
5630:Schechter, Eric
5627:
5623:
5618:
5614:
5572:
5568:
5529:
5525:
5486:Weiss, Benjamin
5479:
5475:
5425:Weiss, Benjamin
5418:
5414:
5401:
5397:
5384:
5380:
5349:
5345:
5314:
5310:
5306:
5291:Superadditivity
5276:
5254:
5250:
5248:
5245:
5244:
5221:
5217:
5204:
5202:
5199:
5198:
5176:
5173:
5172:
5152:
5148:
5143:
5140:
5139:
5116:
5112:
5110:
5107:
5106:
5090:
5087:
5086:
5038:
5035:
5034:
5018:
5015:
5014:
4992:
4989:
4988:
4960:
4957:
4956:
4925:
4922:
4921:
4863:
4860:
4859:
4843:
4840:
4839:
4814:
4811:
4810:
4794:
4791:
4790:
4770:
4767:
4766:
4746:
4743:
4742:
4736:
4724:ideal solutions
4720:
4689:
4685:
4683:
4680:
4679:
4638:
4634:
4632:
4629:
4628:
4611:
4607:
4598:
4594:
4583:
4579:
4573:
4569:
4560:
4556:
4544:
4539:
4526:
4521:
4515:
4513:
4510:
4509:
4481:
4477:
4475:
4472:
4471:
4454:
4449:
4436:
4431:
4425:
4422:
4421:
4405:
4402:
4401:
4380:
4376:
4374:
4371:
4370:
4351:
4347:
4341:
4337:
4328:
4324:
4312:
4307:
4294:
4289:
4283:
4277:
4273:
4261:
4257:
4251:
4247:
4238:
4233:
4232:
4230:
4227:
4226:
4204:
4201:
4200:
4184:
4181:
4180:
4169:risk management
4161:
4127:
4096:
4091:
4085:
4047:
4044:
4043:
4018:
4015:
4014:
3970:
3965:
3936:
3931:
3913:
3910:
3909:
3878:
3875:
3874:
3858:
3832:
3829:
3828:
3756:
3753:
3752:
3691:
3688:
3687:
3676:
3653:
3650:
3649:
3616:
3614:
3602:
3596:
3593:
3592:
3562:
3560:
3548:
3542:
3539:
3538:
3519:
3493:
3490:
3489:
3481:
3473:
3461:superadditivity
3430:
3426:
3409:
3406:
3405:
3389:
3386:
3385:
3347:
3343:
3334:
3330:
3315:
3311:
3309:
3306:
3305:
3288:
3284:
3275:
3271:
3256:
3252:
3250:
3247:
3246:
3243:
3205:
3201:
3200:
3196:
3185:
3181:
3180:
3176:
3174:
3171:
3170:
3169:With that, all
3117:
3113:
3093:
3081:
3077:
3072:
3069:
3068:
3007:
2999:
2996:
2995:
2936:
2932:
2924:
2921:
2920:
2903:
2899:
2897:
2894:
2893:
2831:
2827:
2807:
2795:
2791:
2786:
2783:
2782:
2763:
2761:
2758:
2757:
2733:
2729:
2725:
2712:
2708:
2704:
2702:
2698:
2654:
2651:
2650:
2631:
2627:
2625:
2622:
2621:
2572:
2568:
2567:
2563:
2561:
2558:
2557:
2496:
2492:
2491:
2487:
2485:
2482:
2481:
2432:
2428:
2427:
2423:
2421:
2418:
2417:
2398:
2394:
2393:
2389:
2387:
2384:
2383:
2257:
2254:
2253:
2235:
2221:
2218:
2217:
2197:
2193:
2181:
2177:
2165:
2161:
2149:
2145:
2143:
2140:
2139:
2122:
2118:
2103:
2099:
2097:
2094:
2093:
2068:
2060:
2057:
2056:
2024:
2020:
2008:
2004:
1995:
1991:
1989:
1986:
1985:
1949:
1936:
1934:
1931:
1930:
1921:, but only for
1897:
1893:
1884:
1880:
1865:
1861:
1859:
1856:
1855:
1820:
1816:
1814:
1811:
1810:
1790:
1786:
1777:
1773:
1758:
1754:
1752:
1749:
1748:
1745:
1722:
1718:
1707:
1701:
1697:
1688:
1684:
1679:
1671:
1667:
1666:
1662:
1656:
1650:
1647:
1646:
1630:
1629:
1622:
1616:
1615:
1607:
1598:
1594:
1585:
1581:
1572:
1568:
1557:
1553:
1552:
1548:
1537:
1528:
1524:
1515:
1511:
1502:
1498:
1487:
1483:
1482:
1478:
1471:
1463:
1459:
1458:
1454:
1451:
1450:
1442:
1433:
1429:
1420:
1416:
1407:
1403:
1392:
1388:
1387:
1383:
1376:
1368:
1364:
1363:
1359:
1355:
1353:
1350:
1349:
1321:
1317:
1315:
1312:
1311:
1295:
1292:
1291:
1275:
1272:
1271:
1250:
1246:
1238:
1234:
1233:
1229:
1224:
1221:
1220:
1194:
1182:
1178:
1164:
1160:
1158:
1156:
1153:
1152:
1135:
1131:
1129:
1126:
1125:
1103:
1099:
1097:
1091:
1078:
1074:
1072:
1069:
1068:
1049:
1046:
1045:
1022:
1018:
1007:
1003:
995:
991:
990:
986:
984:
982:
979:
978:
956:
953:
952:
935:
931:
923:
919:
918:
914:
909:
906:
905:
885:
881:
867:
863:
861:
855:
849:
846:
845:
828:
824:
810:
806:
804:
798:
792:
789:
788:
787:By definition,
763:
759:
757:
751:
738:
734:
732:
729:
728:
721:
701:
698:
697:
675:
671:
669:
664:
661:
660:
634:
630:
628:
616:
609:
606:
605:
585:
574:
563:
559:
555:
554:
551:
548:
547:
543:
526:
521:
488:
485:
484:
483:with values in
455:
451:
442:
438:
436:
433:
432:
404:
400:
391:
387:
372:
368:
366:
363:
362:
332:
322:
318:
314:
313:
311:
308:
307:
281:
271:
255:
253:
250:
249:
218:
215:
214:
116:
113:
112:
63:
60:
59:
53:
17:
12:
11:
5:
5866:
5856:
5855:
5850:
5845:
5817:
5816:External links
5814:
5813:
5812:
5809:
5799:
5773:
5770:
5768:
5767:
5720:
5693:
5650:
5642:
5621:
5612:
5566:
5539:(4): 323–415.
5523:
5473:
5448:10.1.1.30.3552
5412:
5395:
5378:
5343:
5324:(1): 228–249.
5307:
5305:
5302:
5301:
5300:
5294:
5288:
5283:
5275:
5272:
5257:
5253:
5232:
5229:
5224:
5220:
5216:
5211:
5208:
5186:
5183:
5180:
5160:
5155:
5151:
5147:
5127:
5124:
5119:
5115:
5094:
5066:
5063:
5060:
5057:
5054:
5051:
5048:
5045:
5042:
5022:
5002:
4999:
4996:
4973:
4970:
4967:
4964:
4944:
4941:
4938:
4935:
4932:
4929:
4909:
4906:
4903:
4900:
4897:
4894:
4891:
4888:
4885:
4882:
4879:
4876:
4873:
4870:
4867:
4847:
4827:
4824:
4821:
4818:
4798:
4774:
4750:
4735:
4732:
4728:heat of mixing
4719:
4718:Thermodynamics
4716:
4703:
4700:
4695:
4692:
4688:
4667:
4664:
4661:
4658:
4655:
4652:
4649:
4644:
4641:
4637:
4614:
4610:
4606:
4601:
4597:
4593:
4586:
4582:
4576:
4572:
4566:
4563:
4559:
4555:
4552:
4547:
4542:
4538:
4534:
4529:
4524:
4520:
4487:
4484:
4480:
4457:
4452:
4448:
4444:
4439:
4434:
4430:
4409:
4383:
4379:
4354:
4350:
4344:
4340:
4334:
4331:
4327:
4323:
4320:
4315:
4310:
4306:
4302:
4297:
4292:
4288:
4280:
4276:
4272:
4267:
4264:
4260:
4254:
4250:
4246:
4241:
4214:
4211:
4208:
4188:
4160:
4157:
4131:cost functions
4126:
4123:
4119:quantum analog
4095:
4092:
4090:
4087:
4060:
4057:
4054:
4051:
4031:
4028:
4025:
4022:
4000:
3997:
3994:
3991:
3988:
3985:
3979:
3976:
3973:
3969:
3963:
3960:
3957:
3954:
3951:
3945:
3942:
3939:
3935:
3929:
3926:
3923:
3920:
3917:
3897:
3894:
3891:
3888:
3885:
3882:
3861:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3811:
3808:
3805:
3802:
3799:
3796:
3793:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3769:
3766:
3763:
3760:
3740:
3737:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3663:
3660:
3657:
3637:
3632:
3628:
3625:
3622:
3619:
3611:
3608:
3605:
3601:
3578:
3574:
3571:
3568:
3565:
3557:
3554:
3551:
3547:
3526:
3522:
3518:
3515:
3512:
3509:
3506:
3503:
3500:
3497:
3474:
3472:
3469:
3445:
3442:
3436:
3433:
3429:
3425:
3422:
3419:
3416:
3413:
3393:
3384:provided that
3373:
3370:
3367:
3364:
3361:
3358:
3355:
3350:
3346:
3342:
3337:
3333:
3329:
3324:
3321:
3318:
3314:
3291:
3287:
3283:
3278:
3274:
3270:
3265:
3262:
3259:
3255:
3230:
3227:
3224:
3221:
3214:
3211:
3208:
3204:
3199:
3195:
3188:
3184:
3179:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3120:
3116:
3112:
3109:
3106:
3103:
3100:
3096:
3092:
3089:
3084:
3080:
3076:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
3017:
3013:
3010:
3006:
3003:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2939:
2935:
2931:
2928:
2906:
2902:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2834:
2830:
2826:
2823:
2820:
2817:
2814:
2810:
2806:
2803:
2798:
2794:
2790:
2769:
2766:
2744:
2736:
2732:
2728:
2723:
2720:
2715:
2711:
2707:
2701:
2697:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2634:
2630:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2575:
2571:
2566:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2499:
2495:
2490:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2435:
2431:
2426:
2401:
2397:
2392:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2238:
2234:
2231:
2228:
2225:
2200:
2196:
2192:
2189:
2184:
2180:
2176:
2171:
2168:
2164:
2160:
2155:
2152:
2148:
2125:
2121:
2117:
2114:
2109:
2106:
2102:
2081:
2078:
2075:
2071:
2067:
2064:
2044:
2041:
2038:
2035:
2030:
2027:
2023:
2019:
2014:
2011:
2007:
2003:
1998:
1994:
1976:
1964:
1961:
1956:
1953:
1948:
1943:
1940:
1900:
1896:
1892:
1887:
1883:
1879:
1874:
1871:
1868:
1864:
1840:
1837:
1834:
1831:
1828:
1823:
1819:
1798:
1793:
1789:
1785:
1780:
1776:
1772:
1767:
1764:
1761:
1757:
1733:
1730:
1725:
1721:
1717:
1714:
1710:
1704:
1700:
1696:
1691:
1687:
1682:
1674:
1670:
1665:
1659:
1655:
1654:lim sup
1645:which implies
1628:
1625:
1623:
1621:
1618:
1617:
1614:
1610:
1606:
1601:
1597:
1593:
1588:
1584:
1580:
1575:
1571:
1567:
1560:
1556:
1551:
1547:
1544:
1540:
1536:
1531:
1527:
1523:
1518:
1514:
1510:
1505:
1501:
1497:
1490:
1486:
1481:
1477:
1474:
1472:
1466:
1462:
1457:
1453:
1452:
1449:
1445:
1441:
1436:
1432:
1428:
1423:
1419:
1415:
1410:
1406:
1402:
1395:
1391:
1386:
1382:
1379:
1377:
1371:
1367:
1362:
1358:
1357:
1332:
1329:
1324:
1320:
1299:
1279:
1253:
1249:
1241:
1237:
1232:
1228:
1201:
1197:
1193:
1190:
1185:
1181:
1177:
1172:
1167:
1163:
1138:
1134:
1111:
1106:
1102:
1094:
1090:
1086:
1081:
1077:
1053:
1033:
1030:
1025:
1021:
1017:
1010:
1006:
998:
994:
989:
966:
963:
960:
938:
934:
926:
922:
917:
913:
888:
884:
880:
875:
870:
866:
858:
854:
853:lim sup
831:
827:
823:
818:
813:
809:
801:
797:
796:lim inf
771:
766:
762:
754:
750:
746:
741:
737:
722:
708:
705:
683:
678:
674:
668:
642:
637:
633:
625:
622:
619:
615:
588:
583:
580:
577:
571:
566:
562:
558:
538:
534:Michael Fekete
525:
522:
520:
517:
504:
501:
498:
495:
492:
472:
469:
466:
463:
458:
454:
450:
445:
441:
407:
403:
399:
394:
390:
386:
381:
378:
375:
371:
341:
338:
335:
330:
325:
321:
317:
290:
285:
280:
275:
270:
265:
262:
259:
237:
234:
231:
228:
225:
222:
189:
186:
183:
180:
177:
174:
171:
168:
165:
162:
159:
156:
153:
150:
147:
144:
141:
138:
135:
132:
129:
126:
123:
120:
107:that are both
79:
76:
73:
70:
67:
52:
49:
15:
9:
6:
4:
3:
2:
5865:
5854:
5851:
5849:
5846:
5844:
5841:
5840:
5838:
5831:
5830:
5828:
5824:
5810:
5807:
5803:
5800:
5797:
5796:0-387-05672-6
5793:
5789:
5788:
5783:
5779:
5776:
5775:
5763:
5759:
5755:
5751:
5747:
5743:
5739:
5735:
5731:
5724:
5716:
5712:
5708:
5704:
5697:
5688:
5683:
5678:
5673:
5669:
5665:
5661:
5654:
5648:, p.314,12.25
5645:
5639:
5635:
5631:
5625:
5616:
5606:
5601:
5596:
5591:
5587:
5583:
5582:
5577:
5570:
5562:
5558:
5554:
5550:
5546:
5542:
5538:
5534:
5527:
5519:
5515:
5510:
5505:
5501:
5497:
5496:
5491:
5487:
5483:
5477:
5468:
5464:
5459:
5454:
5449:
5444:
5440:
5436:
5435:
5430:
5426:
5422:
5416:
5408:
5407:
5399:
5392:
5391:0-89871-380-3
5388:
5382:
5375:
5372:
5369:(The same as
5366:
5362:
5358:
5354:
5347:
5339:
5335:
5331:
5327:
5323:
5319:
5312:
5308:
5298:
5295:
5292:
5289:
5287:
5284:
5281:
5278:
5277:
5271:
5255:
5251:
5230:
5227:
5222:
5218:
5214:
5209:
5206:
5184:
5181:
5178:
5158:
5153:
5149:
5145:
5125:
5122:
5117:
5113:
5092:
5084:
5080:
5064:
5061:
5058:
5055:
5052:
5049:
5046:
5043:
5040:
5020:
5000:
4997:
4994:
4985:
4968:
4962:
4939:
4933:
4930:
4927:
4904:
4898:
4892:
4886:
4883:
4877:
4874:
4871:
4865:
4845:
4822:
4816:
4796:
4788:
4772:
4764:
4748:
4741:
4731:
4729:
4725:
4715:
4701:
4698:
4693:
4690:
4686:
4662:
4659:
4656:
4653:
4647:
4642:
4639:
4635:
4612:
4608:
4604:
4599:
4595:
4591:
4584:
4580:
4574:
4570:
4564:
4561:
4557:
4553:
4550:
4545:
4540:
4536:
4532:
4527:
4522:
4518:
4507:
4503:
4485:
4482:
4478:
4455:
4450:
4446:
4442:
4437:
4432:
4428:
4407:
4399:
4381:
4377:
4352:
4348:
4342:
4338:
4332:
4329:
4325:
4321:
4318:
4313:
4308:
4304:
4300:
4295:
4290:
4286:
4278:
4274:
4270:
4265:
4258:
4252:
4248:
4244:
4239:
4212:
4209:
4206:
4186:
4178:
4174:
4170:
4166:
4156:
4152:
4150:
4146:
4142:
4140:
4136:
4132:
4122:
4120:
4116:
4112:
4108:
4104:
4100:
4086:
4083:
4081:
4080:superadditive
4076:
4074:
4055:
4049:
4026:
4020:
3995:
3992:
3989:
3983:
3977:
3974:
3971:
3967:
3961:
3955:
3949:
3943:
3940:
3937:
3933:
3927:
3921:
3915:
3895:
3892:
3886:
3880:
3846:
3843:
3837:
3834:
3827:
3822:
3809:
3806:
3800:
3794:
3791:
3785:
3782:
3779:
3773:
3770:
3764:
3758:
3735:
3729:
3726:
3720:
3717:
3714:
3708:
3705:
3699:
3693:
3685:
3681:
3675:
3661:
3655:
3635:
3630:
3623:
3617:
3609:
3606:
3603:
3576:
3569:
3563:
3549:
3524:
3507:
3504:
3498:
3495:
3487:
3479:
3468:
3464:
3462:
3457:
3443:
3440:
3434:
3431:
3427:
3420:
3414:
3411:
3391:
3368:
3365:
3362:
3356:
3353:
3348:
3344:
3340:
3335:
3331:
3327:
3322:
3319:
3316:
3312:
3289:
3285:
3281:
3276:
3272:
3268:
3263:
3260:
3257:
3253:
3242:
3228:
3225:
3222:
3219:
3212:
3209:
3206:
3202:
3197:
3193:
3186:
3182:
3177:
3167:
3144:
3138:
3132:
3129:
3123:
3118:
3114:
3110:
3107:
3101:
3090:
3087:
3082:
3078:
3048:
3042:
3039:
3036:
3030:
3024:
3021:
3015:
3011:
3008:
3004:
3001:
2975:
2969:
2966:
2963:
2957:
2951:
2948:
2942:
2937:
2933:
2929:
2926:
2904:
2900:
2870:
2864:
2861:
2858:
2852:
2846:
2843:
2837:
2832:
2828:
2824:
2821:
2815:
2804:
2801:
2796:
2792:
2767:
2764:
2742:
2734:
2730:
2726:
2721:
2718:
2713:
2709:
2705:
2699:
2695:
2692:
2689:
2683:
2677:
2674:
2671:
2665:
2659:
2656:
2648:
2632:
2628:
2618:
2598:
2595:
2592:
2589:
2586:
2580:
2573:
2569:
2564:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2516:
2513:
2510:
2507:
2504:
2497:
2493:
2488:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2440:
2433:
2429:
2424:
2399:
2395:
2390:
2380:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2340:
2337:
2334:
2331:
2328:
2319:
2316:
2310:
2307:
2304:
2298:
2295:
2289:
2286:
2280:
2274:
2271:
2268:
2262:
2259:
2251:
2232:
2229:
2226:
2223:
2214:
2198:
2194:
2190:
2187:
2182:
2178:
2174:
2169:
2166:
2162:
2158:
2153:
2150:
2146:
2123:
2119:
2115:
2112:
2107:
2104:
2100:
2079:
2076:
2073:
2069:
2065:
2062:
2042:
2039:
2036:
2033:
2028:
2025:
2021:
2017:
2012:
2009:
2005:
2001:
1996:
1992:
1982:
1975:
1962:
1959:
1954:
1951:
1946:
1941:
1938:
1928:
1924:
1920:
1916:
1898:
1894:
1890:
1885:
1881:
1877:
1872:
1869:
1866:
1862:
1852:
1838:
1835:
1832:
1829:
1826:
1821:
1817:
1796:
1791:
1787:
1783:
1778:
1774:
1770:
1765:
1762:
1759:
1755:
1744:
1731:
1728:
1723:
1719:
1715:
1712:
1708:
1702:
1698:
1694:
1689:
1685:
1680:
1672:
1668:
1663:
1657:
1643:
1626:
1624:
1619:
1612:
1608:
1599:
1595:
1591:
1586:
1582:
1573:
1569:
1565:
1558:
1554:
1549:
1545:
1542:
1538:
1529:
1525:
1521:
1516:
1512:
1503:
1499:
1495:
1488:
1484:
1479:
1475:
1473:
1464:
1460:
1455:
1447:
1443:
1434:
1430:
1426:
1421:
1417:
1408:
1404:
1400:
1393:
1389:
1384:
1380:
1378:
1369:
1365:
1360:
1347:
1344:
1330:
1327:
1322:
1318:
1297:
1277:
1269:
1268:residue class
1251:
1239:
1235:
1230:
1218:
1213:
1199:
1195:
1191:
1188:
1183:
1179:
1175:
1170:
1165:
1161:
1136:
1132:
1109:
1104:
1100:
1092:
1084:
1079:
1075:
1065:
1051:
1031:
1028:
1023:
1019:
1015:
1008:
1004:
996:
992:
987:
964:
961:
958:
936:
924:
920:
915:
902:
886:
882:
878:
873:
868:
864:
856:
829:
825:
821:
816:
811:
807:
799:
785:
769:
764:
760:
752:
744:
739:
735:
720:
703:
681:
676:
672:
659:
640:
635:
631:
617:
604:
581:
578:
575:
569:
564:
560:
556:
537:
535:
531:
516:
499:
496:
493:
470:
467:
464:
461:
456:
452:
448:
443:
439:
429:
427:
423:
405:
401:
397:
392:
388:
384:
379:
376:
373:
369:
361:
357:
339:
336:
333:
328:
323:
319:
315:
306:
301:
288:
283:
278:
273:
268:
263:
260:
257:
235:
232:
229:
226:
223:
212:
209:
205:
200:
187:
181:
175:
172:
166:
160:
157:
151:
148:
145:
139:
136:
133:
130:
127:
124:
121:
110:
106:
103:
100:
96:
93:
77:
71:
68:
65:
58:
48:
46:
45:Additive maps
42:
38:
34:
30:
26:
25:subadditivity
22:
5820:
5819:
5785:
5778:György Pólya
5737:
5733:
5723:
5706:
5702:
5696:
5687:10419/257929
5667:
5663:
5653:
5633:
5624:
5615:
5585:
5579:
5569:
5536:
5532:
5526:
5502:(1): 1–141.
5499:
5493:
5476:
5438:
5432:
5415:
5405:
5398:
5381:
5373:
5370:
5356:
5352:
5346:
5321:
5317:
5311:
5082:
4986:
4738:A factorial
4737:
4721:
4162:
4153:
4149:average cost
4143:
4128:
4097:
4084:
4077:
4072:
3823:
3683:
3679:
3677:
3477:
3475:
3465:
3458:
3244:
3168:
2649:
2619:
2381:
2252:
2215:
1983:
1980:
1926:
1922:
1918:
1914:
1853:
1746:
1644:
1348:
1345:
1214:
1066:
977:, such that
903:
786:
726:
539:
527:
430:
425:
421:
355:
302:
211:real numbers
208:non-negative
201:
104:
94:
54:
41:square roots
24:
18:
5802:Einar Hille
5782:Gábor Szegő
5740:(3): 1–38.
5610:Theorem 1.1
5471:Theorem 6.1
5441:(1): 1–24.
5359:: 152–163.
4785:, then all
4151:functions.
4115:von Neumann
356:subadditive
204:square root
90:, having a
51:Definitions
21:mathematics
5837:Categories
5823:PlanetMath
5772:References
4987:For every
4838:of length-
3537:the limit
3486:measurable
3484:For every
2480:, then to
2092:, we have
1929:such that
1151:such that
519:Properties
360:inequality
354:is called
5754:0004-5411
5670:(3): 91.
5595:1209.6179
5588:: 59–81.
5561:117100302
5553:1385-0172
5518:0021-7670
5467:0021-2172
5443:CiteSeerX
5338:186223729
5252:γ
5228:≤
5219:γ
5150:γ
5146:∼
5123:≥
5114:γ
4998:≥
4931:
4884:≤
4687:ρ
4654:−
4648:∈
4636:ρ
4609:σ
4596:σ
4592:≤
4581:σ
4571:σ
4558:ρ
4537:σ
4519:σ
4479:ρ
4447:σ
4429:σ
4349:σ
4339:σ
4326:ρ
4305:σ
4287:σ
4263:Δ
4259:σ
4245:≡
4210:−
4177:normality
4125:Economics
3928:≥
3893:≥
3856:→
3850:∞
3792:−
3771:≥
3727:−
3706:≥
3659:∞
3656:−
3556:∞
3553:→
3517:→
3511:∞
3471:Functions
3432:−
3415:ϕ
3412:∫
3392:ϕ
3357:ϕ
3328:≤
3269:≤
3148:∞
3133:
3111:
3102:∩
3043:
3025:
3005:
2970:
2952:
2930:
2865:
2847:
2825:
2816:∩
2756:then let
2696:
2678:
2660:
2602:∞
2590:
2538:
2526:
2511:
2462:
2450:
2362:
2350:
2335:
2311:
2290:
2281:∈
2263:
2233:∈
2213:, etc.
2188:≤
2159:≤
2113:≤
1960:≤
1947:≤
1878:≤
1833:
1771:≥
1732:ϵ
1724:∗
1695:≤
1627:⋯
1620:⋯
1592:−
1546:≤
1522:−
1476:≤
1427:−
1381:≤
1331:ϵ
1323:∗
1192:ϵ
1184:∗
1080:∗
1032:ϵ
1024:∗
959:ϵ
951:, and an
887:∗
879:≤
830:∗
822:≥
740:∗
707:∞
704:−
624:∞
621:→
587:∞
524:Sequences
385:≤
337:≥
269:≤
248:we have:
233:≥
221:∀
158:≤
131:∈
119:∀
75:→
69::
5632:(1997).
5488:(1987).
5427:(2000).
5274:See also
4740:language
4506:variance
3751:. Hence
3478:Theorem:
3012:′
2768:′
2055:. Since
1044:for all
420:for all
305:sequence
102:codomain
57:function
29:elements
5762:7232681
4787:factors
4500:is the
4159:Finance
4099:Entropy
4094:Entropy
1270:modulo
658:infimum
532:due to
99:ordered
97:and an
31:of the
5794:
5760:
5752:
5640:
5559:
5551:
5516:
5465:
5445:
5389:
5336:
4765:is in
4369:where
1067:Since
601:, the
109:closed
92:domain
33:domain
5758:S2CID
5664:Risks
5590:arXiv
5557:S2CID
5334:S2CID
5304:Notes
5083:super
5081:is a
4920:, so
3873:with
1978:Proof
724:Proof
603:limit
530:lemma
37:norms
5792:ISBN
5780:and
5750:ISSN
5638:ISBN
5549:ISSN
5514:ISSN
5463:ISSN
5387:ISBN
5215:<
4763:word
4105:and
4042:and
3607:>
3166:.
2994:and
2672:>
2617:.
1925:and
1917:and
1716:<
1176:<
1016:>
962:>
727:Let
424:and
39:and
5804:. "
5742:doi
5711:doi
5682:hdl
5672:doi
5600:doi
5586:124
5541:doi
5504:doi
5453:doi
5439:115
5361:doi
5326:doi
4928:log
4235:VaR
4173:VaR
4167:in
3678:If
3600:inf
3546:lim
3139:1.5
3031:1.5
2958:1.5
2853:1.5
2727:1.5
2706:1.5
2684:1.5
2593:1.5
2529:1.5
2514:1.5
2453:1.5
2416:to
2353:1.5
2296:1.5
1851:.)
1830:log
1215:By
1089:inf
1064:.
749:inf
719:.)
667:inf
614:lim
494:0.5
19:In
5839::
5784:.
5756:.
5748:.
5738:56
5736:.
5732:.
5705:.
5680:.
5666:.
5662:.
5598:.
5584:.
5578:.
5555:.
5547:.
5535:.
5512:.
5500:48
5498:.
5492:.
5484:;
5461:.
5451:.
5437:.
5431:.
5423:;
5374:14
5357:55
5355:.
5332:.
5322:17
5320:.
4984:.
4420:,
4121:.
4082:.
3824:A
3674:)
3130:ln
3108:ln
3040:ln
3022:ln
3002:ln
2967:ln
2949:ln
2927:ln
2862:ln
2844:ln
2822:ln
2693:ln
2675:ln
2657:ln
2587:ln
2535:ln
2523:ln
2508:ln
2459:ln
2447:ln
2359:ln
2347:ln
2332:ln
2308:ln
2287:ln
2260:ln
1963:2.
1212:.
1085::=
901:.
784:.
745::=
536:.
303:A
43:.
23:,
5829:.
5798:.
5764:.
5744::
5717:.
5713::
5707:6
5690:.
5684::
5674::
5668:7
5646:.
5608:.
5602::
5592::
5563:.
5543::
5537:2
5520:.
5506::
5469:.
5455::
5393:.
5367:.
5363::
5340:.
5328::
5256:2
5231:1
5223:k
5210:k
5207:1
5185:1
5182:=
5179:n
5159:n
5154:k
5126:0
5118:k
5093:n
5065:k
5062:,
5059:.
5056:.
5053:.
5050:,
5047:2
5044:,
5041:1
5021:n
5001:1
4995:k
4972:)
4969:n
4966:(
4963:A
4943:)
4940:n
4937:(
4934:A
4908:)
4905:n
4902:(
4899:A
4896:)
4893:m
4890:(
4887:A
4881:)
4878:n
4875:+
4872:m
4869:(
4866:A
4846:n
4826:)
4823:n
4820:(
4817:A
4797:L
4773:L
4749:L
4702:1
4699:=
4694:y
4691:x
4666:]
4663:1
4660:,
4657:1
4651:[
4643:y
4640:x
4613:y
4605:+
4600:x
4585:y
4575:x
4565:y
4562:x
4554:2
4551:+
4546:2
4541:y
4533:+
4528:2
4523:x
4486:y
4483:x
4456:2
4451:y
4443:,
4438:2
4433:x
4408:p
4382:p
4378:z
4353:y
4343:x
4333:y
4330:x
4322:2
4319:+
4314:2
4309:y
4301:+
4296:2
4291:x
4279:p
4275:z
4271:=
4266:V
4253:p
4249:z
4240:p
4213:p
4207:1
4187:V
4073:f
4059:)
4056:y
4053:(
4050:f
4030:)
4027:x
4024:(
4021:f
3999:)
3996:y
3993:+
3990:x
3987:(
3984:f
3978:y
3975:+
3972:x
3968:x
3962:+
3959:)
3956:0
3953:(
3950:f
3944:y
3941:+
3938:x
3934:y
3925:)
3922:x
3919:(
3916:f
3896:0
3890:)
3887:0
3884:(
3881:f
3860:R
3853:)
3847:,
3844:0
3841:[
3838::
3835:f
3810:0
3807:=
3804:)
3801:y
3798:(
3795:f
3789:)
3786:y
3783:+
3780:0
3777:(
3774:f
3768:)
3765:0
3762:(
3759:f
3739:)
3736:y
3733:(
3730:f
3724:)
3721:y
3718:+
3715:x
3712:(
3709:f
3703:)
3700:x
3697:(
3694:f
3684:f
3680:f
3662:.
3636:.
3631:t
3627:)
3624:t
3621:(
3618:f
3610:0
3604:t
3577:t
3573:)
3570:t
3567:(
3564:f
3550:t
3525:,
3521:R
3514:)
3508:,
3505:0
3502:(
3499::
3496:f
3444:t
3441:d
3435:2
3428:t
3424:)
3421:t
3418:(
3372:)
3369:m
3366:+
3363:n
3360:(
3354:+
3349:m
3345:a
3341:+
3336:n
3332:a
3323:m
3320:+
3317:n
3313:a
3290:m
3286:a
3282:+
3277:n
3273:a
3264:m
3261:+
3258:n
3254:a
3229:.
3226:.
3223:.
3220:,
3213:1
3210:+
3207:k
3203:n
3198:a
3194:,
3187:k
3183:n
3178:a
3154:)
3151:]
3145:,
3142:)
3136:(
3127:[
3124:+
3119:k
3115:n
3105:(
3099:)
3095:Z
3091:m
3088:+
3083:k
3079:n
3075:(
3055:]
3052:)
3049:3
3046:(
3037:,
3034:)
3028:(
3019:[
3016:+
3009:n
2982:]
2979:)
2976:3
2973:(
2964:,
2961:)
2955:(
2946:[
2943:+
2938:k
2934:n
2905:k
2901:n
2880:)
2877:]
2874:)
2871:3
2868:(
2859:,
2856:)
2850:(
2841:[
2838:+
2833:k
2829:n
2819:(
2813:)
2809:Z
2805:m
2802:+
2797:k
2793:n
2789:(
2765:n
2743:)
2735:k
2731:n
2722:m
2719:+
2714:k
2710:n
2700:(
2690:+
2687:)
2681:(
2669:)
2666:2
2663:(
2633:k
2629:n
2605:)
2599:+
2596:,
2584:[
2581:+
2574:k
2570:n
2565:a
2544:]
2541:3
2532:,
2520:[
2517:+
2505:+
2498:k
2494:n
2489:a
2468:]
2465:3
2456:,
2444:[
2441:+
2434:k
2430:n
2425:a
2400:k
2396:n
2391:a
2368:]
2365:3
2356:,
2344:[
2341:+
2338:s
2329:=
2326:]
2323:)
2320:s
2317:3
2314:(
2305:,
2302:)
2299:s
2293:(
2284:[
2278:)
2275:t
2272:+
2269:s
2266:(
2237:N
2230:t
2227:,
2224:s
2199:m
2195:a
2191:3
2183:m
2179:a
2175:+
2170:m
2167:2
2163:a
2154:m
2151:3
2147:a
2124:m
2120:a
2116:2
2108:m
2105:2
2101:a
2080:2
2077:=
2074:m
2070:/
2066:m
2063:2
2043:.
2040:.
2037:.
2034:,
2029:m
2026:3
2022:a
2018:,
2013:m
2010:2
2006:a
2002:,
1997:m
1993:a
1955:n
1952:m
1942:2
1939:1
1927:n
1923:m
1919:n
1915:m
1899:m
1895:a
1891:+
1886:n
1882:a
1873:m
1870:+
1867:n
1863:a
1839:!
1836:n
1827:=
1822:n
1818:a
1797:.
1792:m
1788:a
1784:+
1779:n
1775:a
1766:m
1763:+
1760:n
1756:a
1729:+
1720:s
1713:m
1709:/
1703:m
1699:a
1690:k
1686:n
1681:/
1673:k
1669:n
1664:a
1658:k
1613:m
1609:/
1605:)
1600:1
1596:n
1587:3
1583:n
1579:(
1574:m
1570:a
1566:+
1559:1
1555:n
1550:a
1543:m
1539:/
1535:)
1530:2
1526:n
1517:3
1513:n
1509:(
1504:m
1500:a
1496:+
1489:2
1485:n
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1465:3
1461:n
1456:a
1448:m
1444:/
1440:)
1435:1
1431:n
1422:2
1418:n
1414:(
1409:m
1405:a
1401:+
1394:1
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1328:+
1319:s
1298:m
1278:m
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1240:k
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1231:a
1227:(
1200:2
1196:/
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1180:s
1171:m
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1133:a
1110:n
1105:n
1101:a
1093:n
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1052:k
1029:+
1020:s
1009:k
1005:n
997:k
993:n
988:a
965:0
937:k
933:)
925:k
921:n
916:a
912:(
883:s
874:n
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857:n
826:s
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808:a
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761:a
753:n
736:s
682:n
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673:a
641:n
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632:a
618:n
582:1
579:=
576:n
570:}
565:n
561:a
557:{
503:]
500:1
497:,
491:[
471:.
468:.
465:.
462:,
457:2
453:a
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444:1
440:a
426:n
422:m
406:m
402:a
398:+
393:n
389:a
380:m
377:+
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340:1
334:n
329:}
324:n
320:a
316:{
289:.
284:y
279:+
274:x
264:y
261:+
258:x
236:0
230:y
227:,
224:x
188:.
185:)
182:y
179:(
176:f
173:+
170:)
167:x
164:(
161:f
155:)
152:y
149:+
146:x
143:(
140:f
137:,
134:A
128:y
125:,
122:x
105:B
95:A
78:B
72:A
66:f
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