4166:
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.
4128:. 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
1652:
4165:
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
4378:
4182:. 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
1362:
4636:
4022:
4239:
2765:
1753:
665:
3477:
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.
4522:
1053:
310:
1647:{\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}}}
910:
853:
1221:
3175:
2901:
610:
209:
3600:
1984:
46:
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
5252:
1367:
3393:
2222:
1133:
793:
3657:
4479:
3546:
363:
3882:
4152:. 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.
2626:
3250:
1818:
3313:
2064:
1922:
705:
429:
3831:
3760:
3465:
4929:
2389:
2259:
2565:
5180:
5147:
2147:
257:
1352:
1275:
960:
492:
4687:
986:
99:
1860:
4723:
3076:
5279:
4509:
3917:
3003:
2489:
5086:
4964:
3683:
2663:
2425:
728:
5022:
2790:
2101:
4405:
3413:
2928:
2656:
1160:
4993:
4847:
4080:
4051:
5206:
4234:
5114:
5042:
4867:
4818:
4794:
4770:
4429:
4208:
3922:
1319:
1299:
1073:
524:
5822:
N.H. Bingham, A.J. Ostaszewski. "Generic subadditive functions." Proceedings of
American Mathematical Society, vol. 136, no. 12 (2008), pp. 4257–4266.
3470:
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
1659:
262:
4373:{\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}}}}
125:
618:
991:
4190:
of risk factors. The
Gaussian VaR ensures subadditivity: for example, the Gaussian VaR of a two unitary long positions portfolio
4129:
858:
801:
5652:
1165:
375:
5327:
Fekete, M. (1923). "Über die
Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten".
3081:
2795:
560:
439:. This is a special case of subadditive function, if a sequence is interpreted as a function on the set of natural numbers.
3551:
17:
5591:
5505:
1943:
5211:
2266:
3318:
2152:
1081:
741:
4631:{\displaystyle {\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}+2\rho _{xy}\sigma _{x}\sigma _{y}}}\leq \sigma _{x}+\sigma _{y}}
4145:
3605:
5806:
5797:
5401:
4434:
3502:
320:
4408:
3841:
2570:
4512:
5853:
3183:
1761:
5858:
5444:
3259:
1998:
1868:
673:
5671:"Bigger Is Not Always Safer: A Critical Analysis of the Subadditivity Assumption for Coherent Risk Measures"
3765:
3700:
5863:
4872:
3418:
2631:
Though we don't have continuous variables, we can still cover enough integers to complete the proof. Let
2230:
1227:
4236:
is, assuming that the mean portfolio value variation is zero and the VaR is defined as a negative loss,
2494:
5542:
Gromov, Misha (1999). "Topological
Invariants of Dynamical Systems and Spaces of Holomorphic Maps: I".
5152:
5119:
5089:
2106:
442:
Note that while a concave sequence is subadditive, the converse is false. For example, randomly assign
227:
4773:
1324:
1233:
918:
445:
4641:
965:
72:
5458:
5396:
Michael J. Steele. "Probability theory and combinatorial optimization". SIAM, Philadelphia (1997).
1823:
370:
4692:
38:
is a property of a function that states, roughly, that evaluating the function for the sum of two
5290:
5257:
4484:
3887:
2933:
2430:
5492:
5453:
5310: – Property of geometry, also used to generalize the notion of "distance" in metric spaces
5047:
4934:
3662:
2396:
67:
1321:. This sequence, continued for long enough, would be forced by subadditivity to dip below the
710:
5587:"An analogue of Fekete's lemma for subadditive functions on cancellative amenable semigroups"
5001:
4175:
3008:
119:
5362:
de Bruijn, N.G.; Erdös, P. (1952). "Some linear and some quadratic recursion formulas. II".
5293: – Difference in properties of one mole of substance in a mixture vs. an ideal solution
2069:
4383:
3398:
2906:
2634:
1992:
Continue the proof as before, until we have just used the infinite pigeonhole principle.
1138:
668:
102:
43:
4969:
4823:
4056:
4027:
4017:{\displaystyle f(x)\geq \textstyle {\frac {y}{x+y}}f(0)+\textstyle {\frac {x}{x+y}}f(x+y)}
8:
5307:
5185:
4213:
4187:
4125:
4117:
3496:
613:
540:
5816:
2770:
5837:
5768:
5600:
5567:
5431:
5344:
5099:
5027:
4852:
4803:
4779:
4755:
4414:
4193:
4155:
4113:
2393:
If we were dealing with continuous variables, then we can use subadditivity to go from
1304:
1284:
1278:
1058:
47:
5630:
Hille 1948, Theorem 6.6.1. (Measurability is stipulated in Sect. 6.2 "Preliminaries".)
5375:
497:
5802:
5788:
5760:
5648:
5571:
5559:
5524:
5473:
5397:
5348:
4121:
5792:
5772:
5752:
5721:
5692:
5682:
5610:
5551:
5514:
5463:
5371:
5336:
5296:
4149:
3836:
1758:
The analogue of Fekete's lemma holds for superadditive sequences as well, that is:
2760:{\displaystyle \ln(2)>\ln(1.5)+\ln \left({\frac {1.5n_{k}+m}{1.5n_{k}}}\right)}
5740:
5725:
5301:
4750:
4179:
4090:
5640:
5496:
5435:
4966:
is subadditive, and hence Fekete's lemma can be used to estimate the growth of
4738:
4734:
3078:
touch in the middle. Thus, by repeating this process, we cover the entirety of
544:
5615:
5586:
5555:
5847:
5764:
5563:
5528:
5477:
4183:
3471:
109:
39:
5756:
5585:
Ceccherini-Silberstein, Tullio; Krieger, Fabrice; Coornaert, Michel (2014).
4689:
and, in particular, it equals the sum of the individual risk exposures when
4820:. In combinatorics on words, a common problem is to determine the number
4159:
218:
55:
51:
5415:
1865:
There are extensions of Fekete's lemma that do not require the inequality
1748:{\displaystyle \limsup _{k}a_{n_{k}}/n_{k}\leq a_{m}/m<s^{*}+\epsilon }
5812:
5687:
5670:
221:
214:
31:
5697:
5833:
5519:
5500:
5468:
5439:
5340:
4141:
5741:"Improved bounds on the average length of longest common subsequences"
5584:
4797:
539:
A useful result pertaining to subadditive sequences is the following
660:{\displaystyle \displaystyle \lim _{n\to \infty }{\frac {a_{n}}{n}}}
4725:
which is the case of no diversification effects on portfolio risk.
4516:
315:
112:
5712:
Shur, Arseny (2012). "Growth properties of power-free languages".
5605:
5417:
CBMS Lectures on
Probability Theory and Combinatorial Optimization
5501:"Entropy and isomorphism theorems for actions of amenable groups"
4109:
1820:(The limit then may be positive infinity: consider the sequence
1048:{\displaystyle {\frac {a_{n_{k}}}{n_{k}}}>s^{*}+\epsilon }
4733:
Subadditivity occurs in the thermodynamic properties of non-
3693:
is a subadditive function, and if 0 is in its domain, then
4140:
Subadditivity is an essential property of some particular
3919:
is also subadditive. To see this, one first observes that
305:{\displaystyle {\sqrt {x+y}}\leq {\sqrt {x}}+{\sqrt {y}}.}
5832:
This article incorporates material from subadditivity on
5281:, however, is only known to be between 0.788 and 0.827.
905:{\displaystyle \limsup _{n}{\frac {a_{n}}{n}}\leq s^{*}}
848:{\displaystyle \liminf _{n}{\frac {a_{n}}{n}}\geq s^{*}}
2930:, it's easy to see (draw a picture) that the intervals
1216:{\displaystyle {\frac {a_{m}}{m}}<s^{*}+\epsilon /2}
4638:
Thus the
Gaussian VaR is subadditive for any value of
3975:
3941:
3697:(0) ≥ 0. To see this, take the inequality at the top.
3421:
3170:{\displaystyle (n_{k}+m\mathbb {Z} )\cap (\ln n_{k}+)}
2896:{\displaystyle (n_{k}+m\mathbb {Z} )\cap (\ln n_{k}+)}
1946:
1365:
605:{\displaystyle {\left\{a_{n}\right\}}_{n=1}^{\infty }}
204:{\displaystyle \forall x,y\in A,f(x+y)\leq f(x)+f(y).}
5260:
5214:
5188:
5155:
5122:
5102:
5050:
5030:
5004:
4972:
4937:
4875:
4855:
4826:
4806:
4782:
4758:
4695:
4644:
4525:
4487:
4437:
4417:
4386:
4242:
4216:
4196:
4059:
4030:
3925:
3890:
3844:
3768:
3703:
3665:
3608:
3595:{\displaystyle \lim _{t\to \infty }{\frac {f(t)}{t}}}
3554:
3505:
3401:
3321:
3262:
3186:
3084:
3011:
2936:
2909:
2798:
2773:
2666:
2637:
2573:
2497:
2433:
2399:
2269:
2233:
2155:
2109:
2072:
2001:
1979:{\textstyle {\frac {1}{2}}\leq {\frac {m}{n}}\leq 2.}
1871:
1826:
1764:
1662:
1327:
1307:
1287:
1236:
1168:
1141:
1084:
1061:
994:
968:
921:
861:
804:
744:
713:
676:
622:
621:
563:
500:
448:
378:
323:
265:
230:
128:
75:
4515:
between the two individual positions returns. Since
4174:
Subadditivity is one of the desirable properties of
526:; then the sequence is subadditive but not concave.
5247:{\displaystyle {\frac {1}{k}}<\gamma _{k}\leq 1}
4481:are the individual positions returns variances and
5819:". American Mathematical Society, New York (1948).
5430:
5273:
5246:
5200:
5174:
5141:
5108:
5080:
5036:
5016:
4987:
4958:
4923:
4861:
4841:
4812:
4788:
4764:
4717:
4681:
4630:
4503:
4473:
4423:
4399:
4372:
4228:
4202:
4074:
4045:
4016:
3911:
3876:
3825:
3754:
3677:
3651:
3594:
3540:
3459:
3407:
3388:{\displaystyle a_{n+m}\leq a_{n}+a_{m}+\phi (n+m)}
3387:
3307:
3244:
3169:
3070:
2997:
2922:
2895:
2784:
2759:
2650:
2620:
2559:
2483:
2419:
2383:
2253:
2217:{\displaystyle a_{3m}\leq a_{2m}+a_{m}\leq 3a_{m}}
2216:
2141:
2095:
2058:
1978:
1916:
1854:
1812:
1747:
1646:
1346:
1313:
1293:
1269:
1215:
1154:
1128:{\displaystyle s^{*}:=\inf _{n}{\frac {a_{n}}{n}}}
1127:
1067:
1047:
980:
954:
904:
847:
788:{\displaystyle s^{*}:=\inf _{n}{\frac {a_{n}}{n}}}
787:
722:
699:
659:
604:
518:
486:
423:
357:
304:
251:
203:
93:
5413:
3415:is an increasing function such that the integral
5845:
5838:Creative Commons Attribution/Share-Alike License
3652:{\displaystyle \inf _{t>0}{\frac {f(t)}{t}}.}
3610:
3556:
1664:
1099:
863:
806:
759:
677:
624:
4474:{\displaystyle \sigma _{x}^{2},\sigma _{y}^{2}}
5361:
4737:and mixtures like the excess molar volume and
4186:models which do not rely on the assumption of
4099:
3541:{\displaystyle f:(0,\infty )\to \mathbb {R} ,}
2567:, and so on, which covers the entire interval
358:{\displaystyle \left\{a_{n}\right\}_{n\geq 1}}
5491:
3877:{\displaystyle f:[0,\infty )\to \mathbb {R} }
122:under addition, with the following property:
5801:, vol. 1. Springer-Verlag, New York (1976).
4024:. Then looking at the sum of this bound for
2621:{\displaystyle a_{n_{k}}+[\ln 1.5,+\infty )}
58:are special cases of subadditive functions.
5544:Mathematical Physics, Analysis and Geometry
2792:be the smallest number in the intersection
5668:
5662:
4089:The negative of a subadditive function is
3252:are forced down as in the previous proof.
1357:In more detail, by subadditivity, we have
5696:
5686:
5639:
5614:
5604:
5518:
5467:
5457:
5149:, such that the expected length grows as
4744:
3870:
3531:
3450:
3245:{\displaystyle a_{n_{k}},a_{n_{k+1}},...}
3105:
2819:
2261:, we can use subadditivity on them if
2247:
5645:Handbook of Analysis and its Foundations
4869:words in a factorial language. Clearly
1813:{\displaystyle a_{n+m}\geq a_{n}+a_{m}.}
915:If not, then there exists a subsequence
5387:.) See also Steele 1997, Theorem 1.9.2.
3308:{\displaystyle a_{n+m}\leq a_{n}+a_{m}}
2059:{\displaystyle a_{m},a_{2m},a_{3m},...}
1917:{\displaystyle a_{n+m}\leq a_{n}+a_{m}}
1277:, whose indices all belong to the same
700:{\displaystyle \inf {\frac {a_{n}}{n}}}
424:{\displaystyle a_{n+m}\leq a_{n}+a_{m}}
27:Property of some mathematical functions
14:
5846:
5738:
5541:
5326:
3826:{\displaystyle f(0)\geq f(0+y)-f(y)=0}
1301:, and so they advance by multiples of
5711:
5705:
5364:Nederl. Akad. Wetensch. Proc. Ser. A
5044:uniformly at random on the alphabet
4124:in a generalized formulation due to
3755:{\displaystyle f(x)\geq f(x+y)-f(y)}
3460:{\textstyle \int \phi (t)t^{-2}\,dt}
5817:Functional analysis and semi-groups
4924:{\displaystyle A(m+n)\leq A(m)A(n)}
2384:{\displaystyle \ln(s+t)\in =\ln s+}
2254:{\displaystyle s,t\in \mathbb {N} }
24:
4273:
4146:necessary and sufficient condition
3860:
3669:
3566:
3521:
3158:
2612:
2560:{\displaystyle a_{n_{k}}+\ln 1.5+}
717:
634:
597:
231:
129:
25:
5875:
5826:
5798:Problems and Theorems in Analysis
5175:{\displaystyle \sim \gamma _{k}n}
5142:{\displaystyle \gamma _{k}\geq 0}
5116:, and thus there exists a number
4728:
2142:{\displaystyle a_{2m}\leq 2a_{m}}
1230:, there exists a sub-subsequence
252:{\displaystyle \forall x,y\geq 0}
4409:cumulative distribution function
5732:
5633:
5024:, sample two strings of length
4158:are represented by subadditive
3467:converges (near the infinity).
1354:slope line, a contradiction.
1347:{\displaystyle s^{*}+\epsilon }
1270:{\displaystyle (a_{n_{k}})_{k}}
1228:infinitary pigeonhole principle
955:{\displaystyle (a_{n_{k}})_{k}}
557:For every subadditive sequence
487:{\displaystyle a_{1},a_{2},...}
5836:, which is licensed under the
5739:Lueker, George S. (May 2009).
5624:
5592:Journal d'Analyse Mathématique
5578:
5535:
5506:Journal d'Analyse Mathématique
5485:
5424:
5407:
5390:
5355:
5320:
5304: – Property of a function
4982:
4976:
4953:
4947:
4918:
4912:
4906:
4900:
4891:
4879:
4836:
4830:
4682:{\displaystyle \rho _{xy}\in }
4676:
4661:
4069:
4063:
4040:
4034:
4009:
3997:
3969:
3963:
3935:
3929:
3900:
3894:
3866:
3863:
3851:
3814:
3808:
3799:
3787:
3778:
3772:
3749:
3743:
3734:
3722:
3713:
3707:
3637:
3631:
3583:
3577:
3563:
3527:
3524:
3512:
3474:and subadditivity is present.
3434:
3428:
3382:
3370:
3164:
3161:
3152:
3146:
3137:
3115:
3109:
3085:
3065:
3062:
3056:
3044:
3038:
3029:
2992:
2989:
2983:
2971:
2965:
2956:
2890:
2887:
2884:
2878:
2866:
2860:
2851:
2829:
2823:
2799:
2697:
2691:
2679:
2673:
2658:be large enough, such that
2615:
2594:
2554:
2530:
2478:
2454:
2378:
2354:
2336:
2333:
2324:
2312:
2303:
2294:
2288:
2276:
1615:
1589:
1545:
1519:
1450:
1424:
1258:
1237:
981:{\displaystyle \epsilon >0}
943:
922:
631:
513:
501:
224:as domain and codomain: since
195:
189:
180:
174:
165:
153:
94:{\displaystyle f\colon A\to B}
85:
61:
13:
1:
5782:
5647:. San Diego: Academic Press.
5445:Israel Journal of Mathematics
5376:10.1016/S1385-7258(52)50021-0
5088:. The expected length of the
4407:is the inverse of the normal
1855:{\displaystyle a_{n}=\log n!}
529:
5726:10.1016/j.cosrev.2012.09.001
5440:"Mean topological dimension"
5182:. By checking the case with
4718:{\displaystyle \rho _{xy}=1}
4135:
4112:plays a fundamental role in
3481:
3315:may be weakened as follows:
534:
66:A subadditive function is a
7:
5284:
5274:{\displaystyle \gamma _{2}}
4100:Examples in various domains
4082:, will finally verify that
2227:By the assumption, for any
10:
5880:
5420:. University of Cambridge.
5414:Michael J. Steele (2011).
5254:. The exact value of even
5090:longest common subsequence
4513:linear correlation measure
4504:{\displaystyle \rho _{xy}}
4169:
4148:for the verification of a
4104:
3912:{\displaystyle f(0)\geq 0}
2998:{\displaystyle \ln n_{k}+}
2484:{\displaystyle a_{n_{k}}+}
552:Fekete's Subadditive Lemma
5616:10.1007/s11854-014-0027-4
5329:Mathematische Zeitschrift
5081:{\displaystyle 1,2,...,k}
4959:{\displaystyle \log A(n)}
4800:of that word are also in
3678:{\displaystyle -\infty .}
2420:{\displaystyle a_{n_{k}}}
855:. So it suffices to show
5314:
4210:at the confidence level
3256:Moreover, the condition
723:{\displaystyle -\infty }
5757:10.1145/1516512.1516519
5714:Computer Science Review
5669:Rau-Bredow, H. (2019).
5556:10.1023/A:1009841100168
5291:Apparent molar property
5017:{\displaystyle k\geq 1}
3602:exists and is equal to
3071:{\displaystyle \ln n'+}
2903:. By the assumption on
5275:
5248:
5202:
5176:
5143:
5110:
5096:-additive function of
5082:
5038:
5018:
4989:
4960:
4925:
4863:
4843:
4814:
4790:
4766:
4745:Combinatorics on words
4719:
4683:
4632:
4505:
4475:
4425:
4401:
4374:
4230:
4204:
4176:coherent risk measures
4144:. It is, generally, a
4076:
4047:
4018:
3913:
3878:
3827:
3756:
3679:
3653:
3596:
3542:
3461:
3409:
3389:
3309:
3246:
3171:
3072:
2999:
2924:
2897:
2786:
2761:
2652:
2622:
2561:
2485:
2421:
2385:
2255:
2218:
2143:
2097:
2096:{\displaystyle 2m/m=2}
2060:
1995:Consider the sequence
1980:
1918:
1856:
1814:
1749:
1648:
1348:
1315:
1295:
1271:
1217:
1156:
1129:
1069:
1049:
982:
956:
906:
849:
789:
724:
701:
661:
606:
520:
488:
425:
359:
306:
253:
205:
95:
5854:Mathematical analysis
5276:
5249:
5203:
5177:
5144:
5111:
5083:
5039:
5019:
4990:
4961:
4926:
4864:
4844:
4815:
4791:
4767:
4720:
4684:
4633:
4506:
4476:
4426:
4411:at probability level
4402:
4400:{\displaystyle z_{p}}
4375:
4231:
4205:
4077:
4048:
4019:
3914:
3879:
3828:
3757:
3680:
3654:
3597:
3543:
3499:subadditive function
3462:
3410:
3408:{\displaystyle \phi }
3390:
3310:
3247:
3172:
3073:
3000:
2925:
2923:{\displaystyle n_{k}}
2898:
2787:
2762:
2653:
2651:{\displaystyle n_{k}}
2623:
2562:
2486:
2422:
2386:
2256:
2219:
2149:. Similarly, we have
2144:
2098:
2061:
1981:
1919:
1857:
1815:
1750:
1649:
1349:
1316:
1296:
1272:
1218:
1157:
1155:{\displaystyle a_{m}}
1130:
1070:
1050:
983:
957:
907:
850:
790:
725:
707:. (The limit may be
702:
662:
607:
521:
489:
426:
360:
307:
254:
217:function, having the
206:
96:
5859:Sequences and series
5688:10.3390/risks7030091
5258:
5212:
5186:
5153:
5120:
5100:
5048:
5028:
5002:
4988:{\displaystyle A(n)}
4970:
4935:
4873:
4853:
4842:{\displaystyle A(n)}
4824:
4804:
4780:
4756:
4741:or excess enthalpy.
4693:
4642:
4523:
4519:is always positive,
4485:
4435:
4415:
4384:
4240:
4214:
4194:
4075:{\displaystyle f(y)}
4057:
4046:{\displaystyle f(x)}
4028:
3923:
3888:
3842:
3766:
3701:
3663:
3606:
3552:
3503:
3419:
3399:
3319:
3260:
3184:
3082:
3009:
2934:
2907:
2796:
2771:
2664:
2635:
2571:
2495:
2431:
2397:
2267:
2231:
2153:
2107:
2070:
1999:
1944:
1869:
1824:
1762:
1660:
1363:
1325:
1305:
1285:
1234:
1166:
1139:
1082:
1059:
992:
966:
919:
859:
802:
742:
711:
674:
619:
561:
498:
446:
376:
369:if it satisfies the
321:
263:
228:
126:
73:
18:Subadditive function
5493:Ornstein, Donald S.
5432:Lindenstrauss, Elon
5308:Triangle inequality
5201:{\displaystyle n=1}
4560:
4542:
4470:
4452:
4328:
4310:
4229:{\displaystyle 1-p}
4118:statistical physics
3493: —
601:
555: —
5864:Types of functions
5745:Journal of the ACM
5520:10.1007/BF02790325
5469:10.1007/BF02810577
5382:Indagationes Math.
5341:10.1007/BF01504345
5271:
5244:
5198:
5172:
5139:
5106:
5078:
5034:
5014:
4985:
4956:
4921:
4859:
4839:
4810:
4786:
4772:is one where if a
4762:
4715:
4679:
4628:
4546:
4528:
4501:
4471:
4456:
4438:
4421:
4397:
4370:
4314:
4296:
4226:
4200:
4156:Economies of scale
4114:information theory
4072:
4043:
4014:
4013:
4012:
3909:
3874:
3823:
3752:
3675:
3659:(The limit may be
3649:
3624:
3592:
3570:
3538:
3491:
3457:
3405:
3385:
3305:
3242:
3167:
3068:
2995:
2920:
2893:
2785:{\displaystyle n'}
2782:
2757:
2648:
2618:
2557:
2481:
2417:
2381:
2251:
2214:
2139:
2093:
2056:
1990:
1976:
1914:
1852:
1810:
1745:
1672:
1644:
1642:
1344:
1311:
1291:
1267:
1213:
1152:
1135:, there exists an
1125:
1107:
1065:
1045:
978:
952:
902:
871:
845:
814:
785:
767:
736:
720:
697:
657:
656:
638:
602:
564:
553:
516:
484:
421:
355:
302:
249:
213:An example is the
201:
91:
5654:978-0-12-622760-4
5223:
5208:, we easily have
5109:{\displaystyle n}
5037:{\displaystyle n}
4862:{\displaystyle n}
4813:{\displaystyle L}
4789:{\displaystyle L}
4765:{\displaystyle L}
4600:
4424:{\displaystyle p}
4368:
4247:
4203:{\displaystyle V}
4122:quantum mechanics
3992:
3958:
3644:
3609:
3590:
3555:
3487:
2751:
1988:
1968:
1955:
1663:
1314:{\displaystyle m}
1294:{\displaystyle m}
1184:
1123:
1098:
1068:{\displaystyle k}
1024:
887:
862:
830:
805:
783:
758:
734:
695:
654:
623:
551:
297:
287:
277:
16:(Redirected from
5871:
5777:
5776:
5736:
5730:
5729:
5720:(5–6): 187–208.
5709:
5703:
5702:
5700:
5690:
5666:
5660:
5658:
5637:
5631:
5628:
5622:
5620:
5618:
5608:
5582:
5576:
5575:
5539:
5533:
5532:
5522:
5489:
5483:
5481:
5471:
5461:
5428:
5422:
5421:
5411:
5405:
5394:
5388:
5379:
5359:
5353:
5352:
5324:
5297:Choquet integral
5280:
5278:
5277:
5272:
5270:
5269:
5253:
5251:
5250:
5245:
5237:
5236:
5224:
5216:
5207:
5205:
5204:
5199:
5181:
5179:
5178:
5173:
5168:
5167:
5148:
5146:
5145:
5140:
5132:
5131:
5115:
5113:
5112:
5107:
5087:
5085:
5084:
5079:
5043:
5041:
5040:
5035:
5023:
5021:
5020:
5015:
4994:
4992:
4991:
4986:
4965:
4963:
4962:
4957:
4930:
4928:
4927:
4922:
4868:
4866:
4865:
4860:
4848:
4846:
4845:
4840:
4819:
4817:
4816:
4811:
4795:
4793:
4792:
4787:
4771:
4769:
4768:
4763:
4724:
4722:
4721:
4716:
4708:
4707:
4688:
4686:
4685:
4680:
4657:
4656:
4637:
4635:
4634:
4629:
4627:
4626:
4614:
4613:
4601:
4599:
4598:
4589:
4588:
4579:
4578:
4559:
4554:
4541:
4536:
4527:
4510:
4508:
4507:
4502:
4500:
4499:
4480:
4478:
4477:
4472:
4469:
4464:
4451:
4446:
4430:
4428:
4427:
4422:
4406:
4404:
4403:
4398:
4396:
4395:
4379:
4377:
4376:
4371:
4369:
4367:
4366:
4357:
4356:
4347:
4346:
4327:
4322:
4309:
4304:
4295:
4293:
4292:
4280:
4279:
4267:
4266:
4254:
4253:
4248:
4245:
4235:
4233:
4232:
4227:
4209:
4207:
4206:
4201:
4150:natural monopoly
4120:, as well as in
4086:is subadditive.
4081:
4079:
4078:
4073:
4052:
4050:
4049:
4044:
4023:
4021:
4020:
4015:
3993:
3991:
3977:
3959:
3957:
3943:
3918:
3916:
3915:
3910:
3883:
3881:
3880:
3875:
3873:
3837:concave function
3832:
3830:
3829:
3824:
3761:
3759:
3758:
3753:
3684:
3682:
3681:
3676:
3658:
3656:
3655:
3650:
3645:
3640:
3626:
3623:
3601:
3599:
3598:
3593:
3591:
3586:
3572:
3569:
3547:
3545:
3544:
3539:
3534:
3494:
3466:
3464:
3463:
3458:
3449:
3448:
3414:
3412:
3411:
3406:
3394:
3392:
3391:
3386:
3363:
3362:
3350:
3349:
3337:
3336:
3314:
3312:
3311:
3306:
3304:
3303:
3291:
3290:
3278:
3277:
3251:
3249:
3248:
3243:
3229:
3228:
3227:
3226:
3203:
3202:
3201:
3200:
3176:
3174:
3173:
3168:
3133:
3132:
3108:
3097:
3096:
3077:
3075:
3074:
3069:
3025:
3004:
3002:
3001:
2996:
2952:
2951:
2929:
2927:
2926:
2921:
2919:
2918:
2902:
2900:
2899:
2894:
2847:
2846:
2822:
2811:
2810:
2791:
2789:
2788:
2783:
2781:
2766:
2764:
2763:
2758:
2756:
2752:
2750:
2749:
2748:
2735:
2728:
2727:
2714:
2657:
2655:
2654:
2649:
2647:
2646:
2627:
2625:
2624:
2619:
2590:
2589:
2588:
2587:
2566:
2564:
2563:
2558:
2514:
2513:
2512:
2511:
2490:
2488:
2487:
2482:
2450:
2449:
2448:
2447:
2426:
2424:
2423:
2418:
2416:
2415:
2414:
2413:
2390:
2388:
2387:
2382:
2260:
2258:
2257:
2252:
2250:
2223:
2221:
2220:
2215:
2213:
2212:
2197:
2196:
2184:
2183:
2168:
2167:
2148:
2146:
2145:
2140:
2138:
2137:
2122:
2121:
2102:
2100:
2099:
2094:
2083:
2065:
2063:
2062:
2057:
2043:
2042:
2027:
2026:
2011:
2010:
1985:
1983:
1982:
1977:
1969:
1961:
1956:
1948:
1924:to hold for all
1923:
1921:
1920:
1915:
1913:
1912:
1900:
1899:
1887:
1886:
1861:
1859:
1858:
1853:
1836:
1835:
1819:
1817:
1816:
1811:
1806:
1805:
1793:
1792:
1780:
1779:
1754:
1752:
1751:
1746:
1738:
1737:
1722:
1717:
1716:
1704:
1703:
1694:
1689:
1688:
1687:
1686:
1671:
1653:
1651:
1650:
1645:
1643:
1622:
1614:
1613:
1601:
1600:
1588:
1587:
1575:
1574:
1573:
1572:
1552:
1544:
1543:
1531:
1530:
1518:
1517:
1505:
1504:
1503:
1502:
1481:
1480:
1479:
1478:
1457:
1449:
1448:
1436:
1435:
1423:
1422:
1410:
1409:
1408:
1407:
1386:
1385:
1384:
1383:
1353:
1351:
1350:
1345:
1337:
1336:
1320:
1318:
1317:
1312:
1300:
1298:
1297:
1292:
1276:
1274:
1273:
1268:
1266:
1265:
1256:
1255:
1254:
1253:
1222:
1220:
1219:
1214:
1209:
1198:
1197:
1185:
1180:
1179:
1170:
1161:
1159:
1158:
1153:
1151:
1150:
1134:
1132:
1131:
1126:
1124:
1119:
1118:
1109:
1106:
1094:
1093:
1074:
1072:
1071:
1066:
1054:
1052:
1051:
1046:
1038:
1037:
1025:
1023:
1022:
1013:
1012:
1011:
1010:
996:
987:
985:
984:
979:
961:
959:
958:
953:
951:
950:
941:
940:
939:
938:
911:
909:
908:
903:
901:
900:
888:
883:
882:
873:
870:
854:
852:
851:
846:
844:
843:
831:
826:
825:
816:
813:
794:
792:
791:
786:
784:
779:
778:
769:
766:
754:
753:
729:
727:
726:
721:
706:
704:
703:
698:
696:
691:
690:
681:
667:is equal to the
666:
664:
663:
658:
655:
650:
649:
640:
637:
611:
609:
608:
603:
600:
595:
584:
583:
579:
578:
556:
525:
523:
522:
519:{\displaystyle }
517:
493:
491:
490:
485:
471:
470:
458:
457:
430:
428:
427:
422:
420:
419:
407:
406:
394:
393:
364:
362:
361:
356:
354:
353:
342:
338:
337:
311:
309:
308:
303:
298:
293:
288:
283:
278:
267:
258:
256:
255:
250:
210:
208:
207:
202:
100:
98:
97:
92:
21:
5879:
5878:
5874:
5873:
5872:
5870:
5869:
5868:
5844:
5843:
5829:
5785:
5780:
5737:
5733:
5710:
5706:
5667:
5663:
5655:
5641:Schechter, Eric
5638:
5634:
5629:
5625:
5583:
5579:
5540:
5536:
5497:Weiss, Benjamin
5490:
5486:
5436:Weiss, Benjamin
5429:
5425:
5412:
5408:
5395:
5391:
5360:
5356:
5325:
5321:
5317:
5302:Superadditivity
5287:
5265:
5261:
5259:
5256:
5255:
5232:
5228:
5215:
5213:
5210:
5209:
5187:
5184:
5183:
5163:
5159:
5154:
5151:
5150:
5127:
5123:
5121:
5118:
5117:
5101:
5098:
5097:
5049:
5046:
5045:
5029:
5026:
5025:
5003:
5000:
4999:
4971:
4968:
4967:
4936:
4933:
4932:
4874:
4871:
4870:
4854:
4851:
4850:
4825:
4822:
4821:
4805:
4802:
4801:
4781:
4778:
4777:
4757:
4754:
4753:
4747:
4735:ideal solutions
4731:
4700:
4696:
4694:
4691:
4690:
4649:
4645:
4643:
4640:
4639:
4622:
4618:
4609:
4605:
4594:
4590:
4584:
4580:
4571:
4567:
4555:
4550:
4537:
4532:
4526:
4524:
4521:
4520:
4492:
4488:
4486:
4483:
4482:
4465:
4460:
4447:
4442:
4436:
4433:
4432:
4416:
4413:
4412:
4391:
4387:
4385:
4382:
4381:
4362:
4358:
4352:
4348:
4339:
4335:
4323:
4318:
4305:
4300:
4294:
4288:
4284:
4272:
4268:
4262:
4258:
4249:
4244:
4243:
4241:
4238:
4237:
4215:
4212:
4211:
4195:
4192:
4191:
4180:risk management
4172:
4138:
4107:
4102:
4096:
4058:
4055:
4054:
4029:
4026:
4025:
3981:
3976:
3947:
3942:
3924:
3921:
3920:
3889:
3886:
3885:
3869:
3843:
3840:
3839:
3767:
3764:
3763:
3702:
3699:
3698:
3687:
3664:
3661:
3660:
3627:
3625:
3613:
3607:
3604:
3603:
3573:
3571:
3559:
3553:
3550:
3549:
3530:
3504:
3501:
3500:
3492:
3484:
3472:superadditivity
3441:
3437:
3420:
3417:
3416:
3400:
3397:
3396:
3358:
3354:
3345:
3341:
3326:
3322:
3320:
3317:
3316:
3299:
3295:
3286:
3282:
3267:
3263:
3261:
3258:
3257:
3254:
3216:
3212:
3211:
3207:
3196:
3192:
3191:
3187:
3185:
3182:
3181:
3180:With that, all
3128:
3124:
3104:
3092:
3088:
3083:
3080:
3079:
3018:
3010:
3007:
3006:
2947:
2943:
2935:
2932:
2931:
2914:
2910:
2908:
2905:
2904:
2842:
2838:
2818:
2806:
2802:
2797:
2794:
2793:
2774:
2772:
2769:
2768:
2744:
2740:
2736:
2723:
2719:
2715:
2713:
2709:
2665:
2662:
2661:
2642:
2638:
2636:
2633:
2632:
2583:
2579:
2578:
2574:
2572:
2569:
2568:
2507:
2503:
2502:
2498:
2496:
2493:
2492:
2443:
2439:
2438:
2434:
2432:
2429:
2428:
2409:
2405:
2404:
2400:
2398:
2395:
2394:
2268:
2265:
2264:
2246:
2232:
2229:
2228:
2208:
2204:
2192:
2188:
2176:
2172:
2160:
2156:
2154:
2151:
2150:
2133:
2129:
2114:
2110:
2108:
2105:
2104:
2079:
2071:
2068:
2067:
2035:
2031:
2019:
2015:
2006:
2002:
2000:
1997:
1996:
1960:
1947:
1945:
1942:
1941:
1932:, but only for
1908:
1904:
1895:
1891:
1876:
1872:
1870:
1867:
1866:
1831:
1827:
1825:
1822:
1821:
1801:
1797:
1788:
1784:
1769:
1765:
1763:
1760:
1759:
1756:
1733:
1729:
1718:
1712:
1708:
1699:
1695:
1690:
1682:
1678:
1677:
1673:
1667:
1661:
1658:
1657:
1641:
1640:
1633:
1627:
1626:
1618:
1609:
1605:
1596:
1592:
1583:
1579:
1568:
1564:
1563:
1559:
1548:
1539:
1535:
1526:
1522:
1513:
1509:
1498:
1494:
1493:
1489:
1482:
1474:
1470:
1469:
1465:
1462:
1461:
1453:
1444:
1440:
1431:
1427:
1418:
1414:
1403:
1399:
1398:
1394:
1387:
1379:
1375:
1374:
1370:
1366:
1364:
1361:
1360:
1332:
1328:
1326:
1323:
1322:
1306:
1303:
1302:
1286:
1283:
1282:
1261:
1257:
1249:
1245:
1244:
1240:
1235:
1232:
1231:
1205:
1193:
1189:
1175:
1171:
1169:
1167:
1164:
1163:
1146:
1142:
1140:
1137:
1136:
1114:
1110:
1108:
1102:
1089:
1085:
1083:
1080:
1079:
1060:
1057:
1056:
1033:
1029:
1018:
1014:
1006:
1002:
1001:
997:
995:
993:
990:
989:
967:
964:
963:
946:
942:
934:
930:
929:
925:
920:
917:
916:
896:
892:
878:
874:
872:
866:
860:
857:
856:
839:
835:
821:
817:
815:
809:
803:
800:
799:
798:By definition,
774:
770:
768:
762:
749:
745:
743:
740:
739:
732:
712:
709:
708:
686:
682:
680:
675:
672:
671:
645:
641:
639:
627:
620:
617:
616:
596:
585:
574:
570:
566:
565:
562:
559:
558:
554:
537:
532:
499:
496:
495:
494:with values in
466:
462:
453:
449:
447:
444:
443:
415:
411:
402:
398:
383:
379:
377:
374:
373:
343:
333:
329:
325:
324:
322:
319:
318:
292:
282:
266:
264:
261:
260:
229:
226:
225:
127:
124:
123:
74:
71:
70:
64:
28:
23:
22:
15:
12:
11:
5:
5877:
5867:
5866:
5861:
5856:
5828:
5827:External links
5825:
5824:
5823:
5820:
5810:
5784:
5781:
5779:
5778:
5731:
5704:
5661:
5653:
5632:
5623:
5577:
5550:(4): 323–415.
5534:
5484:
5459:10.1.1.30.3552
5423:
5406:
5389:
5354:
5335:(1): 228–249.
5318:
5316:
5313:
5312:
5311:
5305:
5299:
5294:
5286:
5283:
5268:
5264:
5243:
5240:
5235:
5231:
5227:
5222:
5219:
5197:
5194:
5191:
5171:
5166:
5162:
5158:
5138:
5135:
5130:
5126:
5105:
5077:
5074:
5071:
5068:
5065:
5062:
5059:
5056:
5053:
5033:
5013:
5010:
5007:
4984:
4981:
4978:
4975:
4955:
4952:
4949:
4946:
4943:
4940:
4920:
4917:
4914:
4911:
4908:
4905:
4902:
4899:
4896:
4893:
4890:
4887:
4884:
4881:
4878:
4858:
4838:
4835:
4832:
4829:
4809:
4785:
4761:
4746:
4743:
4739:heat of mixing
4730:
4729:Thermodynamics
4727:
4714:
4711:
4706:
4703:
4699:
4678:
4675:
4672:
4669:
4666:
4663:
4660:
4655:
4652:
4648:
4625:
4621:
4617:
4612:
4608:
4604:
4597:
4593:
4587:
4583:
4577:
4574:
4570:
4566:
4563:
4558:
4553:
4549:
4545:
4540:
4535:
4531:
4498:
4495:
4491:
4468:
4463:
4459:
4455:
4450:
4445:
4441:
4420:
4394:
4390:
4365:
4361:
4355:
4351:
4345:
4342:
4338:
4334:
4331:
4326:
4321:
4317:
4313:
4308:
4303:
4299:
4291:
4287:
4283:
4278:
4275:
4271:
4265:
4261:
4257:
4252:
4225:
4222:
4219:
4199:
4171:
4168:
4142:cost functions
4137:
4134:
4130:quantum analog
4106:
4103:
4101:
4098:
4071:
4068:
4065:
4062:
4042:
4039:
4036:
4033:
4011:
4008:
4005:
4002:
3999:
3996:
3990:
3987:
3984:
3980:
3974:
3971:
3968:
3965:
3962:
3956:
3953:
3950:
3946:
3940:
3937:
3934:
3931:
3928:
3908:
3905:
3902:
3899:
3896:
3893:
3872:
3868:
3865:
3862:
3859:
3856:
3853:
3850:
3847:
3822:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3751:
3748:
3745:
3742:
3739:
3736:
3733:
3730:
3727:
3724:
3721:
3718:
3715:
3712:
3709:
3706:
3674:
3671:
3668:
3648:
3643:
3639:
3636:
3633:
3630:
3622:
3619:
3616:
3612:
3589:
3585:
3582:
3579:
3576:
3568:
3565:
3562:
3558:
3537:
3533:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3485:
3483:
3480:
3456:
3453:
3447:
3444:
3440:
3436:
3433:
3430:
3427:
3424:
3404:
3395:provided that
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3361:
3357:
3353:
3348:
3344:
3340:
3335:
3332:
3329:
3325:
3302:
3298:
3294:
3289:
3285:
3281:
3276:
3273:
3270:
3266:
3241:
3238:
3235:
3232:
3225:
3222:
3219:
3215:
3210:
3206:
3199:
3195:
3190:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3131:
3127:
3123:
3120:
3117:
3114:
3111:
3107:
3103:
3100:
3095:
3091:
3087:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3024:
3021:
3017:
3014:
2994:
2991:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
2958:
2955:
2950:
2946:
2942:
2939:
2917:
2913:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2845:
2841:
2837:
2834:
2831:
2828:
2825:
2821:
2817:
2814:
2809:
2805:
2801:
2780:
2777:
2755:
2747:
2743:
2739:
2734:
2731:
2726:
2722:
2718:
2712:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2645:
2641:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2596:
2593:
2586:
2582:
2577:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2510:
2506:
2501:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2453:
2446:
2442:
2437:
2412:
2408:
2403:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2249:
2245:
2242:
2239:
2236:
2211:
2207:
2203:
2200:
2195:
2191:
2187:
2182:
2179:
2175:
2171:
2166:
2163:
2159:
2136:
2132:
2128:
2125:
2120:
2117:
2113:
2092:
2089:
2086:
2082:
2078:
2075:
2055:
2052:
2049:
2046:
2041:
2038:
2034:
2030:
2025:
2022:
2018:
2014:
2009:
2005:
1987:
1975:
1972:
1967:
1964:
1959:
1954:
1951:
1911:
1907:
1903:
1898:
1894:
1890:
1885:
1882:
1879:
1875:
1851:
1848:
1845:
1842:
1839:
1834:
1830:
1809:
1804:
1800:
1796:
1791:
1787:
1783:
1778:
1775:
1772:
1768:
1744:
1741:
1736:
1732:
1728:
1725:
1721:
1715:
1711:
1707:
1702:
1698:
1693:
1685:
1681:
1676:
1670:
1666:
1665:lim sup
1656:which implies
1639:
1636:
1634:
1632:
1629:
1628:
1625:
1621:
1617:
1612:
1608:
1604:
1599:
1595:
1591:
1586:
1582:
1578:
1571:
1567:
1562:
1558:
1555:
1551:
1547:
1542:
1538:
1534:
1529:
1525:
1521:
1516:
1512:
1508:
1501:
1497:
1492:
1488:
1485:
1483:
1477:
1473:
1468:
1464:
1463:
1460:
1456:
1452:
1447:
1443:
1439:
1434:
1430:
1426:
1421:
1417:
1413:
1406:
1402:
1397:
1393:
1390:
1388:
1382:
1378:
1373:
1369:
1368:
1343:
1340:
1335:
1331:
1310:
1290:
1264:
1260:
1252:
1248:
1243:
1239:
1212:
1208:
1204:
1201:
1196:
1192:
1188:
1183:
1178:
1174:
1149:
1145:
1122:
1117:
1113:
1105:
1101:
1097:
1092:
1088:
1064:
1044:
1041:
1036:
1032:
1028:
1021:
1017:
1009:
1005:
1000:
977:
974:
971:
949:
945:
937:
933:
928:
924:
899:
895:
891:
886:
881:
877:
869:
865:
864:lim sup
842:
838:
834:
829:
824:
820:
812:
808:
807:lim inf
782:
777:
773:
765:
761:
757:
752:
748:
733:
719:
716:
694:
689:
685:
679:
653:
648:
644:
636:
633:
630:
626:
599:
594:
591:
588:
582:
577:
573:
569:
549:
545:Michael Fekete
536:
533:
531:
528:
515:
512:
509:
506:
503:
483:
480:
477:
474:
469:
465:
461:
456:
452:
418:
414:
410:
405:
401:
397:
392:
389:
386:
382:
352:
349:
346:
341:
336:
332:
328:
301:
296:
291:
286:
281:
276:
273:
270:
248:
245:
242:
239:
236:
233:
200:
197:
194:
191:
188:
185:
182:
179:
176:
173:
170:
167:
164:
161:
158:
155:
152:
149:
146:
143:
140:
137:
134:
131:
118:that are both
90:
87:
84:
81:
78:
63:
60:
26:
9:
6:
4:
3:
2:
5876:
5865:
5862:
5860:
5857:
5855:
5852:
5851:
5849:
5842:
5841:
5839:
5835:
5821:
5818:
5814:
5811:
5808:
5807:0-387-05672-6
5804:
5800:
5799:
5794:
5790:
5787:
5786:
5774:
5770:
5766:
5762:
5758:
5754:
5750:
5746:
5742:
5735:
5727:
5723:
5719:
5715:
5708:
5699:
5694:
5689:
5684:
5680:
5676:
5672:
5665:
5659:, p.314,12.25
5656:
5650:
5646:
5642:
5636:
5627:
5617:
5612:
5607:
5602:
5598:
5594:
5593:
5588:
5581:
5573:
5569:
5565:
5561:
5557:
5553:
5549:
5545:
5538:
5530:
5526:
5521:
5516:
5512:
5508:
5507:
5502:
5498:
5494:
5488:
5479:
5475:
5470:
5465:
5460:
5455:
5451:
5447:
5446:
5441:
5437:
5433:
5427:
5419:
5418:
5410:
5403:
5402:0-89871-380-3
5399:
5393:
5386:
5383:
5380:(The same as
5377:
5373:
5369:
5365:
5358:
5350:
5346:
5342:
5338:
5334:
5330:
5323:
5319:
5309:
5306:
5303:
5300:
5298:
5295:
5292:
5289:
5288:
5282:
5266:
5262:
5241:
5238:
5233:
5229:
5225:
5220:
5217:
5195:
5192:
5189:
5169:
5164:
5160:
5156:
5136:
5133:
5128:
5124:
5103:
5095:
5091:
5075:
5072:
5069:
5066:
5063:
5060:
5057:
5054:
5051:
5031:
5011:
5008:
5005:
4996:
4979:
4973:
4950:
4944:
4941:
4938:
4915:
4909:
4903:
4897:
4894:
4888:
4885:
4882:
4876:
4856:
4833:
4827:
4807:
4799:
4783:
4775:
4759:
4752:
4742:
4740:
4736:
4726:
4712:
4709:
4704:
4701:
4697:
4673:
4670:
4667:
4664:
4658:
4653:
4650:
4646:
4623:
4619:
4615:
4610:
4606:
4602:
4595:
4591:
4585:
4581:
4575:
4572:
4568:
4564:
4561:
4556:
4551:
4547:
4543:
4538:
4533:
4529:
4518:
4514:
4496:
4493:
4489:
4466:
4461:
4457:
4453:
4448:
4443:
4439:
4418:
4410:
4392:
4388:
4363:
4359:
4353:
4349:
4343:
4340:
4336:
4332:
4329:
4324:
4319:
4315:
4311:
4306:
4301:
4297:
4289:
4285:
4281:
4276:
4269:
4263:
4259:
4255:
4250:
4223:
4220:
4217:
4197:
4189:
4185:
4181:
4177:
4167:
4163:
4161:
4157:
4153:
4151:
4147:
4143:
4133:
4131:
4127:
4123:
4119:
4115:
4111:
4097:
4094:
4092:
4091:superadditive
4087:
4085:
4066:
4060:
4037:
4031:
4006:
4003:
4000:
3994:
3988:
3985:
3982:
3978:
3972:
3966:
3960:
3954:
3951:
3948:
3944:
3938:
3932:
3926:
3906:
3903:
3897:
3891:
3857:
3854:
3848:
3845:
3838:
3833:
3820:
3817:
3811:
3805:
3802:
3796:
3793:
3790:
3784:
3781:
3775:
3769:
3746:
3740:
3737:
3731:
3728:
3725:
3719:
3716:
3710:
3704:
3696:
3692:
3686:
3672:
3666:
3646:
3641:
3634:
3628:
3620:
3617:
3614:
3587:
3580:
3574:
3560:
3535:
3518:
3515:
3509:
3506:
3498:
3490:
3479:
3475:
3473:
3468:
3454:
3451:
3445:
3442:
3438:
3431:
3425:
3422:
3402:
3379:
3376:
3373:
3367:
3364:
3359:
3355:
3351:
3346:
3342:
3338:
3333:
3330:
3327:
3323:
3300:
3296:
3292:
3287:
3283:
3279:
3274:
3271:
3268:
3264:
3253:
3239:
3236:
3233:
3230:
3223:
3220:
3217:
3213:
3208:
3204:
3197:
3193:
3188:
3178:
3155:
3149:
3143:
3140:
3134:
3129:
3125:
3121:
3118:
3112:
3101:
3098:
3093:
3089:
3059:
3053:
3050:
3047:
3041:
3035:
3032:
3026:
3022:
3019:
3015:
3012:
2986:
2980:
2977:
2974:
2968:
2962:
2959:
2953:
2948:
2944:
2940:
2937:
2915:
2911:
2881:
2875:
2872:
2869:
2863:
2857:
2854:
2848:
2843:
2839:
2835:
2832:
2826:
2815:
2812:
2807:
2803:
2778:
2775:
2753:
2745:
2741:
2737:
2732:
2729:
2724:
2720:
2716:
2710:
2706:
2703:
2700:
2694:
2688:
2685:
2682:
2676:
2670:
2667:
2659:
2643:
2639:
2629:
2609:
2606:
2603:
2600:
2597:
2591:
2584:
2580:
2575:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2527:
2524:
2521:
2518:
2515:
2508:
2504:
2499:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2451:
2444:
2440:
2435:
2410:
2406:
2401:
2391:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2351:
2348:
2345:
2342:
2339:
2330:
2327:
2321:
2318:
2315:
2309:
2306:
2300:
2297:
2291:
2285:
2282:
2279:
2273:
2270:
2262:
2243:
2240:
2237:
2234:
2225:
2209:
2205:
2201:
2198:
2193:
2189:
2185:
2180:
2177:
2173:
2169:
2164:
2161:
2157:
2134:
2130:
2126:
2123:
2118:
2115:
2111:
2090:
2087:
2084:
2080:
2076:
2073:
2053:
2050:
2047:
2044:
2039:
2036:
2032:
2028:
2023:
2020:
2016:
2012:
2007:
2003:
1993:
1986:
1973:
1970:
1965:
1962:
1957:
1952:
1949:
1939:
1935:
1931:
1927:
1909:
1905:
1901:
1896:
1892:
1888:
1883:
1880:
1877:
1873:
1863:
1849:
1846:
1843:
1840:
1837:
1832:
1828:
1807:
1802:
1798:
1794:
1789:
1785:
1781:
1776:
1773:
1770:
1766:
1755:
1742:
1739:
1734:
1730:
1726:
1723:
1719:
1713:
1709:
1705:
1700:
1696:
1691:
1683:
1679:
1674:
1668:
1654:
1637:
1635:
1630:
1623:
1619:
1610:
1606:
1602:
1597:
1593:
1584:
1580:
1576:
1569:
1565:
1560:
1556:
1553:
1549:
1540:
1536:
1532:
1527:
1523:
1514:
1510:
1506:
1499:
1495:
1490:
1486:
1484:
1475:
1471:
1466:
1458:
1454:
1445:
1441:
1437:
1432:
1428:
1419:
1415:
1411:
1404:
1400:
1395:
1391:
1389:
1380:
1376:
1371:
1358:
1355:
1341:
1338:
1333:
1329:
1308:
1288:
1280:
1279:residue class
1262:
1250:
1246:
1241:
1229:
1224:
1210:
1206:
1202:
1199:
1194:
1190:
1186:
1181:
1176:
1172:
1147:
1143:
1120:
1115:
1111:
1103:
1095:
1090:
1086:
1076:
1062:
1042:
1039:
1034:
1030:
1026:
1019:
1015:
1007:
1003:
998:
975:
972:
969:
947:
935:
931:
926:
913:
897:
893:
889:
884:
879:
875:
867:
840:
836:
832:
827:
822:
818:
810:
796:
780:
775:
771:
763:
755:
750:
746:
731:
714:
692:
687:
683:
670:
651:
646:
642:
628:
615:
592:
589:
586:
580:
575:
571:
567:
548:
546:
542:
527:
510:
507:
504:
481:
478:
475:
472:
467:
463:
459:
454:
450:
440:
438:
434:
416:
412:
408:
403:
399:
395:
390:
387:
384:
380:
372:
368:
350:
347:
344:
339:
334:
330:
326:
317:
312:
299:
294:
289:
284:
279:
274:
271:
268:
246:
243:
240:
237:
234:
223:
220:
216:
211:
198:
192:
186:
183:
177:
171:
168:
162:
159:
156:
150:
147:
144:
141:
138:
135:
132:
121:
117:
114:
111:
107:
104:
88:
82:
79:
76:
69:
59:
57:
56:Additive maps
53:
49:
45:
41:
37:
36:subadditivity
33:
19:
5831:
5830:
5796:
5789:György Pólya
5748:
5744:
5734:
5717:
5713:
5707:
5698:10419/257929
5678:
5674:
5664:
5644:
5635:
5626:
5596:
5590:
5580:
5547:
5543:
5537:
5513:(1): 1–141.
5510:
5504:
5487:
5449:
5443:
5426:
5416:
5409:
5392:
5384:
5381:
5367:
5363:
5357:
5332:
5328:
5322:
5093:
4997:
4749:A factorial
4748:
4732:
4173:
4164:
4160:average cost
4154:
4139:
4108:
4095:
4088:
4083:
3834:
3694:
3690:
3688:
3488:
3486:
3476:
3469:
3255:
3179:
2660:
2630:
2392:
2263:
2226:
1994:
1991:
1937:
1933:
1929:
1925:
1864:
1757:
1655:
1359:
1356:
1225:
1077:
988:, such that
914:
797:
737:
550:
538:
441:
436:
432:
366:
313:
222:real numbers
219:non-negative
212:
115:
105:
65:
52:square roots
35:
29:
5813:Einar Hille
5793:Gábor Szegő
5751:(3): 1–38.
5621:Theorem 1.1
5482:Theorem 6.1
5452:(1): 1–24.
5370:: 152–163.
4796:, then all
4162:functions.
4126:von Neumann
367:subadditive
215:square root
101:, having a
62:Definitions
32:mathematics
5848:Categories
5834:PlanetMath
5783:References
4998:For every
4849:of length-
3548:the limit
3497:measurable
3495:For every
2491:, then to
2103:, we have
1940:such that
1162:such that
530:Properties
371:inequality
365:is called
5765:0004-5411
5681:(3): 91.
5606:1209.6179
5599:: 59–81.
5572:117100302
5564:1385-0172
5529:0021-7670
5478:0021-2172
5454:CiteSeerX
5349:186223729
5263:γ
5239:≤
5230:γ
5161:γ
5157:∼
5134:≥
5125:γ
5009:≥
4942:
4895:≤
4698:ρ
4665:−
4659:∈
4647:ρ
4620:σ
4607:σ
4603:≤
4592:σ
4582:σ
4569:ρ
4548:σ
4530:σ
4490:ρ
4458:σ
4440:σ
4360:σ
4350:σ
4337:ρ
4316:σ
4298:σ
4274:Δ
4270:σ
4256:≡
4221:−
4188:normality
4136:Economics
3939:≥
3904:≥
3867:→
3861:∞
3803:−
3782:≥
3738:−
3717:≥
3670:∞
3667:−
3567:∞
3564:→
3528:→
3522:∞
3482:Functions
3443:−
3426:ϕ
3423:∫
3403:ϕ
3368:ϕ
3339:≤
3280:≤
3159:∞
3144:
3122:
3113:∩
3054:
3036:
3016:
2981:
2963:
2941:
2876:
2858:
2836:
2827:∩
2767:then let
2707:
2689:
2671:
2613:∞
2601:
2549:
2537:
2522:
2473:
2461:
2373:
2361:
2346:
2322:
2301:
2292:∈
2274:
2244:∈
2224:, etc.
2199:≤
2170:≤
2124:≤
1971:≤
1958:≤
1889:≤
1844:
1782:≥
1743:ϵ
1735:∗
1706:≤
1638:⋯
1631:⋯
1603:−
1557:≤
1533:−
1487:≤
1438:−
1392:≤
1342:ϵ
1334:∗
1203:ϵ
1195:∗
1091:∗
1043:ϵ
1035:∗
970:ϵ
962:, and an
898:∗
890:≤
841:∗
833:≥
751:∗
718:∞
715:−
635:∞
632:→
598:∞
535:Sequences
396:≤
348:≥
280:≤
259:we have:
244:≥
232:∀
169:≤
142:∈
130:∀
86:→
80::
5643:(1997).
5499:(1987).
5438:(2000).
5285:See also
4751:language
4517:variance
3762:. Hence
3489:Theorem:
3023:′
2779:′
2066:. Since
1055:for all
431:for all
316:sequence
113:codomain
68:function
40:elements
5773:7232681
4798:factors
4511:is the
4170:Finance
4110:Entropy
4105:Entropy
1281:modulo
669:infimum
543:due to
110:ordered
108:and an
42:of the
5805:
5771:
5763:
5651:
5570:
5562:
5527:
5476:
5456:
5400:
5347:
4776:is in
4380:where
1078:Since
612:, the
120:closed
103:domain
44:domain
5769:S2CID
5675:Risks
5601:arXiv
5568:S2CID
5345:S2CID
5315:Notes
5094:super
5092:is a
4931:, so
3884:with
1989:Proof
735:Proof
614:limit
541:lemma
48:norms
5803:ISBN
5791:and
5761:ISSN
5649:ISBN
5560:ISSN
5525:ISSN
5474:ISSN
5398:ISBN
5226:<
4774:word
4116:and
4053:and
3618:>
3177:.
3005:and
2683:>
2628:.
1936:and
1928:and
1727:<
1187:<
1027:>
973:>
738:Let
435:and
50:and
5815:. "
5753:doi
5722:doi
5693:hdl
5683:doi
5611:doi
5597:124
5552:doi
5515:doi
5464:doi
5450:115
5372:doi
5337:doi
4939:log
4246:VaR
4184:VaR
4178:in
3689:If
3611:inf
3557:lim
3150:1.5
3042:1.5
2969:1.5
2864:1.5
2738:1.5
2717:1.5
2695:1.5
2604:1.5
2540:1.5
2525:1.5
2464:1.5
2427:to
2364:1.5
2307:1.5
1862:.)
1841:log
1226:By
1100:inf
1075:.
760:inf
730:.)
678:inf
625:lim
505:0.5
30:In
5850::
5795:.
5767:.
5759:.
5749:56
5747:.
5743:.
5716:.
5691:.
5677:.
5673:.
5609:.
5595:.
5589:.
5566:.
5558:.
5546:.
5523:.
5511:48
5509:.
5503:.
5495:;
5472:.
5462:.
5448:.
5442:.
5434:;
5385:14
5368:55
5366:.
5343:.
5333:17
5331:.
4995:.
4431:,
4132:.
4093:.
3835:A
3685:)
3141:ln
3119:ln
3051:ln
3033:ln
3013:ln
2978:ln
2960:ln
2938:ln
2873:ln
2855:ln
2833:ln
2704:ln
2686:ln
2668:ln
2598:ln
2546:ln
2534:ln
2519:ln
2470:ln
2458:ln
2370:ln
2358:ln
2343:ln
2319:ln
2298:ln
2271:ln
1974:2.
1223:.
1096::=
912:.
795:.
756::=
547:.
314:A
54:.
34:,
5840:.
5809:.
5775:.
5755::
5728:.
5724::
5718:6
5701:.
5695::
5685::
5679:7
5657:.
5619:.
5613::
5603::
5574:.
5554::
5548:2
5531:.
5517::
5480:.
5466::
5404:.
5378:.
5374::
5351:.
5339::
5267:2
5242:1
5234:k
5221:k
5218:1
5196:1
5193:=
5190:n
5170:n
5165:k
5137:0
5129:k
5104:n
5076:k
5073:,
5070:.
5067:.
5064:.
5061:,
5058:2
5055:,
5052:1
5032:n
5012:1
5006:k
4983:)
4980:n
4977:(
4974:A
4954:)
4951:n
4948:(
4945:A
4919:)
4916:n
4913:(
4910:A
4907:)
4904:m
4901:(
4898:A
4892:)
4889:n
4886:+
4883:m
4880:(
4877:A
4857:n
4837:)
4834:n
4831:(
4828:A
4808:L
4784:L
4760:L
4713:1
4710:=
4705:y
4702:x
4677:]
4674:1
4671:,
4668:1
4662:[
4654:y
4651:x
4624:y
4616:+
4611:x
4596:y
4586:x
4576:y
4573:x
4565:2
4562:+
4557:2
4552:y
4544:+
4539:2
4534:x
4497:y
4494:x
4467:2
4462:y
4454:,
4449:2
4444:x
4419:p
4393:p
4389:z
4364:y
4354:x
4344:y
4341:x
4333:2
4330:+
4325:2
4320:y
4312:+
4307:2
4302:x
4290:p
4286:z
4282:=
4277:V
4264:p
4260:z
4251:p
4224:p
4218:1
4198:V
4084:f
4070:)
4067:y
4064:(
4061:f
4041:)
4038:x
4035:(
4032:f
4010:)
4007:y
4004:+
4001:x
3998:(
3995:f
3989:y
3986:+
3983:x
3979:x
3973:+
3970:)
3967:0
3964:(
3961:f
3955:y
3952:+
3949:x
3945:y
3936:)
3933:x
3930:(
3927:f
3907:0
3901:)
3898:0
3895:(
3892:f
3871:R
3864:)
3858:,
3855:0
3852:[
3849::
3846:f
3821:0
3818:=
3815:)
3812:y
3809:(
3806:f
3800:)
3797:y
3794:+
3791:0
3788:(
3785:f
3779:)
3776:0
3773:(
3770:f
3750:)
3747:y
3744:(
3741:f
3735:)
3732:y
3729:+
3726:x
3723:(
3720:f
3714:)
3711:x
3708:(
3705:f
3695:f
3691:f
3673:.
3647:.
3642:t
3638:)
3635:t
3632:(
3629:f
3621:0
3615:t
3588:t
3584:)
3581:t
3578:(
3575:f
3561:t
3536:,
3532:R
3525:)
3519:,
3516:0
3513:(
3510::
3507:f
3455:t
3452:d
3446:2
3439:t
3435:)
3432:t
3429:(
3383:)
3380:m
3377:+
3374:n
3371:(
3365:+
3360:m
3356:a
3352:+
3347:n
3343:a
3334:m
3331:+
3328:n
3324:a
3301:m
3297:a
3293:+
3288:n
3284:a
3275:m
3272:+
3269:n
3265:a
3240:.
3237:.
3234:.
3231:,
3224:1
3221:+
3218:k
3214:n
3209:a
3205:,
3198:k
3194:n
3189:a
3165:)
3162:]
3156:,
3153:)
3147:(
3138:[
3135:+
3130:k
3126:n
3116:(
3110:)
3106:Z
3102:m
3099:+
3094:k
3090:n
3086:(
3066:]
3063:)
3060:3
3057:(
3048:,
3045:)
3039:(
3030:[
3027:+
3020:n
2993:]
2990:)
2987:3
2984:(
2975:,
2972:)
2966:(
2957:[
2954:+
2949:k
2945:n
2916:k
2912:n
2891:)
2888:]
2885:)
2882:3
2879:(
2870:,
2867:)
2861:(
2852:[
2849:+
2844:k
2840:n
2830:(
2824:)
2820:Z
2816:m
2813:+
2808:k
2804:n
2800:(
2776:n
2754:)
2746:k
2742:n
2733:m
2730:+
2725:k
2721:n
2711:(
2701:+
2698:)
2692:(
2680:)
2677:2
2674:(
2644:k
2640:n
2616:)
2610:+
2607:,
2595:[
2592:+
2585:k
2581:n
2576:a
2555:]
2552:3
2543:,
2531:[
2528:+
2516:+
2509:k
2505:n
2500:a
2479:]
2476:3
2467:,
2455:[
2452:+
2445:k
2441:n
2436:a
2411:k
2407:n
2402:a
2379:]
2376:3
2367:,
2355:[
2352:+
2349:s
2340:=
2337:]
2334:)
2331:s
2328:3
2325:(
2316:,
2313:)
2310:s
2304:(
2295:[
2289:)
2286:t
2283:+
2280:s
2277:(
2248:N
2241:t
2238:,
2235:s
2210:m
2206:a
2202:3
2194:m
2190:a
2186:+
2181:m
2178:2
2174:a
2165:m
2162:3
2158:a
2135:m
2131:a
2127:2
2119:m
2116:2
2112:a
2091:2
2088:=
2085:m
2081:/
2077:m
2074:2
2054:.
2051:.
2048:.
2045:,
2040:m
2037:3
2033:a
2029:,
2024:m
2021:2
2017:a
2013:,
2008:m
2004:a
1966:n
1963:m
1953:2
1950:1
1938:n
1934:m
1930:n
1926:m
1910:m
1906:a
1902:+
1897:n
1893:a
1884:m
1881:+
1878:n
1874:a
1850:!
1847:n
1838:=
1833:n
1829:a
1808:.
1803:m
1799:a
1795:+
1790:n
1786:a
1777:m
1774:+
1771:n
1767:a
1740:+
1731:s
1724:m
1720:/
1714:m
1710:a
1701:k
1697:n
1692:/
1684:k
1680:n
1675:a
1669:k
1624:m
1620:/
1616:)
1611:1
1607:n
1598:3
1594:n
1590:(
1585:m
1581:a
1577:+
1570:1
1566:n
1561:a
1554:m
1550:/
1546:)
1541:2
1537:n
1528:3
1524:n
1520:(
1515:m
1511:a
1507:+
1500:2
1496:n
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1476:3
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1459:m
1455:/
1451:)
1446:1
1442:n
1433:2
1429:n
1425:(
1420:m
1416:a
1412:+
1405:1
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1339:+
1330:s
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1289:m
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1242:a
1238:(
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1207:/
1200:+
1191:s
1182:m
1177:m
1173:a
1148:m
1144:a
1121:n
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1112:a
1104:n
1087:s
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1040:+
1031:s
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1016:n
1008:k
1004:n
999:a
976:0
948:k
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936:k
932:n
927:a
923:(
894:s
885:n
880:n
876:a
868:n
837:s
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811:n
781:n
776:n
772:a
764:n
747:s
693:n
688:n
684:a
652:n
647:n
643:a
629:n
593:1
590:=
587:n
581:}
576:n
572:a
568:{
514:]
511:1
508:,
502:[
482:.
479:.
476:.
473:,
468:2
464:a
460:,
455:1
451:a
437:n
433:m
417:m
413:a
409:+
404:n
400:a
391:m
388:+
385:n
381:a
351:1
345:n
340:}
335:n
331:a
327:{
300:.
295:y
290:+
285:x
275:y
272:+
269:x
247:0
241:y
238:,
235:x
199:.
196:)
193:y
190:(
187:f
184:+
181:)
178:x
175:(
172:f
166:)
163:y
160:+
157:x
154:(
151:f
148:,
145:A
139:y
136:,
133:x
116:B
106:A
89:B
83:A
77:f
20:)
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