On law of large numbers for L-R fuzzy numbers

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保险翻译

案例2
• 案例：某银行向保险公司投保火险附加盗窃险，在投保单上写明24小时有警卫值班，保险公司予以承保并以此作为减费的条件。后银行被窃，经调查某日24小时内有半小时警卫不在岗。问保险公司是否承担赔偿责任？
•
1999年1月，江西某公司将184吨价值100万余元
的棉浆粕向保险公司投保水路货物运输综合保险，运输工具为“赣南昌货0236”，轮，航线上海至南昌，交保费1177．6元。同年1月13日18时30分，货轮航行至黄浦江106灯浮附近，为避免与他船碰撞，驾
• 1. Insurability • Risk which can be insured by private companies typically shares seven common characteristics: • 1.1 Large number of similar exposure units
Principles of Insurance
• 1.6 Limited risk of catastrophically large losses: Insurable losses are ideally independent and non-catastrophic, meaning that the losses do not happen all at once and individual losses are not severe enough to bankrupt the insure; • Loss 全部损失 • Losses 部分损失
Legal Principles
• Benefit insurance: the insurance company doesn't have the right of recovery from the party who caused the injury and is to compensate the Insured regardless of the fact that Insured had already sued the negligent party for the damages. • 财产保险的被保险人在保险事故发生时对保险标的应当有保险利益。如果投保人以不具有保险利益的标的投保，保险人可单方面宣布保险合同无效；保险标的发生保险责任事故，被保险人不得因保险而获得不属于保险利益限度内的额外利益。

G期望讲义

E[ϕ(−∞
ϕ(x) exp(−
x2 )dx, 2σ 2
but if ϕ is a concave function, the above σ 2 must be replaced by σ 2 . If σ = σ = σ , then N ({0} × [σ 2 , σ 2 ]) = N (0, σ 2 ) which is a classical normal distribution. This result provides a new way to explain a well-known puzzle: many practitioners, e.g., traders and risk oﬃcials in ﬁnancial markets can widely use normal distributions without serious data analysis or even with data inconsistence. In many typical situations E[ϕ(X )] can be calculated by using normal distributions with careful choice of parameters, but it is also a high risk calculation if the reasoning behind has not been understood.
Shige PENG Institute of Mathematics Shandong University 250100, Jinan, China peng@ Version: ﬁrst edition
2
Preface
This book is focused on the recent developments on problems of probability model under uncertainty by using the notion of nonlinear expectations and, in particular, sublinear expectations. Roughly speaking, a nonlinear expectation E is a monotone and constant preserving functional deﬁned on a linear space of random variables. We are particularly interested in sublinear expectations, i.e., E[X + Y ] ≤ E[X ] + E[Y ] for all random variables X , Y and E[λX ] = λE[X ] if λ ≥ 0. A sublinear expectation E can be represented as the upper expectation of a subset of linear expectations {Eθ : θ ∈ Θ}, i.e., E[X ] = supθ∈Θ Eθ [X ]. In most cases, this subset is often treated as an uncertain model of probabilities {Pθ : θ ∈ Θ} and the notion of sublinear expectation provides a robust way to measure a risk loss X . In fact, the sublinear expectation theory provides many rich, ﬂexible and elegant tools. A remarkable point of view is that we emphasize the term “expectation” rather than the well-accepted classical notion “probability” and its non-additive counterpart “capacity”. A technical reason is that in general the information contained in a nonlinear expectation E will be lost if one consider only its corresponding “non-additive probability” or “capacity” P(A) = E[1A ]. Philosophically, the notion of expectation has its direct meaning of “mean”, “average” which is not necessary to be derived from the corresponding “relative frequency” which is the origin of the probability measure. For example, when a person gets a sample {x1 , · · · , xN } from a random variable X , he can directly 1 1 use X = N xi to calculate its mean. In general he uses ϕ(X ) = N ϕ(xi ) for the mean of ϕ(X ). We will discuss in detail this issue after the overview of our new law of large numbers (LLN) and central limit theorem (CLT). A theoretical foundation of the above expectation framework is our new LLN and CLT under sublinear expectations. Classical LLN and CLT have been widely used in probability theory, statistics, data analysis as well as in many practical situations such as ﬁnancial pricing and risk management. They provide a strong and convincing way to explain why in practice normal distributions are so widely utilized. But often a serious problem is that, in general, the “i.i.d.” condition is diﬃcult to be satisﬁed. In practice, for the most real-time processes and data for which the classical trials and samplings become impossible, the uncertainty of probabilities and distributions can not be neglected. In fact the abuse of normal distributions in ﬁnance and many other industrial or commercial i

参考文献

参考文献[1]任玉珑．电力工程概预算原理．重庆：水利电力出版社[2]金昀．面对WTO如何深化电力工程造价管理．上海电力，2002年第2期[3]王杰．电力工程造价管理分析.内蒙古电力技术，2005年第23卷第4期[4]梁森荣．了解美国电力工程造价管理的启示．管理论述[5]董士波．全过程工程造价管理与全生命周期工程造价管理之比较．经济师，2003．12[6]严玲，尹贻林．工程造价导论．天津大学出版社，2004．9[7]程鸿群，姬晓辉，陆菊春．工程造价管理．武汉大学出版社，2004．4[8]Electronic Tagging－Functional Specifications Version 1.67,TransactionInformation System Working Group,2005．4[9]李琼．国外电网投资多元化及其启示．电力技术经济，2003．1[10]谢识予．经济博弈论．上海：复旦大学出版社，2002[11]张维迎.博弈论与信息经济学．上海人民出版社,2006[12]谭忠富等．电力市场环境下发电企业的风险识别．中国电业，2000．1[13]江伟，黄文杰．博弈论在工程招投标中的应用分析．工业技术经济学，第23卷第1期，2004．2[14]LIANG Zhi-feng,Extensive Strategic Game:Concept,Transformability andExistence of Nash Equilibrium ．Proceedings of 2002 International Conference on Management Science ＆ Engineering(I),October22-24,2002[15]秦旋．工程监理制度下的委托代理博弈分析．中国软科学，2004．4[16]Nigd J．Smith．Managing Risk in Construction project TCP．London1999[17]Sead H．AL-Jibouri Monitoring systems and their effectiveness for projectcost control in construction[J] ．International Journal of Project Management,2003(21):145-154[18]卢现祥．寻租经济学导论[M]．北京：中国财政经济出版社，2000[19]Quentin W Fleming．Cost/Schedule Control System Criteria[J] ．Chicago:ProbusPublishing Company,1992:255-274[20]Kreps D．Game Theory and Economic Modeling，Oxford Univer—sity Press，1990[21]罗捷．电力工程造价管理改革浅见．电力技术经济，2005年第17卷第1期[22]Sang Yeol Joo.Weak laws of large numbers for fuzzy randomvariables:FSS147(2004)453-646[23]王祯显．土木工程管理中的模糊数学方法．长沙：湖南大学出版社，1989[24]蔡雷，徐扬．模糊多属性决策方法及其在投资决策中的应用，西南科技大学学报，2004，21[25]周宏安，李炳杰．基于相对隶属度的模糊信息多目标决策法，陕西理工学院学报，2004，20[26]Sang Yeol Joo.Strong convergence for weighted sums of fuzzy randomsets:Information sciences 176(2006)1086-1099[27]Sang Yeol Joo,Yun Kyung Kim.on Chung’s type law of large numbers for fuzzyrandom variables:Statistics & Probability Letters,74[28]戚安邦．建设项目全过程造价管理理论与方法．天津：天津人民出版社[29]Mark Konchar，Victor Sanvido．Comparison of U．S．Project DeliverySystems[J]．Journal of Construction Engineering and Management．1998，124(6)：110—113[30]Life Cycle Cost Analysis Guidelines 2002.Deparment of Natual Resources[31]建设工程工程量清单计价规范GB50500-2003．中国计划出版社, 2003．北京[32]曾永健．对工程造价体制改革的探索．电力建设，2003．12[33]马毅颖，王宏超．电力投资体制如何走向市场主导的多元化．中国农村水电及电气化信息网，2004．8[34]朱建君．建设项目全要素集成造价管理模型研究．基建优化，2005年第26卷第6期[35]刘华敏,杨丽霞.电力工程造价的管理与控制电力建设,2001．3。

模糊逻辑工具箱

Glossary术语表Adaptive Neuro-Fuzzy Inference System(ANFIS) A technique for automatically tuning Sugeno-type inference systems based on training data.Foreword（前言）The past few years have witnessed a rapid growth in the number and variety of applications of fuzzy logic. The applications range from consumer products such as cameras, camcorders, washing machines, and microwave ovens to industrial process control, medical instrumentation, decision-support systems, and portfolio selection.过去几年间，模糊逻辑无论是在应用数量上还是应用种类上都呈现快速增长的趋势。

其应用范围从消费产品，例如照相机，便携式摄像机，洗衣机，及微波炉到工业过程控制，医药器具，决策支持系统，以及部长职务选举等To understand the reasons for the growing use of fuzzy logic it is necessary, first, to clarify what is meant by fuzzy logic.为了理解模糊逻辑为何能得以如此快速使用，首先，有必要理清什么是模糊逻辑。

Fuzzy logic has two different meanings. In a narrow sense, fuzzy logic is a logical system, which is an extension of multivalued logic. But in a wider sense, which is in predominant use today, fuzzy logic (FL) is almost synonymous with the theory of fuzzy sets, a theory which relates to classes of objects with unsharp boundaries in which membership is a matter of degree. In this perspective, fuzzy logic in its narrow sense is a branch of FL. What is important to recognize is that, even in its narrow sense, the agenda of fuzzy logic is very different both in spirit and substance from the agendas of traditional multivalued logical systems.模糊逻辑有两种含义。

Law of large numbers

Law of large numbersAn illustration of the Law of Large Numbers using die rolls. As the number of die rollsincreases, the average of the values of all the rolls approaches 3.5.In probability theory, the law of large numbers (LLN ) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.For example, a single roll of a six-sided die produces one of the numbers 1, 2, 3, 4, 5, 6, each with equal probability. Therefore, theexpected value of a single dice roll isAccording to the law of large numbers, if a large number of dice are rolled, the average of their values (sometimes called the sample mean) is likely to be close to 3.5, with the accuracy increasing as more dice are rolled.It follows from the law of large numbers that the empirical probability of success in a series of Bernoulli trials will converge to the theoretical probability. For a Bernoulli random variable, the expected value is the theoretical probability of success, and the average of n such variables (assuming they are i.i.d.) is precisely the relative frequency.For example, a fair coin toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to 1/2. Therefore, according to the law of large numbers, the proportion of heads in a "large" number of coin flips "should be" roughly 1/2. In particular, the proportion of heads after n flips will almost surely converge to 1/2 as n approaches infinity.Though the proportion of heads (and tails) approaches 1/2, almost surely the absolute (nominal) difference in the number of heads and tails will become large as the number of flips becomes large. That is, the probability that the absolute difference is a small number, approaches zero as the number of flips becomes large. Also, almost surely the ratio of the absolute difference to the number of flips will approach zero. Intuitively, expected absolute difference grows, but at a slower rate than the number of flips, as the number of flips grows.The LLN is important because it "guarantees" stable long-term results for random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. It is important to remember that the LLN only applies (as the name indicates) when a large number of observations are considered. There is no principle that a small number of observations will converge to the expected value or that a streak of one value will immediately be "balanced" by the others. See the Gambler's fallacy.HistoryDiffusion is an example of the law of large numbers, applied to chemistry. Initially, there are solute molecules on the left side of a barrier (purple line) and none on the right. The barrier is removed,and the solute diffuses to fill the whole container. Top: With a single molecule, the motion appears to be quite random. Middle: With more molecules, there is clearly a trend where the solute fills the container more and more uniformly, but there are also random fluctuations.Bottom: With an enormous number of solute molecules (too many to see), the randomness is essentially gone: The solute appears to move smoothly and systematically from high-concentration areas to low-concentration areas. In realistic situations, chemists can describe diffusion as a deterministic macroscopic phenomenon (see Fick'slaws), despite its underlying random nature.The Italian mathematician Gerolamo Cardano (1501–1576) stated without proof that the accuracies of empirical statistics tend to improve with the number of trials.[1] This was then formalized as a law of large numbers. A special form of the LLN (for a binary random variable) was first proved by Jacob Bernoulli.[2] It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his Ars Conjectandi (The Art of Conjecturing) in 1713. He named this his "Golden Theorem" but it became generally known as "Bernoulli's Theorem". This should not be confused with the principle in physics with the same name,named after Jacob Bernoulli's nephew Daniel Bernoulli. In 1835, S.D. Poisson further described it under the name "La loi des grands nombres" ("The law of large numbers").[3] Thereafter, it was known under both names, but the "Law of large numbers" is most frequently used.After Bernoulli and Poisson published their efforts,other mathematicians also contributed to refinement of the law, including Chebyshev, Markov, Borel, Cantelli and Kolmogorov and Khinchin (who finally provided a complete proof of the LLN for the arbitrary randomvariables). These further studies have given rise to two prominent forms of the LLN. One is called the "weak" law and the other the "strong" law. These forms do not describe different laws but instead refer to different ways of describing the mode of convergence of the cumulative sample means to the expected value, and the strong form implies the weak.FormsTwo different versions of the Law of Large Numbers are described below; they are called the Strong Law of Large Numbers , and the Weak Law of Large Numbers . Both versions of the law state that – with virtual certainty – thesample averageconverges to the expected valuewhere X 1, X 2, ... is an infinite sequence of i.i.d. random variables with finite expected value E(X 1) = E(X 2) = ... = µ <∞.An assumption of finite variance Var(X 1) = Var(X 2) = ... = σ2 < ∞ is not necessary . Large or infinite variance will make the convergence slower, but the LLN holds anyway. This assumption is often used because it makes the proofs easier and shorter.The difference between the strong and the weak version is concerned with the mode of convergence being asserted.For interpretation of these modes, see Convergence of random variables.Weak lawSimulation illustrating the Law of Large Numbers. Each frame, you flip a coin that is red on one side and blue on the other, and put a dot in the corresponding column. A pie chart shows the proportion of red and blue so far. Notice that the proportion varies a lot at first, but graduallyapproaches 50%.The weak law of large numbers states that the sample average converges in probability towards the expected value [4][proof]That is to say that for any positive number ε,Interpreting this result, the weak law essentially states that for any nonzero margin specified, no matter how small,with a sufficiently large sample there will be a very high probability that the average of the observations will be close to the expected value, that is, within the margin.Convergence in probability is also called weak convergence of random variables. This version is called the weak law because random variables may converge weakly (in probability) as above without converging strongly (almost surely) as below.Strong lawThe strong law of large numbers states that the sample average converges almost surely to the expected value [5]That is,The proof is more complex than that of the weak law. This law justifies the intuitive interpretation of the expected value of a random variable as the “long-term average when sampling repeatedly”.Almost sure convergence is also called strong convergence of random variables. This version is called the strong law because random variables which converge strongly (almost surely) are guaranteed to converge weakly (in probability). The strong law implies the weak law.The strong law of large numbers can itself be seen as a special case of the pointwise ergodic theorem.Moreover, if the summands are independent but not identically distributed, thenhas a finite second moment andprovided that each XkThis statement is known as Kolmogorov’s strong law, see e.g. Sen & Singer (1993, Theorem 2.3.10). Differences between the weak law and the strong lawThe weak law states that for a specified large n, the average is likely to be near μ. Thus, it leaves open the possibility that happens an infinite number of times, although at infrequent intervals.The strong law shows that this almost surely will not occur. In particular, it implies that with probability 1, we have that for any ε > 0 the inequality holds for all large enough n.[6]Uniform law of large numbersSuppose f(x,θ) is some function defined for θ∈ Θ, and continuous in θ. Then for any fixed θ, the sequence {f(X,θ),1 ,θ), …} will be a sequence of independent and identically distributed random variables, such that the sample f(X2mean of this sequence converges in probability to E[f(X,θ)]. This is the pointwise (in θ) convergence.The uniform law of large numbers states the conditions under which the convergence happens uniformly in θ. If [7]1.Θ is compact,2.f(x,θ) is continuous at each θ∈ Θ for almost all x’s,3.there exists a dominating function d(x) such that E[d(X)] < ∞, andThen E[f(X,θ)] is continuous in θ, andBorel's law of large numbersBorel's law of large numbers, named after Émile Borel, states that if an experiment is repeated a large number of times, independently under identical conditions, then the proportion of times that any specified event occurs approximately equals the probability of the event's occurrence on any particular trial; the larger the number of repetitions, the better the approximation tends to be. More precisely, if E denotes the event in question, p its probability of occurrence, and Nn(E) the number of times E occurs in the first n trials, then with probability one,This theorem makes rigorous the intuitive notion of probability as the long-run relative frequency of an event's occurrence. It is a special case of any of several more general laws of large numbers in probability theory.ProofGiven X1, X2, ... an infinite sequence of i.i.d. random variables with finite expected value E(X1) = E(X2) = ... = µ < ∞,we are interested in the convergence of the sample averageThe weak law of large numbers states:Theorem:Proof using Chebyshev's inequalityThis proof uses the assumption of finite variance (for all ). The independence of the random variables implies no correlation between them, and we have thatThe common mean μ of the sequence is the mean of the sample average:Using Chebyshev's inequality on results inThis may be used to obtain the following:As n approaches infinity, the expression approaches 1. And by definition of convergence in probability (see Convergence of random variables), we have obtainedProof using convergence of characteristic functionsBy Taylor's theorem for complex functions, the characteristic function of any random variable, X , with finite mean μ,can be written asAll X 1, X 2, ... have the same characteristic function, so we will simply denote this φX .Among the basic properties of characteristic functions there areThese rules can be used to calculate the characteristic function ofin terms of φX :The limit e it μ is the characteristic function of the constant random variable μ, and hence by the Lévy continuity theorem, converges in distribution to μ:μ is a constant, which implies that convergence in distribution to μ and convergence in probability to μ are equivalent. (See Convergence of random variables) This implies thatThis proof states, in fact, that the sample mean converges in probability to the derivative of the characteristic function at the origin, as long as this exists.Notes[1]Mlodinow, L. The Drunkard's Walk. New York: Random House, 2008. p. 50.[2]Jakob Bernoulli, Ars Conjectandi: Usum & Applicationem Praecedentis Doctrinae in Civilibus, Moralibus & Oeconomicis , 1713, Chapter 4,(Translated into English by Oscar Sheynin)[3]Hacking, Ian. (1983) "19th-century Cracks in the Concept of Determinism"[4]Loève 1977, Chapter 1.4, page 14[5]Loève 1977, Chapter 17.3, page 251[6]Ross (2009)[7]Newey & McFadden 1994, Lemma 2.4References•Grimmett, G. R. and Stirzaker, D. R. (1992). Probability and Random Processes, 2nd Edition . Clarendon Press,Oxford. ISBN 0-19-853665-8.•Richard Durrett (1995). Probability: Theory and Examples, 2nd Edition . Duxbury Press.•Martin Jacobsen (1992). Videregående Sandsynlighedsregning (Advanced Probability Theory) 3rd Edition.HCØ-tryk, Copenhagen. ISBN 87-91180-71-6.•Loève, Michel (1977). Probability theory 1 (4th ed.). Springer Verlag.•Newey, Whitney K.; McFadden, Daniel (1994). Large sample estimation and hypothesis testing . Handbook of econometrics, vol.IV, Ch.36. Elsevier Science. pp. 2111–2245.•Ross, Sheldon (2009). A first course in probability (8th ed.). Prentice Hall press. ISBN 978-0136033134.•Sen, P. K; Singer, J. M. (1993). Large sample methods in statistics . Chapman & Hall, Inc.External links•Weisstein, Eric W., " Weak Law of Large Numbers (/ WeakLawofLargeNumbers.html)" from MathWorld.•Weisstein, Eric W., " Strong Law of Large Numbers (/ StrongLawofLargeNumbers.html)" from MathWorld.•Animations for the Law of Large Numbers (/prob:law_of_large_numbers) by Yihui Xie using the R package animation (/package=animation)Article Sources and Contributors8Article Sources and ContributorsLaw of large numbers Source: /w/index.php?oldid=409632062 Contributors: 3mta3, ABCD, Aastrup, Alcmaeonid, Ali Esfandiari, Andrea Ambrosio, Archelon, Atfyfe, Athenean, AxelBoldt, Bearian, Beland, Billjefferys, Bjcairns, Blue Tie, Breadbox, Burn, Burnte, Cedders, Chephy, Cherkash, Christian75, Cleared as filed, Cretog8, Czap42, Dan33gopo,DanielCD, Daqu, Daughter of Mímir, Dcoetzee, DerHexer, DidgeGuy, Dondegroovily, Dougluce, Elwikipedista, Eric Kvaalen, Fangz, Flloater, Func, Gazpacho, Giftlite, Green caterpillar,GregorB, Grendelkhan, Grick, GuntisOzols, Happily ever after, Headbomb, Hede2000, Iwaterpolo, J04n, Jayanta Sen, Jim.belk, Jusjih, Kaanstivonausfern, LOL, Lihaas, MER-C, MarkSweep, Maurice Carbonaro, McKay, Melcombe, Mets501, Michael Hardy, Mickraus, MicroBio Hawk, Mister Magotchi, Moreschi, Mundo tarantino, Mysid, Numbo3, O18, Oleg Alexandrov,OliAtlason, OoberMick, OwenX, PRRfan, Paolo.dL, Patrick, Paul August, Paul Pogonyshev, Phil Boswell, Phobos11, Pkfff, PlutarcoNaranjo, Pokipsy76, Psiphiorg, Qiqi.wang, Qwfp, Rar,Rick.G, Ritchy, Rockoprem, Rsgurevich, Salgueiro, Sanchom, Santhosh Thota, Sbyrnes321, Schmeitgeist, Schutz, Scott Ritchie, Shanes, SheffieldSteel, Shivan Bird, Sligocki, Soup man,Steel2009, Stpasha, Sławomir Biały, Teorth, That Guy, From That Show!, The Anome, TheObtuseAngleOfDoom, Tiagofassoni, TitanOne, TonyW, Trevorgoodchild, Uvaphdman, VMS Mosaic, Viclick, Voidvector, WikipedianProlific, Xaven, Xieyihui, Александър, 187 anonymous editsImage Sources, Licenses and ContributorsFile:Largenumbers.svg Source: /w/index.php?title=File:Largenumbers.svg License: Creative Commons Zero Contributors: User:NYKevinFile:DiffusionMicroMacro.gif Source: /w/index.php?title=File:DiffusionMicroMacro.gif License: Public Domain Contributors: User:Sbyrnes321File:Lawoflargenumbersanimation.gif Source: /w/index.php?title=File:Lawoflargenumbersanimation.gif License: Public Domain Contributors: User:Sbyrnes321LicenseCreative Commons Attribution-Share Alike 3.0 Unported/licenses/by-sa/3.0/。

Algebraic hierarchy of logics unifying fuzzy logic and quantum logic

a rXiv:077.2161v1[mat h.LO ]14J ul27Algebraic hierarchy of logics unifying fuzzy logic and quantum logic —Lecture Notes —Andreas de Vries ∗South Westphalia University of Applied Sciences,Haldener Straße 182,58095Hagen,Germany February 1,2008Abstract In this paper,a short survey about the concepts underlying general logics is given.In particular,a novel rigorous deﬁnition of a fuzzy negation as an operation acting on a lattice to render it into a fuzzy logic is presented.Ac-cording to this deﬁnition,a fuzzy negation satisﬁes the weak double negation condition,requiring double negation to be expansive,the antitony condition,being equivalent to the disjunctive De Morgan law and thus warranting com-patibility of negation with the lattice operations,and the Boolean boundary condition stating that the universal bounds of the lattice are the negation of each other.From this perspective,the most general logics are fuzzy logics,containing as special cases paraconsistent (quantum)logics,quantum logics,intuitionistic logics,and Boolean logics,each of which given by its own al-gebraic restrictions.New examples of a non-contradictory logic violating the conjunctive De Morgan law,and of a typical non-orthomodular fuzzy logic along with its explicit lattice representation are given.Contents1Introduction 21.1Notation ...............................32Lattices 32.1Distributive lattices (8)2Andreas de Vries:Algebraic hierarchy of logics 3Logics10 4Fuzzy logics174.1t-norms and the derivation of negations (18)4.2Examples ofﬁnite and discrete fuzzy logics (21)5Quantum logics245.1Subspaces in Hilbert space (25)5.2Quantum mechanics constructed from quantum logic (29)5.3Compatible propositions (31)5.4The logic of quantum registers (33)6Paraconsistent logics and effect structures356.1Paraconsistent quantum logics(PQL) (38)7Implicative lattices and intuitionistic logic397.1Implicative lattices (40)8Boolean algebras458.1Propositional logic (46)8.2First-order logic (48)8.3Modal logic (54)9Discussion55 1IntroductionLogic is as empirical as geometry.We live in a world with a non-classical logic.Hilary Putnam Logic is the science which investigates the principles governing correct or reliable inference.It deals with propositions and their relations to each other.Besides the classical Boolean logic there has been established various generalizations such as modal,intuitionistic,quantum,or fuzzy logic,as well as propositional structures underlying substructural logics which focus on relaxations of structural rules gov-erning validity and provability.The purpose of the present article is to give a brief unifying survey of the algebraic interrelations of logics which does not seem to have been presented before in this comprehensive form.It does not intend to deal, however,with the wide aspect of semantic algebras deriving logics,an importantAndreas de Vries:Algebraic hierarchy of logics3 issue in considerations both of fuzzy logics and of quantum logics.It concentrates on the propositional structure of general logics but noteworthy not on logical cal-culi neither on provability or model theory.Although this work has been greatly inﬂuenced by some classical references such as[5,32],it provides a wider spec-trum by including fuzzy logics,and thus complements modern approaches like [38]in revealing the algebraic hierarchy of logics as propositional structures of substructural logics,especially pointing out the decisive role of negation.The mathematical concept underlying any logic is the notion of the lattice,a partially ordered set with two binary operations forming an algebraic structure.Es-tablishing a lattice with an additional fuzzy negation operator yields a fuzzy logic, and further algebraic requirements such as non-contradiction,paraconsistency,or-thomodularity,or distributivity then specify it to the different classical and non-classical logics as sketched in Figure5.This comprehensive view on the different concepts of logics is enabled by deﬁning a fuzzy negation as a lattice operation satisfying weak double negation,antitony and the Boolean boundary condition. Remarkably,antitony is equivalent to the disjunctive De Morgan law,but does not imply the conjunctive De Morgan law.This deﬁnition is well established in fuzzy logic contexts[16]and generalizes commonly used notions of a negation as an involutive operation[14,17].It is wide enough to include all common fuzzy, quantum,intuitionistic,and classical logics.Consequently,the next section of is paper starts with an outline of lattice theory and subsequently gives the deﬁnition and important properties of a fuzzy logic and its common sublogics.In the following sections,various logics are considered in some more detail,with emphasis on examples of a typical non-orthomodular fuzzy logic and some special logics of quantum registers in unentangled states.1.1NotationEvery investigation,including the present one about logic,has to be communicated by means of language.The language being used is usually called the metalan-guage.It has to be distinguished carefully from the language of the studied logic, the object language There are many different notations existing in the literature,so Table1lists the symbols as they are used in the present text.2LatticesLattice theory is concerned with the properties of a binary relation≦,to be read “precedes or equals,”“is contained in,”“is a part of,”or“is less than or equal to.”This relation is assumed to have certain properties,the most basic of which leads4Andreas de Vries:Algebraic hierarchy of logics A⇔B A if and only if BMetalanguagex=y x equals yx∧y x and y(meet,conjunction) Object languagex→y x implies y,¬x∨y(material implication)Andreas de Vries:Algebraic hierarchy of logics5M5N5O6L723P6Figure1:Hasse diagrams of various posets.All depicted posets unless P6are lattices. circle,placing y higher than x whenever x<y.If x<y and there exists no z∈Xsuch that x<z<y,then a straight line is drawn between x and y.(In this case,oneoften says“y covers x”).A poset X can contain at most one element O∈X which satisﬁes O≦x for allx∈X.For if O and˜O are two such elements,then O≦˜O and also˜O≦O,i.e.,O=˜O by(2).If such an element O exists,it is called the least element of X.Withthe analogous reasoning,the greatest element of X,if it exists,is denoted by I andsatisﬁes x≦I for all x∈X.If both O and I∈X exist,they are called universalbounds of X,since then O≦x≦I for all x∈X.Such an element O is also calledthe zero of the poset L,and I the unit of the poset.In connection with logics,beingspecial posets as we shall see below,O is also called the“absurdity.”In a poset X with a least element O∈X,elements x∈X satisfying O<x suchthat there is no y∈X with O<y<x are called atoms or points of X.In logics wethus can state that“atoms immediately follow from absurdity.”An upper bound of a subset Y⊆X of a poset X is an element a∈X with y≦afor every y∈Y.The least upper bound sup Y is an upper bound contained in everyother upper bound.By(2),sup Y is unique if it exists.Note that Y=P6\{I}in Figure1does not contain an upper bound.The notions of lower bound and greatestlower bound inf Y are deﬁned analogously.Again by(2),inf Y is unique if it exists. Deﬁnition3.A lattice is a poset L such that any elements x,y∈L have a unique greatest lower bound,denoted x∧y,and a unique least upper bound x∨y,i.e.,x∧y=inf{x,y},x∨y=sup{x,y}.(4) The operation∧is also called meet,and the operation∨is called join.A lattice L is complete,when for any set D⊆L the bounds sup D and inf D exist;it isσ-complete,when for any countable set D⊆L the bounds sup D and inf D exist.A lattice is atomic if every element is a join of atoms.♦The lattice condition will later ensure that the logical operations of conjunction (∧)and disjunction(∨)are well-deﬁned for any pairs of propositions.For instance,6Andreas de Vries:Algebraic hierarchy of logics the two atoms of the poset P6in Figure1do not have unique least upper bounds, hence P6is not a lattice.Moreover,it follows thatﬁnite sets of pairwise disjoint (“orthogonal”)propositions also have a well-deﬁned disjunction.The complete-ness condition ensures that the latter is true also for countable sets of pairwise disjoint propositions.This is arguably not an essential requirement for a logic (must logic be inﬁnitary?),but it allows for probability measures to be deﬁned on an inﬁnite lattice or poset,since it is customary to require that the probability of a countable set of disjoint events be well-deﬁned and equal to the countable sum of the probabilities of the disjoint events(σ-additivity of probabilities).Example4.The poset(P(Ω),⊆)in Example2(b)is a lattice,where for any family A={A1,A2,...}of subsets A1,A2,...⊆Ωwe haveinf A= j A j,sup A= j A j.(5) Note that A⊆P(Ω).Especially,we have A∧B=A∩B and A∨B=A∪B for two subsets A,B⊆Ω.Especially,P(Ω)is a complete lattice.♦Example5.For aﬁeld K,let L(K n)={V⊆K n:V is a vector space}be the set of all subspaces of the vector space K n.Moreover,for any set A⊆K n let span A denote the intersection of all subspaces of K n which contain A,i.e.,span A=∩{V⊆L(K n):A⊆V}.(6)span A is also called the“linear hull”of A.Then with≦deﬁned as the usual set inclusion and with the following deﬁnitions for two subspaces V,W⊆K n,V∧W=V∩W,V∨W=span(V∪W),(7)the set L(K n)is a lattice.It has universal bounds O={0}and I=K n.♦Lemma6.Let L be a lattice L,and x,y∈L with x≦y.Then for all z∈L,x∧z≦y∧z.(8) Proof.We have x∧z≦z and x∧z≦x≦y by(4),hence(8)by(4)again.The binary operations∧and∨in lattices have important algebraic properties, some of them analogous to those of ordinary multiplication and addition.Andreas de Vries:Algebraic hierarchy of logics7 Theorem7.In a lattice L,the operations of meet and join satisfy the following laws,whenever the expressions referred to exist.(Idempotent laws)x∧x=x,x∨x=x.(9) (Commutative laws)x∧y=y∧x,x∨y=y∨x.(10) (Associative laws)x∧(y∧z)=(x∧y)∧z,x∨(y∨z)=(x∨y)∨z.(11) (Laws of absorption)x∧(x∨y)=x∨(x∧y)=x.(12)(The laws of absorption are often also called“laws of contraction.”)Moreover,(Consistency)x≦y⇐⇒x∧y=x⇐⇒x∨y=y.(13)Proof.The idempotence and the commutativity laws are evident from(4).The associativity laws(11)follow since x∧(y∧z)and(x∧y)∧z are both equal to sup{x,y,z}whenever all expressions referred to exist.The equivalence between x≦y,x∧y=x,and x∨y=y is easily veriﬁed.Thus x≧y is equivalent to x∧y=y and x∨y=x,and this implies(12).It can be proved that the identities(9)–(12)completely charactarize lattices [5,Theorem I.8].In fact Dedekind,whoﬁrst considered the concept of a lattice (“Dualgruppe”)at the end of the19th century,used(9)–(12)to deﬁne lattices. Theorem8(Principle of Duality).Given any valid formula over a lattice,the dual formula obtained by interchanging≦with≧,and simultaneously∧with∨,is also valid.Proof.Since for any elements x,y of the lattice we have x≦y if and only if y≧x, the poset structure with respect to≧is isomorphic to the poset structure≦,but with∧and∨interchanged.The dual of a lattice is simply its Hasse diagram(Fig.1)turned upside down, illustrating the principle of duality.In fact,the two poset structures≦and≧of a lattice are tied up to each other by the laws of associativity,absorption,and consistency so strongly that they are inescapably dual.The following theorem concerns relations of modality and distributivity which are valid in every lattice. Theorem9.Let L be a lattice.For all x,y,z∈L we then have the“modular inequality”x∨(y∧z)≦(x∨y)∧z if x≦z,(14) and the“distributive inequalities”x∧(y∨z)≧(x∧y)∨(x∧z)(15)x∨(y∧z)≦(x∨y)∧(x∨z)(16)8Andreas de Vries:Algebraic hierarchy of logics Proof.If x≦z,we have with x≦x∨y that x≦(x∨y)∧z.Also y∧z≦y≦x∨y and y∨z≦z.Therefore,y∧z≦(x∨y)∧z,i.e.x∨(y∧z)≦(x∨y)∧z,which is (14).Clearly x∧y≦x,and x∧y≦y≦y∨z;hence x∧y≦x∧(y∨z).Also x∧z≦x, x∧z≦z≦y∨z;hence x∧z≦x∧(y∨z).That is,x∧(y∨z)is an upper bound of x∧y and x∧z,from which(15)follows.The inequality(16)follows from(15)by the principle of duality.2.1Distributive latticesIn many lattices,and thus in many logics,the analogy between the lattice opera-tions∧,∨and the arithmetic operations·,+includes the distributive law x(y+z) =xy+xz.In such lattices,the distributive inequalities(15)and(16)can be sharp-ened to identities.These identities do not hold in all lattices;for instance,they fail in the lattices M5and N5in Figure2.We now study distributivity,which in a lattice is symmetric with respect to∧and∨due to the duality principle(Theorem 8),which is not the case in ordinary algebra,where a+(bc)=(a+b)(a+c)due to the priority precedence of multiplication(·)and addition(+).Ix y zOIyxzOIyzxOM5N5N5Figure2:Non-distributive lattices.Deﬁnition10.A lattice L is called distributive if the following identity holds.x∧(y∨z)=(x∧y)∨(x∧z)for all x,y,z∈L.(17)♦Theorem11.In any lattice L,the identity(17)is equivalent tox∨(y∧z)=(x∨y)∧(x∨z)for all x,y,z∈L.(18) Proof.We prove(17)⇒(18).The converse(18)⇒(17)follows analogously.(x∨y)∧(x∨z)=[(x∨y)∧x]∨[(x∨y)∧z]by(17)=x∨[z∧(x∨y)]by(12),(10)=x∨[(z∧x)∨(z∧y)]by(17)=[x∨[z∧x)]∨(z∧y)by(11)=x∨(z∧y)by(12)Andreas de Vries:Algebraic hierarchy of logics9 However,in nondistributive lattices the truth of(17)for some elements x,y,z does not imply them obeying(18),as the two variants of N5in Figure2show.An important property of distributive lattices is the following.Theorem12.A lattice L is distributive if and only if the following property is satisﬁed for all a,x,y∈L:a∧x=a∧y and a∨x=a∨y imply x=y.(19) Proof.Suppose the lattice to be distributive.Then using repeatedly the Equations (12),(10),and(17),we havex=x∧(a∨x)=x∧(a∨y)=(x∧a)∨(x∧y)=(a∧y)∨(x∧y)=(a∨x)∧y=(a∨y)∧y=y.The converse is proved in[5,§II.7].Expressions involving the symbols∧,∨and elements of a lattice are called lattice polynomialsLemma13.In any lattice L,the sublattice S generated by two elements x and y consists of x,y,u,and v,where u=x∨y and v=x∧y,as in Figure3.Proof.By(12),x∧u=x;by(11),(9),x∨u=x∨(x∨y)=(x∨x)∨y=x∨y= u.The other cases are analogous,using symmetry in x and y and duality.ux yvFigure3:The lattice F2∼=22.For22,we have v=00,x=01,y=10,and u=11.A lattice morphism is a mappingµ:L→K from a lattice L to a lattice K which preserves the meet and join operations,i.e.,µ(x∧y)=µ(x)∧µ(y),µ(x∨y)=µ(x)∨µ(y)for all x,y∈L.Corollary14.Let F2∼=22be the lattice of Figure3,and let a,b∈L be two elements of an arbitrary lattice.Then the mapping x→a,y→b can be extended to a lattice morphismµ:F2→L.10Andreas de Vries:Algebraic hierarchy of logics The preceding results are usually summarized in the statement that F2is the “free lattice”with generators x,y.It has just four elements and is distributive;in fact it is a Boolean lattice.Lattice polynomials in three or more variables can be extremely complicated. However,in a distributive lattice any polynomial can be brought to a normal form, similarly as a real or complex polynomial can be written as a sum of products, p(x1,...,x n)=∑r i=1 ∏s(i)j=1x i j or as a product of its divisors,p(x1,...,x n)=∏δ ∑j∈Tδx j .Theorem15.In a distributive lattice L,every polynomial p:L n→L of n variables is equivalent to a join of meets,and dually:p(x1,...,x n)= α∈A i∈Sαx i = δ∈D j∈Tδx j ,(20) where Sαand Tδare nonempty sets of indices.Proof.Each single x i can be so written,where A(or D,respectively)is the family of sets consisting of the single element set{x i}.On the other hand,we have by (9)–(11)α∈A i∈Sαx i ∨ β∈B i∈Sβx i = γ∈A∪B i∈Sγx i .(21)Using the distributive law,we have similarlyα∈A i∈Sαx i ∨ β∈B i∈Sβx i = γ∈A×B i∈Sα∪Sβx i .(22)The assertion follows from(10)and(17),combined with the relation( S x i)∧( T x i)= S∪T x i,which follows from(9)–(11).Equation(22)is the lattice generalization of the distributive law of ordinary algebra, ∑α∈A xα ∑β∈B yβ =∑(i,j)∈A×B x i y j.3LogicsA general logic is now going to be introduced as a lattice with universal bounds and a special operation,the fuzzy negation.Here the fuzzy negation of a lattice element x is the square root of a unique supremum x′′of x,cf.Eq.(23).Con-sidering the elements of the lattice as propositions,each proposition then impliesAndreas de Vries:Algebraic hierarchy of logics11 its double negation,but not always vice versa.Furthermore,the negation is an-titone,a property which turns out to be equivalent to the disjunctive De Morganlaw,and the Boolean boundary condition for the universal bounds holds true.Ifnothing else is assumed,then the lattice is a fuzzy logic.If in addition the law ofnon-contradiction holds,then it is a logic.In this way,the notion“fuzzy logic”in-cludes the propositional structures of substructural logics focussing on validity andpremise combination[38],inﬁnite-valued fuzzy logics ofŁukasiewiczian type[9,§2.3.2],as well as quantum and distributive logics,in particular Boolean algebras or nonclassical Heyting-Brouwerian(“intuitionistic”)logics[5,§XII.3]in which“tertium non datur”or“reductio ad absurdum”are not valid.A fuzzy logic is notnecessarily distributive,not even orthomodular or paraconsistent.Quantum logicswill turn out to be orthomodular,but not necessarily distributive.Creation comes when you learn to say no.Madonna,The Power of GoodbyeDeﬁnition16.Let L be a lattice with universal bounds0and1,i.e.,0≦x≦1forall x∈L.A mapping′:L→L,x→x′,is called fuzzy negation,if the following relations hold for all x,y∈L:(Weak double negation)x≦(x′)′,(23)(Antitony)y′≦x′if x≦y,(24) (Boolean boundary condition)0′=1,1′=0.(25) The pair(L,′)then is called a fuzzy logic,and the elements x∈L are called propo-sitions.If for the fuzzy negation the“law of non-contradiction”x∧x′=0(26) holds for all x∈L,then it is called(non-contradictory)negation and(L,′)is a logic. As long as misunderstanding is excluded,we shortly write L instead of(L,′). Algebraically,the element x′∈L in a(non-contradictory)logic L is called a pseudo-complement of x∈L;if in addition x∨x′=1,then x′is called a com-plement of x∈L.In general,a lattice is called(pseudo-)complemented if all its elements have(pseudo-)complements.A mapping′:L→L,x→x′in a(pseudo-) complemented lattice L,assigning to each element x a(pseudo-)complement,is called(pseudo-)complementation.If the(pseudo-)complementation is bijective, the lattice is called uniquely(pseudo-)complemented.♦Deﬁned this way,a logic is a special fuzzy logic.In any fuzzy logic L the rela-tion x≦y will be interpreted as the statement“x implies y.”The propositions x∧y12Andreas de Vries:Algebraic hierarchy of logics and x∨y will be interpreted as“x and y”and“x or y,”respectively.The universal bounds of a logic are usually denoted by0and1,the proposition1expresses truth, and the proposition0expresses falsehood or absurdity.Theorem17.Let be L a lattice and′:L→L a mapping satisfying(23)for all x∈L.Then′is antitone if and only if the disjunctive De Morgan law holds,i.e.,(x∨y)′=x′∧y′for all x,y∈L.(27) Proof.Supposeﬁrst the antitony of′,and let u:=x∨y and v:=x′∧y′for arbitrary x,y∈L.Then u≧x and u≧y,as well as v≦x′and v≦y′.By the antitony,this means that u′≦x′and u′≦y′,as well as v′≧x′and v′≧y′,i.e.,u′≦x′∧y′=v,(28) as well as v′≧x′∨y′=u.By(23)and the antitony,the last inequality yields v≦v′′≦u′which means together with(28)that u′=v.Assume,on the other hand,the disjunctive De Morgan law(27).Since y=x∨y for x≦y,we have y′=(x∨y)′=x′∧y′with(27),hence y′≦x′.Theorem18.In a general fuzzy logic L the conjunctive De Morgan inequality(x∧y)′≧x′∨y′.(29) holds for all x,y∈L.Proof.Since(x′∨y′)′=x′′∧y′′≧x∧y by(27)and(23)for all x,y∈L,we have x′∨y′≦((x′∨y′)′)′≦(x∧y)′by(23)and the antitony of the negation.Note that the conjunctive De Morgan law(see Eq.(33below)does not neces-sarily hold in a fuzzy logic,even not in a logic.Remark19.If we abandon the Boolean boundary condiditon(25)on a negation, then very little is known about the fuzzy negations of0and1in a general fuzzy logic.By the antitony(24)and by the general lattice property0≦x≦1for all x∈L,we only can derive1′≦x′≦0′for all x∈L.(30) A fuzzy negation with0′=0thus must be constant,i.e.,x′=0for all x∈L.On the other hand,a constant fuzzy negation x′=x0∈L for all x∈L implies x0=0, since otherwise we had0′′=x0>0,contradicting(23).In a logic with the law of non-contradiction x∧x′=0,however,we have1′=0,(31) since1∧y=y for all y∈L.♦Andreas de Vries:Algebraic hierarchy of logics13 Remark20.Sometimes the notions“strong”and“weak negation”are used,espe-cially in the context of logic programming[7]and artiﬁcial intelligence,motivated by the following ideas.Intuitively speaking,strong negation captures the presence of explicit negative information,while weak negation captures the absence of pos-itive information.In computer science,weak negation captures the computational concept of negation-as-failure(or“closed-world negation”).A strong negation(′)can be interpreted as“impossible.”Negating this,in turn,gives a weak“not impossible”assertion,so that x implies(x′)′,i.e,x≦(x′)′, but not vice versa.With this negation,the rule of bivalence x∨x′=1(x is true or impossible)does not necessarily hold(since x is possible as long as it is not recognized as true),but the corresponding x∧x′=0(not both true and impossible) does.Weak negation,in contrast,can be regarded as“unconﬁrmed.”Negating this gives x≦(x′)′(if x is true it is always unconﬁrmed that it is unconﬁrmed), but not vice versa(x′)′≦x(if it is not conﬁrmed that x is unconﬁrmed,then x is certainly true).However,it has x∨x′=1(x is true or unconﬁrmed)holding, since if x is not true it is certainly not conﬁrmed,but not necessarily x∧x′=0 (it is never conﬁrmed that x is true and unconﬁrmed),since x may be true but not conﬁrmed.Deﬁning such kind of weak negation therefore implies that tertium non datur x∨x′=1does hold,but the law of contradiction is not necessarily true, x∧x′≧0.Likewise,the ability to speak of the uncertain apparently may force a weakening of the tertium non datur in some form,losing double negation,and also of the law of contradiction.Both possibilities are enabled by the above concept of a fuzzy logic.However,there is some confusion with the term“strong negation.”Sometimes it simply means the classical negation“false”=“not true.”Negating this gives another strong assertion,(x′)′≦x(if it is false that x is false,then x is true),and vice versa.This has x∨x′=1(x is true or false)and x∧x′=0(x is not both true and false)holding.♦What are the reasons that the conjunctive version(Eq.(33)below)is not im-plied by the antitony of the negation?In fact,it is easily proved that for lattices with a total order(i.e.,∀x,y either x≦y or y≦x)antitony,disjunctive De Morgan law and conjunctive De Morgan law are equivalent.However,in a partially ordered set this is not necessarily true.One of the simplest counterexamples is M5. Example21.Let L=M5denote the modular lattice as in Figure4and deﬁne the operation′:M5→M5byxx′14Andreas de Vries:Algebraic hierarchy of logics1a b cFigure4:The non-distributive modular lattice M5.Then x′′=x for all x=b,but b′′=1>b,and antitony is easily veriﬁed.Since moreover x∧x′=0for all x∈M5,′is a non-contradictory negation and(M5,′)is a logic.However,in contrast to the disjunctive De Morgan law(27),the conjunctive De Morgan law(33)is not valid since,e.g.,(a∧b)′=1but a′∨b′=c.♦Theorem22.(De Morgan’s conjunctive law)If in a fuzzy logic L we have(x′)′=x for all x∈L,then the conjunctive De Morgan law holds,(x∧y)′=x′∨y′.(33)Proof.Let(x′)′=x.By(27)we obtain(x′∨y′)′=x∧y,hence x′∨y′=((x′∨y′)′)′=(x∧y)′.For a non-contradictory logic the above theorem has a stronger consequence. Theorem23.(Tertium non datur)In a logic L,a negation with(x′)′=x for all x ∈L implies the law“tertium non datur”,or“law of excluded middle,”x∨x′=1. Proof.By Theorem22,De Morgan’s laws(27)and(33)hold,and hence the prop-erty x∧x′=0for all x∈L implies(x′∨x)′=x′′∧x′=0,i.e.,x′∨x=1by(25). In general,a non-contradictory negation satisfying the law tertium non datur is called ortho-negation[38],complemented negation,or involutive negation,and (L,′)a complemented logic.Example24.[41,§2.2]The tertium non datur is a highly nontrivial assumption. An example for its nonconstructive feature is a proof of the following proposition:“There exist irrational numbers x,y∈R\Q with x y∈Q.”Proof:Either√2∈Q,√2√2√2=2∈Q,i.e.,x=√2,y=√i.e.,x=y=√2Andreas de Vries:Algebraic hierarchy of logics 15intuitionistic logic fuzzy negation law of non-contradiction distributivity lattice ց−→fuzzy logic ց−→ցlogic ց−→ցx ≦x ′′րdistributive logic x =x ′′ցmodal logic paraconsis-tent logic −→quantum logic −→Boolean logic 2,3ր∃,∀−→ﬁrst-order logic րրparaconistency orthomodularityFigure 5:The algebraic hierarchy of logics.In particular,a Boolean logic is a special quantum logic,a quantum logic is a special fuzzy logic.By Theorem 28,a logic with the law tertium non datur is a quantum logic.Establishing Boolean logic with the quantiﬁers ∃and ∀yields ﬁrst-order logic,and with the quantiﬁers 2and 3modal logic.Deﬁnition 25.A paraconsistent logic is a fuzzy logic satisfying the paraconsis-tency conditionx =y if x ≦y and x ′∧y =0.(34)An intuitionistic logic is a distributive logic in which there exists propositions x <(x ′)′.A quantum logic ,or orthologic ,is a logic satisfying the orthomodular identityx ∨(x ′∧y )=y if x ≦y ,(35)A Boolean logic is a complemented distributive logic.♦Therefore we obtain the algebraic structure of logics in Figure 5.Every dis-tributive complemented lattice is othomodular,since interchanging x and y and setting z =x ′for x ≦y in the distributive law (17)with x ∧y =x ,x ∧x ′=0und y =x ∨y yields (35).Example 26.(The logics BN 4and MO 1)A given a lattice may yield the propo-sitional structure for more than one fuzzy logics,depending on the negation.AtbnfFigure 6:The lattice F 2∼=22,providing the propositional structure of the logics BN 4and MO 1.simple example is the Boolean lattice 22.Deﬁning two negations x ∼and x ⊥by the following tables,xx ∼tb n f f n b t (36)16Andreas de Vries:Algebraic hierarchy of logicswe obtain the logics[38,§8.2],[41,§2.4]BN4=({f,n,b,t},∼),MO1=({f,n,b,t},⊥),(37) Both negations may be illustrated geometrically,supposing f,n,b,t as the four points on the unit circle S1in the plane R2,viz.,f=(0,−1),n=(1,0),b=(−1,0), t=(0,1).The negation∼then corresponds to the reﬂection in the horizontal line {(x,0)}through the origin,whereas the negation⊥corresponds to a rotation aroundthe origin by the angleπ.The main differences between the logics BN4and MO1 are that BN4is contradictory(e.g.,b∧b∼=b>f)and that the tertium non datur does not hold in BN4(b∨b∼=b<t),whereas MO1is a classical Boolean logic. The fusion and material implication of BN4read:∗ff f n n bf n t t →fn t n tbf n f t(38)The many-valued logic BN4considered by Dunn and Belnap was the result of research on relevance logic,but it also has signiﬁcance for computer science appli-cations.The truth degrees may be interpreted as indicating,e.g.,with respect to a database query for some particular state of affairs,that there is no information con-cerning this state of affairs(n=/0),information saying that the state of affairs fails (f={0}),information saying that the state of affairs obtains(t={1}),conﬂicting information saying that the state of affairs obtains as well as fails(b={0,1}).♦Example27.The lattice L(V n)of all linear subspaces of an n-dimensional vectorspace V n(Example5)is uniquely complemented since the orthogonal complement V⊥of any subspace V satisﬁes V∧V⊥={0}and V∨V⊥=V n.Note that O={0} and I=V n are the universal bounds of L(V n).Moreover,it is orthomodular,andthe complementation is involutive,(x⊥)⊥=x.Therefore,L(V n)is an involutively complemented logic in which De Morgan’s laws hold.♦Theorem28.A logic is a quantum logic if and only if x=x′′.Proof.In a quantum logic we have by y=x′′in(35)that x′′=x∨(x′∧x′′)=x∨0= x.Conversely,if x=x′′,then from(27),(33)and(15)we deduce(x∨(x′∧y))′=x′∧(x′∧y)′=x′∧(x∨y′)≧(x′∧x)∨(x′∧y′)=(x∨y)′,(39) hence x∨(x′∧y)≧x∨y.But x∨(x′∧y)≦(x∨x′)∧(x∨y)by(16),and since with Theorem23x∨x′=1,we have x∨(x′∧y)≦x∨y.We conclude x∨(x′∧y)= x∨y=y for x≦y,i.e.,(35)holds.。

Chapter 18Risk Management and Insurance

The law of large numbers
The law of large numbers assumes that while any policyholder may suffer a loss, only a few of the policyholders will actually suffer a loss. Therefore, if every policyholder pays a small amount, there will be enough money to pay for the damages suffered by the few policyholders.
A Basic Business Reader
Chapter 18 Risk Management and Insurance
Objectives
After studying this chapter you will be able to:

Define risk and describe strategies of risk handling. Discuss risks that businesses confront and choice of risk-handling alternatives. Explain the law of large numbers and the law of adverse selection. Discuss types of insurance coverage and risky areas for the insurance industry.
Risk and risk management
What is risk?

Chaos… of a dripping faucet - University of Roches

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Logistic Map: Comparing Successive Drops
The “Spectrum” of Drop Frequency
Time(sec) # Drops
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DROPPER A (1.9 mm) 21.41 DROPS/SEC
Logistic Map: Comparing Successive Drops
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DROPPER B (0.45 mm) 11.10 DROPS/SEC
Logistic Map: Comparing Successive Drops
The “Spectrum” of Drop Frequency
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CHAOS… OF A DRIPPING
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Dan Allan and
Adam Bublitz
RECONSTRUCTION, PHASE 2
Consequence of trying to fix missed data
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英汉文化比较与翻译智慧树知到课后章节答案2023年下武汉学院

英汉文化比较与翻译智慧树知到课后章节答案2023年下武汉学院武汉学院绪论单元测试1.She has beauty still, and, if it be not in its heyday, it is not yet in its autumn.( )A:她依旧美丽，如果不是美艳动人，至少也算风韵犹存。

B:她依然很美，如果不是芳华正茂，也还不到迟暮之年。

C:她仍旧光彩照人，如果不是美如冠玉，也还不到色褪香消的时候。

D:她依旧楚楚动人，如果不是千娇百媚，也还不到薄暮之秋。

答案:她依然很美，如果不是芳华正茂，也还不到迟暮之年。

2.His best jokes fell flat. ( )A:他的好笑话没用了。

B:他那些最能逗人笑的笑话都不灵了。

C:他那好笑的笑话都没能引人发笑。

D:他那些拿手笑话苍白无力。

答案:他那些最能逗人笑的笑话都不灵了。

3.Which of the following statements are true? ( )A:Language is the carrier and the mirror of culture.B:Language is the motivation of culture development.C:Language is strongly influenced and shaped by culture.D:Language will not undergo corresponding alterations with the change ofculture.答案:Language is the carrier and the mirror of culture.;Language is the motivation of culture development.;Language is strongly influenced and shaped by culture.4.Which of the following statements are not true? ( )A:The task of a translator is to only to convey the meaning of the sourcelanguage.B:Translation is not only a cross-linguistic communication.C:Translation is also an intercultural and cross-social communicative activity.D:Translation is only a transfer of language, but not the transfer of cultures.答案:The task of a translator is to only to convey the meaning of thesource language.;Translation is only a transfer of language, but not the transfer ofcultures.5.Culture is that complex whole which includes knowledge, beliefs, art, morals,law, custom and any other capabilities and habits acquired by a man as amember of society. （）A:错 B:对答案:对第一章测试1.What are features of idioms?（）A:Syntactic StabilityB:IdiomaticityC:Semantic UnityD:Semantic Opacity答案:Syntactic Stability;Idiomaticity;Semantic Unity;Semantic Opacity2.Semantic Opacity refers to the extensive usage of idiomatic expressionsamong people. （）A:错 B:对答案:错3.Literal translation could retain the images in the target language and belongsto foreignization.（）A:错 B:对答案:对4.What is the translation method for“have other fish to d ry 另有公干”（）A:Literal Translation Plus AnnotationsB:Liberal TranslationC:TransliterationD:Literal Translation答案:Liberal Translation5.What is the translation method for “四面楚歌to be besieged on all sides(“楚”is the name of a stateduring the period of Warring States)”?（）A:Literal Translation Plus AnnotationsB:Literal TranslationC:TransliterationD:Liberal Translation答案:Literal Translation Plus Annotations第二章测试1.Onomatopoeia is the formation or use of （） such as buzz or murmur thatimitate the sounds associated with the objects or actions they refer to.A:actionsB:soundsC:charactersD:words答案:words2.The part of speech of English onomatopoeia is noun or verb, which works as（）.A:the subject B:object C:attributiveD:predicate答案:the subject;object;predicate3.In translating the sentence ”There were no sound but that of the tread of themen and the footsteps of the two shaggy ponies which drew the van.” wha ttranslation technique is employed in the translated version: “那时候只听见沙沙的脚步声以及拉车的那两匹鬃毛蓬松的小马得得的马蹄声，除此之外，再也听不到别的声音了。

TBBT学习笔记s1

TBBT笔记（B版）102 Big Bran假说2009-09-16 19:49:45 来自: brians01e02 The Big Bran Hypothesis1.普通级词汇dolly：n. 小轮搬运车,手推车fulcrum：n.(杠杆的)支点,支轴vortex：n. 漩涡,旋风entropy：n. 熵transvestite：n. 易装癖者immaculate：adj. 无缺点的,无瑕疵的evening gown：n. (通常带有拖地长裙的)女夜礼服insomnia：n.失眠unorthodox：adj. 非传统的,异端的sinus：n. 鼻窦sleep apnea：睡眠时呼吸暂停otolaryngologist：n. 耳鼻喉科医师proctologist：n. 直肠科医师pelvis：n. 骨盘Intoxicating：adj. 醉人的,使人兴奋的dowels：n.木钉,暗销infrared repeater：n. 红外线中继器photocell：n. 光电池aquarium pump：n. 潜水泵drip tray：n. 除霜水盘sluice：n. 水闸overflow reservoir：n. 蓄水池,储液器heat sink：n. 散热片junkyard：n. 废品旧货栈oxyacetylene torch：n. 氧乙炔炬2.爆炸级词汇Lois Lane：超人前女友Green Lantern：绿灯侠Mandelbrot set of complex numbers：芒德勃罗（Beno?t Mandelbrot，1924-），波兰几何学家，分形理论创始人。

Mandelbrot集又被称为“数学恐龙”，对每一个C，让z0=0代入迭代式：f(z) = z*z + C,经足够多次迭代后函数值不扩散，这样的C所组成的集合为M集。

M集被认为是数学上最为复杂、最美丽的集合之一。

Oppenheimer：奥本海默,1945年带领“曼哈顿计划”洛斯·阿拉莫斯实验室全体科学家成功研制出世界上第一枚原子弹3.爆炸级食品pad thai：泰式炒面Vienna sausages：维也纳香肠Honey Puffs：一种低纤维麦片Big Bran：一种含有糠麸的高纤维麦片4.精选语录Raj: Are there any chopsticks?Sheldon: You don't need chopsticks. This is Thai food.Leonard: Here we go.Sheldon: Thailand has had the fork since the latter half of the 19th century. Interestingly they don't put the fork in their mouth, they use it toput the food on a spoon, which then goes into their mouth.Leonard: Ask him for a napkin, I dare you.听谢博士讲泰国餐具Leonard: Penny, wait.Penny: Yeah?Leonard: Um... If you don't have any other plans, do you want to join us for Thai food and a Superman movie marathon?Penny: A marathon? Wow, how many Superman movies are there?Sheldon: You're kidding, right?Penny: Yeah, I do like the one where Lois Lane falls from the helicopter and Superman swooshes down and catches her. Which one was that?All guys: One.我只看过超人一Sheldon: You realize that scene was rife with scientific inaccuracy.Penny: Yes, I know, men can't fly.Sheldon: No, no. Let's assume that they can. Hmm. Lois Lane is falling, accelerating at an initial rate of 32 feet per second per second. Superman swoops down to save her by reaching out two arms of steel. Miss Lane, who is now traveling at approximately 120 miles an hour, hits them and is immediately sliced into three equal pieces.Leonard: Unless Superman matches her speed and decelerates.Sheldon: In what space, sir? In what space? She's 2 feet above the ground. Yeah, frankly, if he really loved her, he'd let her hit the pavement. It'd be a more merciful death.钢铁手臂三段式Leonard: I guess we'll just bring it up ourselves.Sheldon: I hardly think so.Leonard: Why not?Sheldon: Well, we don't have a dolly or lifting belts or any measurable upper-body strength. Leonard: We don't need strength. We're physicists['fizisist]. We are the intellectual descendants of Archimedes[,a:ki'mi:di:z]. Give me a fulcrum and a lever and I can move the Earth. It's just a matter... I don't have this. I don't have this! I don't have this.Sheldon: Archimedes would be so proud.阿基米德的后代们Leonard: I'm not surprised. A well-known folk cure for insomnia is to break in your neighbor's apartment andclean.Sheldon: Sarcasm?Leonard: You think?Sheldon: Granted, my methods may have been somewhat unorthodox, but I think the end result will be a measurable enhancement to Penny's quality of life.Leonard: You know what, you convinced me. Maybe tonight we should sneak in and shampoo her carpet.Sheldon: You don't think that crosses a line?Leonard: Yes. For God's sake, Sheldon, do I have to hold up a sarcasm sign every time I open my mouth?Sheldon: You have a sarcasm sign?Leonard: No, I do not have a sarcasm sign.讽刺？Leonard: Uh. Here's the thing: ,Penny, just as Oppenheimer came to regret his contributions to the first atomic bomb, so too I regret my participation in what was, at the very least, an error in judgment. The hallmark of the great human experiment is the willingness to recognize one's mistakes. Some mistakes, such as Madam Curie's discovery of radium, turned out to have great scientific potential, even though she would later die a slow, painful death from radiation poisoning. Another example, from the field of Ebola research...Leonard的科学道歉5.地道表达swoop down：猛扑下straighten up：整理,整顿get out of your hair：不打扰了，get in one's hair 打扰某人per se：本质上,本身6.本集八卦他们住在四楼，以前Penny的房间住的是有异装癖的警察Leonard有2600本漫画书，但是跳舞毯上赢不了HowardPenny在农场长大，12岁时组装过拖拉机马达，但是睡觉会打鼾Sheldon有一个Master和两个Ph.D.学位，但是不懂什么是讽刺Howard用俄语泡妞，但是对花生（包括花生油）过敏Raj是个好的倾听者，但是。

the law of large numbers

the law of large numbers大数定律由雅各布·伯努利（1654－1705）提出，他是瑞士数学家、也是概率论的重要奠基人。

频率的稳定性是概率定义的客观基础，而伯努利大数定理以严密的数学形式论证了频率的稳定性。

而大数定律发表于伯努利死后8年，即1713年出版的《猜度术》，正是这本巨著使得概率论从那时起真正成为了数学的一个分支。

大数定律和中心极限定理，是概率论中极其重要的两个极限定理，也是概率学的核心定律。

一、大数定律概述大数定律的定义是，当随机事件发生的次数足够多时，随机事件发生的频率趋近于预期的概率。

可以简单理解为样本数量越多，其平概率越接近于期望值。

大数定律的条件：1、独立重复事件；2、重复次数足够多。

与“大数定律”对应的，就是“小数定律”，小数定律的内容：如果样本数量比较小，那么什么样的极端情况都有可能出现。

但是我们在判断不确定事件发生的概率时，往往会违背大数定律，而不由自主地使用“小数定律”，滥用典型事件，犯以偏概全的错误。

二、与大数定律相关的常见事件保险大数法则是近代保险业赖以建立的数理基础。

保险公司正是利用在个别情形下存在的不确定性将在大数中消失的这种规则性，来分析承保标的发生损失的相对稳定性。

按照大数法则，保险公司承保的每类标的数目必须足够大，否则，缺少一定的数量基础，就不能产生所需要的数量规律。

墨菲定律墨菲定律是大数定律的特殊情况，概念为“凡事有可能会出错，就一定会出错”。

墨菲定律的成立条件：1、事件有大于零的概率；2、样本足够大（比如时间足够长，人数足够多等）。

所以墨菲定律可以算是大数定律的一种特殊情况，概率只要大于0就会发生。

墨菲定律告诉我们，即便一个东西概率很低，只要次数足够多，就一定会发生，而如果这个东西会造成巨大的影响，我们不得不事先做好准备，避免遭受无法承受的打击，“黑天鹅”事件指的就是这类事情。

查理·芒格在《穷查理宝典》提到：”坏事总会发生，我们只是不知道什么时候而已“。

The preference order of fuzzy numbers

Байду номын сангаас
required. We have proved the feature of the proposed index in a particular case. Several examples axe also used to illustrate the feature and applicability in ranking fuzzy numbers. ~) 2002 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - F u z z y numbers, Ranking methods, Signal/Noise ratio (S/N ratio).
T h e Preference Order of Fuzzy N u m b e r s
L.-H. CHEN AND H.-W. LU
Department of Industrial Management Science National Cheng Kung University, Talnan, Taiwan, 701, R.O.C.
ha I ~ m ~ m l
Journal
with q ~ k ~ o M
computers & mathematics
P E R G A M O N Computers and Mathematics with Applications 44 (2002) 1455-1465 /locate/camwa
1. I N T R O D U C T I O N
Ranking fuzzy numbers is important for decision-making problems in a fuzzy environment, since the measurements are imprecise in nature. A number of ranking methods are developed for this purpose. Chen et al. [1] have surveyed the existing methods and classified them into four categories, such as preference relation method [2-14], fuzzy mean and spread method [15], fuzzy scoring method [16-22], and linguistic method [23,24]. Some researchers have also reviewed and compared the existing methods, such as Bortolan et al. [25], Fortemps et al. [26], Kim et al. [27], and Lee et al. [15]. Although most methods are developed for ranking a group of fuzzy numbers according to particular attributes, some limitations are embedded in these methods, such as lack of discrimination, producing counter-intuitive orderings, inconsistent orderings if a new fuzzy number is added, and high complex and cumbersome computational efforts [1,28,29[. In addition, for almost all methods, membership functions of fuzzy numbers should be known before ranking, which may be infeasible in real applications. Among the existing ranking methods, a few methods are efficient, which are based on area measurement [8,9,11,14,22,28,30]. These methods generally determine the order of fuzzy numbers, 0898-1221/02/$ - see front matter (~) 2002 Elsevier Science Ltd. All rights reserved. Typeset by ~.~S-TEX PII: S0898-1221(02)00270-5

of

A note on the law of large numbers for fuzzyvariables∗Robert Full´e r rfuller@ra.abo.ﬁEberhard Triesch triesch@math2.rwth-aachen.de AbstractThis short note a counterexample showing that Williamson’s theorem on the law of large numbers for fuzzy variables under a general triangular norm extension principle isnot valid.The objective of this note is to provide a counterexample to Theorem1in Williamson’s paper[1].To save space,we essentially use the same notation as in[1]and we do not repeat the statement of the theorem.Let t(u,v):=uv(product norm)and deﬁne the sequence of fuzzy numbers(X i)∞i=1by their membership functions as follows:µXi (x):= 1−|x|i,if−1≤x≤1,0,otherwiseThenαXi =βXi=0for all i and Theorem1states that the membership functionsµZNofthe arithmetic means Z N=(1/N) N i=1X i converge pointwise(as N→∞)to the function µgiven byµ(x):= 1,for x=0,0,otherwiseat least on(−1,0)∩(0,1).However,we will show that in the open interval(-1,1),the functionsµZNare bounded from below by some strictly positive function(not depending on N).To see this,recall that the T-arithmetic meansµZNare deﬁned byµZN (z)=supx1+...x N=N zNi=1µX i(x i).For z∈(−1,1),we can thus estimateµZN (z)≥Ni=1µX i(z)=N i=1(1−|z|i)≥∞ i=1(1−|z|i).By Euler’s pentagonal number theorem(see,e.g.,[2],p.312),the inﬁnite product is well known to converge for|z|<1to the following series:f(|z|)=1+∞n=1(−1)n(|z|(3n2−n)/2+|z|(3n2+n)/2).The value of the inﬁnite product is thus positive on the interval(−1,1).∗Theﬁnal version of this paper appeared in:Fuzzy Sets and Systems,55(1993)235-236.References[1]R.C.Williamson,The law of large numbers for fuzzy variables under a general tri-angular norm extension principle,Fuzzy Sets and Systems,41(1991)55-81.[2]T.M.Apostol,Introduction to Analytic Number Theory,(Springer Verlag,Berlin-Heilderberg-New York,1976).。

Consider a stochastic process (1.1)

Asymptotic bounds for inﬁnitely divisible sequencesStanis l aw Kwapie´n∗Jan Rosi´n ski†In memory of Kazimierz UrbanikAbstractWe give asymptotic bounds for sample paths of discrete time inﬁnitely divisible processes and prove the optimality of such bounds.Keywords and phrases.Inﬁnitely divisible processes,boundedness of sample paths,L´e vy pro-cesses,Rademacher series.2000Mathematics Subject Classiﬁcation.Primary:60G17.Secondary:60G151Preliminaries and the main resultConsider a stochastic process(1.1)X(t)= R g(t,s)Z(ds)t∈R,where Z is a L´e vy process and g is a deterministic kernel.Suppose that we sample this process at discrete times t n,which are not far apart from each other.Formally,we state such condition asR R(v2sup n g(t n,s)2∧1)ΠZ(dv)ds<∞,(1.2)whereΠZ is the L´e vy measure of Z(1).Note that(1.2),without the supremum,is necessary for the integral(1.1)to exist(see,e.g.,[6]).We will characterize an extremal sample behavior of X(t n) as n→∞.Roughly speaking,we show that possible heavy tails of X(t)have small inﬂuence on the variation of the sequence(X(t n))and its behavior depends mostly on small jumps of Z(see Examples7and8).We quantify this dependence.We will put and solve this problem in a general framework and then apply the solution to special cases as above.Our work extends and reﬁnes a result of Braverman[2]given for symmetric stable processes.The technique is based on series representations of inﬁnitely divisible processes combined with precise estimates for the tail of a Rademacher series related to an inﬁnitely divisible distribution.We further develop this technique in Section2.Section3contains the proof of the main result and some applications.The proof uses series representations and methods developed in[4]as well.A sequence X=(X n)n∈N is said to be inﬁnitely divisible if for every n∈N the random vector (X1,...,X n)has an inﬁnitely divisible distribution in R n.It follows from Maruyama[5]that every inﬁnitely divisible sequence has the L´e vy-Khintchine representationE e i y,X =exp y,b −1t ∞t((θ−1(s))2∧1)ds 1κθ(log n)<∞ a.s.(1.7)This asymptotic bound of X is sharp in the following sense.For every left continuous non-increasing functionθ:(0,∞)→[0,∞)with (u2∧1)(−dθ(u))∈(0,∞)there exists an inﬁnitely divisible sequence X with L´e vy measureνsatisfying(1.3),(1.4)and(1.5)such thatlim supn→∞|X n|In order to use(1.7)we need information about the asymptotic behavior ofκθ.This is possible, to a large extent,whenθis regularly varying function at zero.We will write f(t) g(t)(t→∞) when C−1g(t)≤f(t)≤Cg(t)for some positive constant C and all t suﬃciently large.Lemma2Supposeθis a regularly varying function with index−αat zero.Ifα<1thenκθis bounded.Ifα=1then(1.9)κθ(t) t1θ−1(s)ds.If1<α<2then(1.10)κθ(t) tθ−1(t).Ifα=2thenκθ(t) t ∞t(θ−1(s))2ds 1α( #(t))−1α n k=1(log(k)t)βk 1For each t >0we consider the following quasi norms of a =(a i )M a (t ):= E |S |t1t a2:a =a +a,(2.1)and H a (t ):=i ≤ta ∗i+√2,(2.2)Then the following inequlities hold for each t >0(see [3])P (|S |>eM a (t ))≤e −t ≤2e 2P (|S |>M a (t ))14H a (t )≤M a (t )≤K a (t )≤H a (t ).(2.4)Finally,letψa (s ):=K −1asWe want to express bounds for the tail of S explicitly as a function ofθ.To this end we introduce the function(2.10)κθ(t)= t0(θ−1(s)∧1)ds+√2,t>0andκθ(0)=0,as in(1.6).The following estimates play the key role.Lemma4Let a=(θ−1(i))i∈N and letψa be given by(2.5).Then for all t>0(2.11)cκθ(t)≤ψ−1a(t)≤Cκθ(t)where c=et i>t(θ−1(i))2 1t ∞t(θ−1(s))2ds+(θ−1(t))2 1t ∞t(θ−1(s))2ds 1tθ−1(t)).It is easy to verify the following bounds for t≥1θ−1(1)+ t1θ−1(s)ds≤(θ−1(1)∨1) t0(θ−1(s)∧1)ds,∞t(θ−1(s))2ds≤((θ−1(1))2∨1) ∞t((θ−1(s))2∧1)dsand√t ∞ i=1(θ−1(i))2 1/2≤√2.We have the following elementary bounds for t<1∞1(θ−1(s))2ds≤((θ−1(1))2∨1) ∞t((θ−1(s))2∧1)ds,(θ−1(1))2≤2((θ−1(1))2∨1) ∞t((θ−1(s))2∧1)ds if t∈(0,1/2],and√2tθ−1(1)≤√3(θ−1(1)∨1)κθ(t)if t<1.(2.13)Combining(2.12)and(2.13)we prove the upper bound in(2.11).Now we will prove the lower bound in(2.11).For t<1we haveκθ(t)= t0(θ−1(s)∧1)ds+√2≤t+√2≤2√θ−1(1)+1 H a(t).For t≥1we obtainκθ(t)≤1+ i≤tθ−1(i)+√2≤ 1t i>t(θ−1(i))2 1tθ−1(t). Since t≥1,we get i≤tθ−1(i)≥t tθ−1(1)+3 H a(t).Using(2.4)we obtain the lower bound in(2.11).The proof of Lemma4is complete.2 Lemma5(i).For every k>0there exists a constant L such thatκθ(kt)≤Lκθ(t)t≥0.(ii).Suppose thatθ1(u)≤kθ2(u)for all u∈(0,u0)and some k,u0>0.Then there exists a constant L such thatκθ1(t)≤Lκθ2(t)t≥0.Proof(i).If k≤1,then by(2.11)and the fact thatψ−1a is nonincreasing,κθ(kt)≤c−1ψ−1a(kt)≤c−1ψ−1a(t)≤Cc−1κθ(t).If k>1,then by the concavity ofψ−1a,ψ−1a(0)=0,and(2.11)we haveκθ(kt)≤c−1ψ−1a(kt)≤c−1kψ−1a(t)≤Cc−1kκθ(t).(ii).Putθ3=kθ2.We haveθ−11(t)≤θ−13(t)for all t>θ3(u0)andθ−11(t)∧1≤(θ−13(t)∧1)/(u0∧1) for all t≤θ3(u0).Using(i)and the deﬁnition ofκθwe haveκθ1(t)≤(u0∧1)−1κθ3(t)=(u0∧1)−1kκθ2(k−1t)≤Lκθ2(t).23Proof of Theorem1and some applicationsProof of the Theorem1:Let N be a Poisson random measure on R N with intensity measure ν.Deﬁne f n:R N→R by f n(x)=x n and f(x):=sup n|f n(x)|= x ∞.PutX n= R N f n(x)(N(d x)−(|f n(x)|∨1)−1ν(d x))n∈N.The sequence X =(X n)is well deﬁned and its characteristic function is the same as of X.There-fore,we may further assume that X =X.Then we decompose X n as folowsX n=a n+ f≤1f n d(N−ν)+ f>1f n dNwherea n=− f>1f n(|f n|∨1)−1dν.From(1.4)ν(f>1)<∞.Hencesupn|a n|≤ν(f>1)<∞andsupn f>1f n dN ≤ f>1sup n|f n|dN= f>1f dN<∞ a.s.because the last integral is aﬁnite sum.Therefore,the asymptotic behavior of X n is determined byY n:= f≤1f n d(N−ν).From now on we may and do assume thatνis concentrated on{f≤1},that is,all f n’s are uniformly bounded by1.Let Y n:= f n d(N −ν),where N is an independent copy of N.Let N=N−N be the symmetrization of N and letY n=Y n−Y n= f n d N(3.1)be the corresponding symmetrization of Y n.We will use the series representation of( Y n)given in [4],ly,let iδs i be a Poisson point process on S:=R N with intensity measure2νindependent of the Rademacher sequence{ i}.Then N d= i iδs i and( Y n)n∈N d=( i i f n(s i))n∈N. Without loss of generality,we may and do assume thatY n=∞ i=1 i f n(s i).Putθ(u)=2ν(f≥u)u>0and let θ−1denote the right continuous inverse of θgiven by(2.7).Recall that(a∗i)i∈N stands for the nonincreasing rearrangement of any sequence(|a i|)i∈N.LetT=sup{i≥1:f∗(s i)>u i}(3.2)whereu i := θ−112i≥ibecause θ( θ−1(t )+)≤t .Applying a large deviations estimate P|M (t )3exp(−αt )forcertain α>0as t →∞,we get that Ee βT <∞for some β>0.(3.5)Now we consider W n in (3.3).Using the contraction principle conditionally {s i }we obtainP (|W n |>t )≤2P i>Ti f ∗(s i ) >t ≤4P ∞ i =2i θ−1(i 2+4P∞i =1i θ−1(i +12≤12P∞ i =1i θ−1(i ) >t 2t >0.(3.6)Using(3.6)and(2.6)for a=( θ−1(i))i∈N we get for every x>0P sup n≥2|W n|2κeθ(log n)≤12∞ n=2exp −ψa(xκeθ(log n)) .Applying(2.11)we getψa(xκθ(log n))≥ψa x C log n.The last inequality follows from the concavity ofψ−1a,provided x≥C.Consequently,P sup n≥2|W n|κeθ(log n) <∞,(3.7)which together with(3.3),(3.4)and(3.5)yieldsE sup n≥2| Y n|κeθ(log n) =E sup n≥2|Y n−E Y(Y n)|κeθ(log n)<∞.Combining this with theﬁrst part of the proof we getsupn≥2|X n|where(Γi)are arrival times in a Poisson process with rate1independent of( in).It is well known that given the condition onθ,series(3.9)converges a.s.,deﬁning an inﬁnitely divisible sequence X=(X n)(see,e.g.[7]).The L´e vy measureνof X is the push-forward of the product ofτand 12δ1 ⊗N by the map(u,x)→u x.Therefore,ν(x: x ∞≥u)=θ(u).By theﬁrst part of Theorem1,lim supn→∞|X n|b n≥L a.s.(3.11)Indeed,by(2.6)for a= θ−1(di) i∈Nn P(|S n|>Lb n)= n P(|S1|>Lb n)≥(4e4)−1 n exp(−ψa(8eLb n)).By Lemmas5and4b n=κθ(log n)≤L1κd−1θ(log n)≤L1c−1ψ−1a(log n).Hencen P(|S n|>Lb n)≥ n exp −ψa(8eLL1c−1ψ−1a(log n)) = n1b n= m n=k b−1n S n e n ∞= ∞ i=1θ−1(di)y i ∞wherey i=mn=k b−1n in e n.(y i)are independent symmetric random vectors with y i ∞=b−1k.By the contraction principle we haveP sup k≤n≤m|S n|.b n≥L|ALetting m→∞and then k→∞and applying(3.11)we get1≤2P lim sup n→∞|X n|>0 a.s.κθ(log n)(see,e.g.,[8]).If also |x|≤1|x|ΠZ(dx)=∞,thenκθ(log n)→∞.In this case one can show by the Hewitt-Savage0-1law applied to the representation(3.9)that|X n|lim supn→∞t ∞t(θ−1(s))2ds) 1)–regulary varying and(θ−1(t))2is(−2αand∞t(θ−1(s ))2ds t (θ−1(t ))2if α<2.(3.14)This and (3.12)imply (1.10).If α=1,then (3.14)and t1θ−1(s )ds ≥13t (θ−1(t ))2for t large enough yield(1.11).The cases α<1and α>2are obvious.2Now we will return to our original question concerning inﬁnitely divisible sequences deﬁned by a stochastic integral.Example 7Let X (t ),t ∈R be a stochastic process as in (1.1)–(1.2),where Z is a L´e vy process with no Gaussian part.For simplicity we assume that E |Z (1)|<∞with E Z (1)=0or Z (1)is symmetric.Then we writeX (t n )=b n +X n where X =(X n )n ∈N satisﬁes (1.3)with νbeing the push-forward of the product of ΠZ and the Lebesgue measure on R by the map (s,x )→(xg (t n ,s ))n ∈N andb n =− RRsign(vg (t n ,s ))I (|vg (t n ,s )|>1)ΠZ (dv )dtor b 0=0when Z is symmetric (cf.[6]).By (1.2)sequence (b n )n ∈N is bounded.Deﬁneθ(u )=RΠZ ({v ∈R :|v |sup n|g (t n ,s )|≥u })ds(3.15)for small u >0,otherwise θcan be arbitrary.By Theorem 1lim supn →∞|X (t n )|(log n )1−1log(log n )<∞ a.s.when α=1.(3.18)(3.17)-(3.18)were obtained by Braverman [2]for symmetric stable processes.Observe that a smallervalue of αproduces less variability in (X (t n )).12Naturally,one may consider more general inﬁnitely divisible processes given by a stochastic integral with respect to an inﬁnitely divisible random measure.The conclusion of Theorem1does not depend on a representation of an inﬁnitely divisible process.Example8Let M be an independently scattered divisible random measure on R withﬁnite control measure m and no Gaussian part.Assume that M is symmetric andE e iuM(A)=e−m(A)ψ(u),whereψ(u)= ∞0(1−cos uv)Π(du)withΠbeing a L´e vy measure.Consider a harmonizable processX(t)= R e itu M(du)t∈R.Condition(1.4)of Theorem1holds for the real and imaginary parts of X(t n)for any sequence (t n)n∈N.In this case we takeθ(u)=m(R)Π([u,∞))and conclude that|X(t n)|lim supn→∞。

高考英语复习资料之难点讲解大难点26 斤斤计较的数字计算题试题

专题26 斤斤计较的数字计算题数字计算题往往涉及多处信息，这就要求学生要处处留心、斤斤计较，方能解答正确。

●难点磁场AThe Grange15 Lamb’s MewsLondonSW19 1PF19th JanuaryDear Ms HawthorneI am writing to express my disappointment with the evening my wife and I spent at your restaurant two nights ago.While the food was,as usual,very good,I regret to inform you that the service from our waiter,Max,was most unsatisfactory.First of all,when we arrived,we discovered our table overlooking the garden had been given away to some other customers.The new table we finally received was by the door and when I complained,Max just laughed.Furthermore,as we were sitting down,he was extremely familiar with my wife,which deeply made her uncomfortable.When our meal came,he served the food casually and at the end cleared the plates in a most rude fashion.Regarding the payment of service,I noticed your menu which says that the matter ofrewarding service is at the willingness of the guest.In addition,I should point out that on other occasions I have left at least 12 percent.This time,I felt right in leaving nothing.I regret having to write this letter,but as a regular customer I am extremely displeased and did not expect to be treated in this manner.I look forward to hearing from you with a satisfactory reply,or,if you prefer,we could meet in person to resolve this matter.Yours sincerely,Rupert Metcalf1.(★★★★)Mr Metcalf had dinner in the restaurant on ________.A.16th JanuaryB.17th JanuaryC.18th JanuaryD.19th January2.How many complaints does Mr Metcalf have in his letter?A.One.B.Two.C.Four.D.Six.BHave you ever noticed,when looking at a map of the world ,that the east coast of South America and the west coast of Africa look as though they might fit together?If you have,you are not alone,In 1965 the English scientist Sir Edward Bullard used a computer to test the fit of the two continents and found that at an ocean depth of 2000 metres the match was very close indeed.It seems too remarkable to be possible,but there is a lot of evidence(证据) to suggest that Africa was once joined to South America.For example,there is a belt of ancientrocks along the east coast of Brazil which is similar to the rocks across the South Atlantic in West Africa.There is further evidence that existing land masses were once linked.The remains of a 400-to-500-million-year-old mountain chain has been found running down the eastern part of Greenland,western Scandinavia,and through north-west Scotland and Ireland,into western Canada, eventually finding their way to north-west Africa. Then there is the evidence from life itself.In various parts of the world today the same animals and plants can be found on land masses separated by,in some cases,thousands of miles of oceans.Did they develop at the same time in two different places?It seems unlikely.Biologists believe that there must have been land bridges which have now sunk beneath the sea.Also fossils(化石) found in sedimentary rocks (fossils are the preserved remains of life forms) have allowed geologists to trace the same plants from South America,and Antarctica(北极) in rocks perhaps 300 million years old.The ice that is now confined to(限制) the polar regions〔北极或者南极地区)has not always been so limited in extent.Indeed,during a period of the Earth’s history known as the permocarboniferous age(二叠石炭纪) about 250~350 million years ago there is evidence from the rocks that there were glaciers covering South America,parts of Africa,India,Australia,as well as Antarctica.On the other hand,in the northern hemisphere(北半球),there were deserts.If the continents sparead out as they are today,it is hard to understand how this could be.So there is considerable evidence to show whole continents moved apart,and naturallymany people have tried to discover how and why whole land masses moved.3.(★★★★★)How many proofs does the author give in the passage?A.3.B.6.C.4.D.7.CThe outburst of joy at the success of China’s national football team in reaching the World Cup finals easily matched the scenes of celebration which attended Beijing’s Olympic bid win.It’s little surprise:football fans had waited 44 years for that moment—three times longer than the seemingly endless World Trade Organization negotiations(会谈)！The 53-year-old Serb coach(教练) Bora Milutinovic is regarded now as nothing less than a saviour by football fans used to years of broken dreams and bitter setbacks. Best known to fans as Milu,he has been believed by many with playing a major role in helping the Chinese team win its tickets to the 2021 World Cup being co-hosted by South Korea and Japan.Having already guided Mexico,Costa Rica,Nigeria and the United States to the World Cup finals,Bora Milutinovic managed to continue his winning streak in China.He made it clear however last Friday in Shenyang Wulihe Stadium that he preferred not to be seen as a miracle worker.Before guiding the Chinese team to the finals,Milu was one of only two coaches in the world to take four national sides to the final stages.The other one is Carlos Alberto u however is the only man to take all for teams to the second round of the finals.And now he was gone on better taking the fifth team to the finals.4.(★★★★)The negotiation of China’s entering the WTO has lasted about ________ years.●案例探究1. JULYMs. Rossi owns the Roadside Motel(汽车游客旅馆)，which has 50 units.She wishes to keep a handy record of the number of units occupied(住房，占有).To do this she uses a calendar and writes the number of units occupied in a small box in the upper right-hand comer of each date.The following questions are based on the occupancy rate for July.1.The total number of units occupied during the week of July 14 through July 20 was ______.2.The average occupancy rate for Thursday was ________.3.Ms. Rossi regards a 90% occupancy rate as excellent.On how many days during the month did the Roadside Motel have a 90% or better rate of occupancy?A.5.B.4.C.6.D.1.命题意图:考察学生的观察才能、综合运用多条信息进展数据处理的才能。

大数定律（LawofLargeNumbers）的原理及Python实现

⼤数定律（LawofLargeNumbers）的原理及Python实现本⽂以抛掷硬币(tossing coins)为例, 来理解⼤数定律(Law of Large Numbers), 并使⽤ Python 语⾔实现.原理⼤数定律, 简单来说, 就是随着抛掷硬币的次数的增多, 正⾯向上出现的⽐例(the ratio of heads)会越来越接近正⾯朝上的概率(the probability of heads).Python 实现在⽰例代码中, 假定正⾯朝上的概率(the probability of heads)为0.51, 模拟进⾏10个系列的硬币投掷(coin tosses), 每个投掷系列, 投掷硬币10000 次, 然后, 将正⾯朝上的⽐例(the ratio of heads)随着投掷次数的变化进⾏显⽰, 并保存到 images/ ⽬录下. 具体代码如下:#-*- coding: utf8 -*-from__future__import print_functionimport numpy as npimport matplotlib.pyplot as pltimport osdef law_of_large_numbers(num_series=10, num_tosses=10000, heads_prob=0.51, display=True):""" Get `num_series` series of biased coin tosses, each of which has `num_tosses` tosses, and the probability of heads in each toss is `heads_prob`."""# 1 when less than heads_prob; 0 when no less than heads_probcoin_tosses = (np.random.rand(num_tosses, num_series) < heads_prob).astype('float32')cumulative_heads_ratio = np.cumsum(coin_tosses, axis=0)/np.arange(1, num_tosses+1).reshape(-1,1)if display:plot_fig(cumulative_heads_ratio, heads_prob)def save_fig(fig_id, dirname="images/", tight_layout=True):print("Saving figure", fig_id)if tight_layout:plt.tight_layout()# First, ensure the directory existsif not os.path.isdir(dirname):os.makedirs(dirname)# Then, save the fig_id imagenameimage_path = "%s.png" % os.path.join(dirname, fig_id)plt.savefig(image_path, format='png', dpi=300)def plot_fig(cumulative_heads_ratio, heads_prob, save=True):# Get the number of tosses in a seriesnum_tosses = cumulative_heads_ratio.shape[0]# Set the width and height in inchesplt.figure(figsize=(8, 3.5))# Plot cumulative heads ratioplt.plot(cumulative_heads_ratio)# Plot the horizontal line of value `heads_prob`, with black dashed linetypeplt.plot([0, num_tosses], [heads_prob, heads_prob], "k--", linewidth=2, label="{}%".format(round(heads_prob*100, 1)))# Plot the horizontal line of value 0.5 with black solid linetypeplt.plot([0, num_tosses], [0.5, 0.5], "k-", label="50.0%")plt.xlabel("Number of coin tosses")plt.ylabel("Heads ratio")plt.legend(loc="lower right")# Set x ranges and y rangesxmin, xmax, ymin, ymax = 0, num_tosses, 0.42, 0.58plt.axis([xmin, xmax, ymin, ymax])if save:save_fig("law_of_large_numbers_plot")plt.show()if__name__ == '__main__':num_series, num_tosses = 10, 10000heads_proba = 0.51law_of_large_numbers(num_series, num_tosses, heads_proba)显⽰结果, 如下图所⽰参考资料[1] Aurélien Géron. Hands-On Machine Learning with Scikit-Learn and TensorFlow. O'Reilly Media, 2017.。

英文描述大数定理

英文描述大数定理1. 引言大数定理（Law of Large Numbers，简称LLN），又称大数极限定理，是概率论中的一条非常重要的理论。

它描述的是在重复实验条件下，随着实验次数的增多，样本的平均值趋于收敛于期望值的现象。

这一定理在现实生活中具有广泛的应用，如统计、金融、风险管理等领域。

2. 传统大数定理与弱大数定理在大数定理的研究中，传统的大数定理是高斯在1820年提出的，也称为大数第一定理（Theorem of Large Numbers）。

它的基本思想是：在相同条件下，如果独立实验次数趋于无限，事件发生的频率趋近于概率。

即大量重复实验的结果，随着实验次数增多，越来越趋近于理论概率值。

这一定理被应用广泛，例如在股市分析中用于分析股票价格变动。

但是，传统的大数定理仅对可列的无限个随机变量成立，即研究的随机变量是形如$X_1,X_2,...$的无限个变量序列，这在现实生活中非常罕见。

因此，人们提出了弱大数定理。

弱大数定理是针对有限个随机变量的情况而提出的。

其核心思想是：对于独立同分布随机变量$X_1,X_2,...,X_n$，样本平均值$\frac{X_1+X_2+...+X_n}{n}$在$n$趋于无穷时，收敛于期望值$E(X)$的概率为1，即：$$\lim_{n\to\infty}P\left(\left|\frac{X_1+X_2+...+X_n}{n}-E(X)\right|<\epsilon\right)=1$$其中，$\epsilon$是任意小正数。

弱大数定理通常是从刻画实验次数趋于无穷时，样本平均值与期望值差距的角度进行研究。

3. 中心极限定理与大数定理中心极限定理是概率论中另一条非常重要的定理。

其意义在于分布在各种分布上的独立同分布随机变量的和，当样本容量$n$足够大时，接近于正态分布。

中心极限定理说明了在许多情况下，样本均值的分布接近于正态分布。

中心极限定理和大数定理有相似的地方。

Lecture3.WeakLawofLargeNumbers

−
µn
> 2
.
(1.5)
We can not apply Markov’s inequality because of lack of integrability. Instead, we note that
(Sn − Snn)/n =
n i=1
(Xi
−
Xin)/n,
and
hence
∀ δ > 0, ∃ M0 > 0, such that for all M > M0 : Note that by Markov’s inequality,
lim
sup
P
n→∞
Sn − SnM > nn 3
P
|µM − µ| > 3
≤ δ.
≤ δ, (1.2)
P
Sn − SnM nn
> 3
3 ≤E
P(|Yn| > ) = E[1{|Yn|> }] ≤ E[|Yn|2 −21{|Yn|> }] ≤ E[Yn2] −2,
1
which tends to 0 if E[Yn2] → 0 as n → ∞. Under the ﬁnite second moment assumption, E[Yn2] is easy to evaluate:
Theorem 1.3 [L1 Weak Law of Large Numbers] Let (Xn)n∈N be a sequence of i.i.d.
random variables with E[X1] = µ ∈ R. Then Sn/n :=
n i=1
Xi/n
converges
in

Funnymoney,fuzzymaths不明的财源，模糊的数字

Funnymoney,fuzzymaths不明的财源，模糊的数字[2012.4.28]Funny money, fuzzy maths不明的财源，模糊的数字The euro zone’s rescue fund欧元区救助基金Funny money, fuzzy maths不明的财源，模糊的数字The IMF’s coffers are fuller, but the euro zone’s “firewall” is still flimsyIMF金库充盈，欧元防火墙依旧薄弱CHRISTINE LAGARDE, the IMF’s managing director, boasted of a “Washington moment”. At its spring meetings in theAmerican capital this month, the fund saw its lending power almost double,thanks to the promise of $430 billion in loans from more than a score of itsmembers. The official goal is to boost a “global firewall” against crisis. The unofficial hope is that a fatter IMF will help ease fears about the euro, by bolstering the €700 billion ($925 billion) that euro-zone economies have pledged in their own rescue funds.IMF总裁克丽丝汀.拉嘉德女士骄傲地宣称“华盛顿时刻”已经实现。

IMF春季会议于本月在华盛顿召开，20多个成员国的注资承诺使IMF可贷资金规模几乎扩大一倍，增加4300亿美元以上。

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is the sequence of t-arithmetic means of the ﬁrst n-terms of {ξi } (deﬁned via a sup-t-norm convolution), mn = and Nes denotes Necessity. M 1 + · · · + Mn , n
n→∞ n→∞
Without lost of generality we can suppose that z < m. Then there exist an index n0 ∈ N and a real number > 0 such that mn − z > for all n ≥ n0 Therefore we get ( lim ξn )(z ) = lim ξn (z ) = lim f [−1] (nf (L((mn − z )/α))) ≤
2
The result
The theorem in question can be stated as follows: Theorem 2.1 Let T be an Archimedean t-norm with an additive generator f and let {ξi = (Mi , α, β )LR , i ∈ N} be a sequence of fuzzy numbers of type L-R. If L and R are twice diﬀerentiable, concave functions, and f is twice diﬀerentiable, strictly convex function and m := lim mn
x1 +x2 =z
sup T (ξ1 (x1 ), ξ2 (x2 )), z ∈ I R
R. Then (according to Let n ∈ N , T be a t-norm and ξ1 , . . . , ξn be fuzzy sets of I Zadeh’s extension principle) their T-arithmetic mean (ξ1 ⊕ · · · ⊕ ξn )/n is deﬁned as ξ1 ⊕ ... ⊕ ξn R. (x) := (ξ1 ⊕ · · · ⊕ ξn )(nx), for all x ∈ I n
On law of large numbers for L-R fuzzy numbers∗
Robert Full´ er † rfuller@ra.abo.ﬁ
Abstract This paper extends the author’s earlier work on the Law of Large Numbers for fuzzy numbers [2] to the case where the fuzzy numbers are of type L-R. Namely, we shall deﬁne a class of Archimedean triangular norms in which the equality lim Nes(mn − ≤ ηn ≤ mn + = 1, for any > 0,
m-α
m
m+β
Figure 1: The limit distribution of ηn , ηn = (m, α, β )LR , if T =’min’.
References
[1] D. Dubois and H. Prade, Additions of Interactive Fuzzy Numbers, IEEE Transactions on Automatic Control, 1981,Vol.26, No.4 926-936. [2] R.Full´ er, A Law of Large Numbers for Fuzzy Numbers, Fuzzy Sets and Systems, 45(1992) 299-303. [3] R. Full´ er and T. Keresztfalvi, t-Norm-based Addition of Fuzzy Intervals, Fuzzy Sets and Systems 51(1992) 155-159. [4] R.Kruse, The Strong Law of Large Numbers for fuzzy Random Variables, Information Sciences, 28(1982) 233-241. [5] M. Miyakoshi and M. Shimbo, A Strong Law of Large Numbers for Fuzzy Random Variables, Fuzzy Sets and Systems, 12(1984) 133-142. [6] M.B. Rao and A. Rashed, Some Comments on Fuzzy Variables, Fuzzy Sets and Systems, 6(1981) 285-292. [7] L.A.Zadeh, Fuzzy Sets as a Basis for a Theory of Possibility, Fuzzy Sets and Systems, 1(1978) 3-28.
Proof. From Theorem 1 [3] it follows that 1 f [−1] nf L[(mn − z )/α] ηn (z ) = f [−1] nf R[(z − mn )/β ] 0 therefore, we have
n→∞
exists and it is ﬁnite, then
n→∞
lim N es(mn − ≤ ηn ≤ mn + ) = 1, 2
for all > 0, where {ηn = (ξ1 ⊕ · · · ⊕ ξn )/n, n ∈ N} is the sequence of T -arithmetic means of the ﬁrst n-terms of {ξi } and mn = M1 + · · · + Mn . n if z = mn if mn − α ≤ z ≤ mn if mn ≤ z ≤ mn + β otherwise
x∈ D
¯ ), where D ¯ is the complement of D, is denoted by N es(ξ |D) The quantity 1 − P os(ξ |D and is interpreted as the grade of necessity of the statement ”D contains the value of ξ ”. If D = [a, b] ⊂ I Rthen instead of N es(ξ |[a, b]) we shall write N es(a ≤ ξ ≤ b) and if D = x, x ∈ I R we write N es(ξ = x). Let {ξn } be a sequence of fuzzy numbers. We say that {ξn } converges pointwise to a fuzzy set ξ (and write limn→∞ ξn = ξ ) if
The ﬁnal version of this paper appeared in: R.Lowen and M.Roubens eds., Proceedings of the Fourth IFSA Congress, Vol. Mathematics, Brussels, 1991 74-77. † Supported by the German Academic Exchange Service (DAAD).
n→∞
R. lim ξn (x) = ξ (x), for all x ∈ I
A triangular norm (t-norm for short) T is said to be Archimedean iﬀ T is continuous and T (x, x) < x, for all x ∈ (0, 1). Every Archimedean t-norm T is representable by a continuous and decreasing function f : [0, 1] × [0, ∞] with f (1) = 0 and T (x, y ) = f [−1] (f (x) + f (y )) where f [−1] is the pseudo-inverse of f , deﬁned by f −1 (y ) if y ∈ [0, f (0)] and f [−1] (y ) = 0 if y ∈ [f (0), ∞]. The function f is the additive generator of T . Let T be a t-norm and let ξ1 , ξ2 be fuzzy sets of the real line, then their T -sum ξ1 ⊕ ξ2 is deﬁned by [1] (ξ1 ⊕ ξ2 )(z ) =
Keywords: Triangular norm, L-R fuzzy number, t-arithmetic mean, possibility, necessity
1
Deﬁnitions
A fuzzy number ξ is a fuzzy set of the real line I R with an unimodal, normalized (i.e. there exists unique a ∈ I R such that ξ (a) = 1) and upper-semicontinuous membership function. An L-R fuzzy number ξ denoted by (M, α, β )LR is deﬁned as ξ (t) = L((M − t)/α) if M − α ≤ t ≤ M , ξ (t) = R((t − M )/β ) if M ≤ t ≤ M + β and ξ (t) = 0