On the existence of statistics intermediate between those of Fermi-Dirac and Bose-Einstein

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[美]R·格伦·哈伯德《宏观经济学》R.GlennHubbard,AnthonyP

Macroeconomics R. GLENN HUBBARD COLUMBIA UNIVERSITY ANTHONY PATRICK O’BRIEN LEHIGH UNIVERSITY MATTHEW RAFFERTY QUINNIPIAC UNIVERSITY Boston Columbus Indianapolis New York San Francisco Upper Saddle RiverAmsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City So Paulo Sydney Hong Kong Seoul Singapore Taipei TokyoAbout the AuthorsGlenn Hubbard Professor Researcher and Policymaker R. Glenn Hubbard is the dean and Russell L. Carson Professor of Finance and Economics in the Graduate School of Business at Columbia University and professor of economics in Columbia’s Faculty of Arts and Sciences. He is also a research associate of the National Bureau of Economic Research and a director of Automatic Data Processing Black Rock Closed- End Funds KKR Financial Corporation and MetLife. Professor Hubbard received his Ph.D. in economics from Harvard University in 1983. From 2001 to 2003 he served as chairman of the White House Council of Economic Advisers and chairman of the OECD Economy Policy Commit- tee and from 1991 to 1993 he was deputy assistant secretary of the U.S. Treasury Department. He currently serves as co-chair of the nonpar-tisan Committee on Capital Markets Regulation and the Corporate Boards Study Group. ProfessorHubbard is the author of more than 100 articles in leading journals including American EconomicReview Brookings Papers on Economic Activity Journal of Finance Journal of Financial EconomicsJournal of Money Credit and Banking Journal of Political Economy Journal of Public EconomicsQuarterly Journal of Economics RAND Journal of Economics and Review of Economics and Statistics.Tony O’Brien Award-Winning Professor and Researcher Anthony Patrick O’Brien is a professor of economics at Lehigh University. He received a Ph.D. from the University of California Berkeley in 1987. He has taught principles of economics money and banking and interme- diate macroeconomics for more than 20 years in both large sections and small honors classes. He received the Lehigh University Award for Distin- guished Teaching. He was formerly the director of the Diamond Center for Economic Education and was named a Dana Foundation Faculty Fel- low and Lehigh Class of 1961 Professor of Economics. He has been a visit- ing professor at the University of California Santa Barbara and Carnegie Mellon University. Professor O’Brien’s research has dealt with such issues as the evolution of the U.S. automobile industry sources of U.S. economiccompetitiveness the development of U.S. trade policy the causes of the Great Depression and thecauses of black–white income differences. His research has been published in leading journals in-cluding American Economic Review Quarterly Journal of Economics Journal of Money Credit andBanking Industrial Relations Journal of Economic History Explorations in Economic History andJournal of PolicyHistory.Matthew Rafferty Professor and Researcher Matthew Christopher Rafferty is a professor of economics and department chairperson at Quinnipiac University. He has also been a visiting professor at Union College. He received a Ph.D. from the University of California Davis in 1997 and has taught intermediate macroeconomics for 15 years in both large and small sections. Professor Rafferty’s research has f ocused on university and firm-financed research and development activities. In particular he is interested in understanding how corporate governance and equity compensation influence firm research and development. His research has been published in leading journals including the Journal of Financial and Quantitative Analysis Journal of Corporate Finance Research Policy and the Southern Economic Journal. He has worked as a consultantfor theConnecticut Petroleum Council on issues before the Connecticut state legislature. He has alsowritten op-ed pieces that have appeared in several newspapers including the New York Times. iii Brief Contents Part 1: Introduction Chapter 1 The Long and Short of Macroeconomics 1 Chapter 2 Measuring the Macroeconomy 23 Chapter 3 The Financial System 59 Part 2: Macroeconomics in the Long Run: Economic Growth Chapter 4 Determining Aggregate Production 105 Chapter 5 Long-Run Economic Growth 143 Chapter 6 Money and Inflation 188 Chapter 7 The Labor Market 231 Part 3: Macroeconomics in the Short Run: Theory and Policy Chapter 8 Business Cycles 271 Chapter 9 IS–MP: A Short-Run Macroeconomic Model 302 Chapter 10 Monetary Policy in the Short Run 363 Chapter 11 Fiscal Policy in the Short Run 407 Chapter 12 Aggregate Demand Aggregate Supply and Monetary Policy 448 Part 4: Extensions Chapter 13 Fiscal Policy and the Government Budget in the Long Run 486 Chapter 14 Consumption and Investment 521 Chapter 15 The Balance of Payments Exchange Rates and Macroeconomic Policy 559 Glossary G-1 Index I-1ivContentsChapter 1 The Long and Short of Macroeconomics 1WHEN YOU ENTER THE JOB MARKET CAN MATTER A LOT ........................................................ 11.1 What Macroeconomics Is About........................................................................... 2 Macroeconomics in the Short Run and in the Long Run .................................................... 2 Long-Run Growth in the United States ............................................................................. 3 Some Countries Have Not Experienced Significant Long-Run Growth ............................... 4 Aging Populations Pose a Challenge to Governments Around the World .......................... 5 Unemployment in the United States ................................................................................. 6 How Unemployment Rates Differ Across Developed Countries ......................................... 7 Inflation Rates Fluctuate Over Time and Across Countries................................................. 7 Econo mic Policy Can Help Stabilize the Economy .. (8)International Factors Have Become Increasingly Important in Explaining Macroeconomic Events................................................................................. 91.2 How Economists Think About Macroeconomics ............................................. 11 What Is the Best Way to Analyze Macroeconomic Issues .............................................. 11 Macroeconomic Models.................................................................................................. 12Solved Problem 1.2: Do Rising Imports Lead to a Permanent Reductionin U.S. Employment. (12)Assumptions Endogenous Variables and Exogenous Variables in EconomicModels ........................................................................................................ 13 Forming and Testing Hypotheses in Economic Models .................................................... 14Making the Connection: What Do People Know About Macroeconomicsand How Do They KnowIt .............................................................................................. 151.3 Key Issues and Questions of Macroeconomics ............................................... 16An Inside Look: Will Consumer Spending Nudge Employers to Hire................................ 18Chapter Summary and Problems ............................................................................. 20 Key Terms and Concepts Review Questions Problems and Applications Data Exercise Theseend-of-chapter resource materials repeat in all chapters.Chapter 2 Measuring the Macroeconomy 23HOW DO WE KNOW WHEN WE ARE IN ARECESSION ........................................................... 23Key Issue andQuestion .................................................................................................... 232.1 GDP: Measuring Total Production and Total Income ..................................... 25 How theGovernment Calculates GDP (25)Production and Income (26)The Circular Flow of Income (27)An Example of Measuring GDP (29)National Income Identities and the Components of GDP (29)vvi CONTENTS Making the Connection: Will Public Employee Pensions Wreck State and Local Government Budgets.................................................................... 31 The Relationship Between GDP and GNP........................................................................ 33 2.2 Real GDP Nominal GDP and the GDP Deflator.............................................. 33 Solved Problem 2.2a: Calculating Real GDP . (34)Price Indexes and the GDP Deflator (35)Solved Problem 2.2b: Calculating the Inflation Rate ..........................................................36 The Chain-Weighted Measure of Real GDP ....................................................................37 Making the Connection: Trying to Hit a Moving Target: Forecasting with “Real-Time Data” .................................................................................. 37 Comparing GDP Across Countries................................................................................... 38 Making the Connection: The Incredible Shrinking Chinese Economy ................................ 39 GDP and National Income .............................................................................................. 40 2.3 Inflation Rates and Interest Rates ....................................................................... 41 The Consumer Price Index .............................................................................................. 42 Making the Connection: Does Indexing Preserve the Purchasing Power of Social Security Payments ................................................................ 43 How Accurate Is theCPI ............................................................................................... 44 The Way the Federal Reserve Measures Inflation ............................................................ 44 InterestRates .................................................................................................................. 45 2.4 Measuring Employment and Unemployment .. (47)Answering the Key Question ............................................................................................ 49 An Inside Look: Weak Construction Market Persists.......................................................... 50 Chapter 3 The Financial System 59 THE WONDERFUL WORLD OFCREDIT ................................................................................... 59 Key Issue and Question .................................................................................................... 59 3.1 Overview of the Financial System ...................................................................... 60 Financial Markets and Financial Intermediaries ................................................................ 61 Making the Connection: Is General Motors Making Cars or Making Loans .................... 62 Making the Connection: Investing in the Worldwide Stock Market . (64)Banking and Securitization (67)The Mortgage Market and the Subprime Lending Disaster (67)Asymmetric Information and Principal–Agent Problems in Financial Markets...................68 3.2 The Role of the Central Bank in the Financial System (69)Central Banks as Lenders of Last Resort ..........................................................................69 Bank Runs Contagion and Asset Deflation ....................................................................70 Making the Connection: Panics Then and Now: The Collapse of the Bank of United States in 1930 and the Collapse of Lehman Brothers in2008 (71)3.3 Determining Interest Rates: The Market for Loanable Funds and the Market forMoney .......................................................................................... 76 Saving and Supply in the Loanable Funds Market ........................................................... 76 Investment and the Demand for Loanable Funds ............................................................ 77 Explaining Movements in Saving Investment and the Real Interest Rate (78)CONTENTS .。

初三英语哲学思考问题单选题40题

初三英语哲学思考问题单选题40题1. When we think about the nature of reality, which of the following statements is correct?A. Reality is only what we can see.B. Reality is determined by our thoughts.C. Reality is independent of human perception.D. Reality changes based on our feelings.答案：C。

本题主要考查对现实本质的哲学理解。

选项A 过于局限，现实不仅仅是我们能看到的。

选项B 是主观唯心主义观点，不符合客观事实。

选项C 符合唯物主义观点，现实是独立于人类感知而存在的。

选项D 现实不会仅仅因为我们的感受而改变。

2. What is the essence of philosophy according to the basic concepts?A. The study of history.B. The exploration of science.C. The reflection on fundamental questions of life and existence.D. The analysis of language.答案：C。

哲学的本质是对生命和存在的基本问题进行反思。

选项 A 历史研究并非哲学的本质。

选项 B 科学探索也不是哲学的本质核心。

选项D 语言分析只是哲学的一个方面，而非本质。

3. In the philosophical view, which one is true about truth?A. Truth is relative and changes over time.B. Truth is absolute and never changes.C. Truth depends on personal belief.D. Truth is something that cannot be known.答案：A。

On the notion of concept I

On the notion of concept IMichael FreundLaLICCUniversity of Paris Sorbonne28rue Serpente75006Paris Franceemail:Michael.Freund@paris4.sorbonne.frAbstractIt is well known that classical set theory is not expressive enough to adequately model categorization and prototype theory.Recent workon compositionality and concept determination showed that the quan-titative solution initially oﬀered by classical fuzzy logic also led toimportant drawbacks.Several qualitative approaches were thereaftertempted,that aimed at modelling membership through ordinal scalesor lattice fuzzy sets.Most of the solutions obtained by these theoreti-cal constructions however are of diﬃcult use in categorization theory.We propose a simple qualitative model in which membership relativeto a given concept f is represented by a function that takes its valuein an abstract set A f equipped with a bounded total order.This func-tion is recursively built through a stratiﬁcation of the set of conceptsat hand based on a notion of complexity.Similarly,the typicality asso-ciated with a concept f will be described using an ordering that takesinto account the characteristic features of f.Once the basic notions ofmembership and typicality are set,the study of compound concepts ispossible and leads to interesting results.In particular,we investigatethe internal structure of concepts,and obtain the characterization ofall smooth subconcepts of a given concept.Keywords categorization,concept,extension,intension,typicality,mem-bership,modular orders,Fuzzy sets,Formal Concepts Analysis.11IntroductionIn this paper we propose a new framework for the study of some basic notions classically used in categorization theory.In particular,we shall be concerned with the problem ofﬁnding a suitable theoretical apparatus to model the notions of membership and typicality that underlie prototype theory.It is well recognized since the work of Eleanor Rosch(17)that membership,for instance,is not an all-or-not matter:the classical set-theoretical or the two-value logic model are of therefore of little use to render count of most of the cognition process.This drove Zadeh and his followers(22)and(23)to propose a representation of concepts by fuzzy sets,membership being mod-elled through a real function with values in the unit interval.Such a repre-sentation nevertheless lead to counterintuitive results:see for instance the seminal papers of Kamp and Partnee and of Osherson and Smith(11)(15) and(16)).At a quite elementary level,for instance,it was observed that the membership degree relative to a compound concept could never be greater than the degree induced by any of its components,a result that cannot be accepted for both theoretical and experimental reasons.Even for elemen-tary concepts,the representation of concepts as quantitative fuzzy sets poses problems:vague concepts like to-be-an-adult or to-lie are given continuous values in the unit interval,but what does it mean to qualify somebody as adult‘with degree.4837’?In particular,as observed by several authors (for instance(13))there is no reason why the same set-the unit interval-should serve as a uniform criterion,being invariably referred to as a measure of membership whatever the concept at hand.True,in practice membership is often evaluated through statistical data,and the membership degree iden-tiﬁed with a simple frequency.But the fact that,say,87individuals out of 100consider a car seat as a piece of furniture by no means involves that,in an agent mind,the membership degree of a car-seat relative to the concept to-be-a-piece-of-furniture is equal.87.These drawbacks led to various solutions which all aimed at replacing the primitive quantitative model by a qualitative one:thus,attention focussed on ordinal scales and on lattice fuzzy sets-see for instance(10)or(23).Fora brief analysis of the most recent work on this area,the reader may refer to(13)or(3).However,we consider that the solutions that were proposed are not fully adapted to model prototype theory,and that they cannot be easily exploited to address the classical questions raised by categorization theory.In a diﬀerent area,Peter G¨a rdenfors(8)or(7)proposed a geometrical2model as a framework for concept theory:a concept is deﬁned as a convex region of a multidimensional space,each dimension corresponding to a basic quality.Convexity is related with a notion of betweenness that is supposed to be meaningful for the relevant quality dimensions:if two objects are ex-emplars of a concept,such will be the case for any object that lies‘between’them.The typical instances of a concept are those which are located‘near the center’of the considered region.This Geometry of Thought,as the au-thor calls it,provides interesting tracks in the analysis of concepts.However, it is mostly based on quantitative notions,which weﬁnd not best appropri-ate to model the cognition process.Furthermore,it does not seem that the distinction between vague and sharp concepts is fully taken into account.For these reasons,we propose to revisit the basic notions linked with cate-gorization theory and treat them from a qualitative point of view.Concerning membership,for instance,and rather than dealing with uniform gradation functions that take their values in the unit interval,we represent member-ship relative to a concept by a function whose set of values depends on the chosen concept.This set is endowed with a total bounded order that can be used to evaluate to which degree a object falls under this concept.We think indeed that such a representation is the most adequate to model notions like: object x plainly falls under the concept f,object x falls deﬁnitely not under the concept f or object x falls more than object y under the concept f.These notions,which are the basis of categorization theory,are also theﬁrsts one should deal with in order to understand the problems that arise with vague concepts:for instance,an agent may consider that an elevator is deﬁnitely less a vehicle than a chairlift,while being unable at the same time to attribute a precise numerical membership degree to any of these items.We propose in this paper an example of construction such an order,by making use of the set of deﬁning features attached to the concepts at hand.Postulating the existence of such a set is part of most of the theories on categorization: see for instance(21),(20),(1),(4)or more recently(2),where a concept is assimilated with a set of properties which things that fall under the concept typically have or are believed to have.These deﬁning features,from the point of view of the agent,help understanding the chosen concept;they are indi-vidually necessary and collectively suﬃcient to decide whether or not an item is an exemplar of this concept.Given a vague concept f,we shall use this associated deﬁning feature set to compare the f-membership of two items in the following way:an object x will be considered as falling less under f than an object y if it falls less than y under the f-deﬁning features.The3circularity of this deﬁnition will be avoided by attributing to each concept a complexity level:the sharp concepts,those for which membership is an all-or-not matter,will be given complexity level0;at level1,we shall rank all the vague concepts whose deﬁning feature set only consists of sharp con-cepts;at level2,we will have the vague concepts whose deﬁning feature set consists of concepts that have complexity level equal to0or1,and so on. This ranking will eventually render possible a recursive deﬁnition of mem-bership,and,consequently,the construction of a membership order among the set of objects at hand.Having represented concepts by means of order-functions poses the prob-lem ofﬁnding an adequate representation of the notion of typicality.Since the work of E.Rosch,a considerable amount of study has been carried out on this notion,and it is now widely accepted that,relative to a given con-cept,objects may be classiﬁed following their degree of typicality.Although a precise and general deﬁnition of this typicality degree is still missing,one generally agrees on the fact that such a degree has to faithfully reﬂect the number of characteristic features attached to the concept at hand,together with the relative pertinence,or the frequency,of these features((14),Chapter 2).Nevertheless the attempts at a rigorous construction are rare,and none of them seem to have gained general recognition.Besides,researchers in this domain restricted themselves to elementary cases,dealing with sharp con-cepts,for which membership is an all-or-not matter,or with concepts with sharp features.In particular,they did not seem to be concerned with situa-tions in which the typicality relative to a concept depends on the membership relative to another concept:in order to determine the relative typicality of a hen as a bird,for instance,they would not consider that it is necessary to ﬁrst evaluate its membership degree relative to the concept to-ﬂy.We think on the contrary that typicality must be determined through membership, and that these two notions are correlatedWe therefore propose the construction of a typicality order,clear and easy to evaluate,that faithfully conforms with our intuition.This order is meant to reﬂect a particular agent’s judgment at a precise time.It is based on the agent’s choice,for each concept,of a an associated characteristic feature set,partially ordered through a salience relation that is meant to evaluate the relative importance of these features.The typicality of two items will be compared by investigating the characteristic features that apply to them,the way they apply to them,and their relative salience.Once we have completed the construction of the typicality order,it will be possible to deﬁne the typical4instances of a concept as those that have maximal order,that is those that fall under all the characteristic features of this concept.This deﬁnition of typicality will then enable us to deﬁne the intension of a concept as the set of features that apply to all typical instances of the concept.Thus,the intension of a concept may be interpreted as the set of characteristic features that agents belonging to a well-deﬁned cultural environment would generally agree to associate with this concept:it enlarges the more subjective notion of characteristic features sets.A coherent theory of typicality must be able to correctly address the problem of compound concepts.We shall show that our formalism pro-vides natural and intuitive answers concerning composed concepts,provided one departs from the idea that the logic of concepts boils down to a simple propositional calculus.Indeed we do not agree with the commonly admit-ted postulate following which the negation of a concept,the conjunction or the disjunction of two concepts should be again a concept:we do not consider that not-to-be-an-apple or(to-be-an-apple)&(to-be-a-pear)are con-cepts.Consequently,we believe that the treatment of such sentences,which clearly goes beyond the limits of the elementary concept theory we are deal-ing with,should be addressed only after a coherent logical framework for categorization has been proposed.In the present work,we shall therefore content ourselves with a language that only admits a single partial operator, the determination connective,which is meant to represent the determination of a principal concept by a secondary one:for instance,the concept to-be-a-green-apple is the determination of the principal concept to-be-an-apple by the secondary one,to-be-green.Concept determination is not compositional, except in some limit cases:this means that neither the membership,nor the typicality relative to a composed concept can be directly evaluated through a computation of the corresponding magnitudes of its components.However, it remains possible to determine the typical order,hence the typical instances of a composed concept,via the typicality orders induced by its components. This result is important as it can be considered as an answer to the compo-sitionality problem.Plan of this paperAfter introducing in section2the framework we are going to work in and recall the distinction between sharp and vague concepts,we shall introduce in section3the membership orders and functions associated with elemen-5tary concepts.In section4,we shall present the determination connective and extend the membership order to compound concepts.We shall then turn to typicality,and build in section5the typicality order associated with elementary and compound concepts.In section6,we show how the notion of smooth subconcepts can be formalized through the determination con-nective,and we propose an interpretation of our results in the language of Formal Concept Analysis.Section7is a conclusion in which we discuss our future work.2Concepts and objectsWe denote by O the universe of discourse,which we may see as the set of all objects,real orﬁctive,that an agent has at his disposal.Together with this set,we suppose given a set F of concepts.These concepts constitute the elementary items on which the agent builds its reasoning process,and they reﬂect its knowledge on the world at a given time.A concept applies to an object if it describes a property that this object possesses,or if it is an attribute of this object.For instance,the concept to-be-a-fruit applies to the object an-apple.We will say indiﬀerently that the concept f applies to the object x,that x falls under f,or that x is an instance of f.In the classical theory,where categories were modelled through set theory,membership rel-ative to a concept was an all-or-none matter:an object could not partially fall under a concept.This perspective was also that of Frege(5),for whom concepts were deﬁned as one-place predicates having a bivalent membership truth function.With prototype theory and the evidence that there existed vague concepts(eg.to-be-a-lie,to-be-an-adult,to-be-employed,to-be-a-sand-heap etc),it became clear that this primitive notion of concepts had to be enlarged and that membership was a question of degree,rather than an all-or-none matter.As observed in(11),“We all have strong intuitions that the concepts encoded by many natural-language predicates are vague;whether something is a chair,or is red,does not seem to be an all-or-none matter but a matter of degree;there may be some clear positive cases and some clear negative cases,but there are many unclear cases in between.”Sharp concepts are deﬁned as those for which membership is an all-or-not matter:an object simply falls or does not fall under such a concept,with-out the possibility of taking intermediate values.To-be-a-human-being,to-be-a-tooth-brush,to-be-an-even-integer may provide examples of sharp con-6cepts.This deﬁnition has nevertheless to be understood as tightly related to a given agent’s point of view,and we shall always consider that we work from a particular subjective perspective,and at a particular time:the same concept may appear as sharp to a non-expert agent while being considered as vague for an expert.For vague concepts,membership is indeed not an all-or-not-matter:such are for instance the concepts to-be-a-lie,to-be-poor, to-be-employed,to-be-a-weapon-of-mass-destruction or to-be-a-mammal.In-deed,politeness sometimes drives us to make compliments that,although not sincere,cannot be considered as real lies;to be poor or to be employed is clearly a matter of degree;a gun is more a WMD than a knife;and the platypus is and is not a mammal.Of course,opinions may diﬀer whether a given concept should be considered as a sharp or a vague one,but,and this is the important point,it is well recognized that both kinds of concepts exist. An interesting suggestion of(1)is that,for noun concepts,the opposition be-tween nominal and non-nominal categories reﬂects the duality between vague and sharp concepts:nominal categories can be deﬁned through their deﬁning features,and may therefore give rise to vague concepts,while non-nominal cannot.Non-nominal categories may be themselves divided between natural kind categories(eg:the category of tigers or of games)and artifact cate-gories(eg:the category of hammers,walls,cars).Note that the distinction between nominal and natural kind concepts is far from being evident:a same concept may be considered as nominal for an expert,and as non-nominal for a non-expert agent.For instance,the concept to-be-a-bird is undoubtedly of a natural kind for a child,but it may turn later to a nominal one once the child has learnt that all and only those animals that have beak and feathers are to be considered as birds.In deciding whether the concept to-be-a-bird is or not a sharp concept,we have therefore toﬁrst analyze which of these two concepts we are referring to:an agent aware that birdhood may be deﬁned through the sum of a certain number of conditions,will consider to-be-a-bird a vague concept:the octopus,for instance will be more a bird than the bat, since the octopus has a beak.On the other hand,for a child,to-be-a-bird is bond to be a sharp concept,and the penguin will simply not be a member of the category,while the bat will.In the present work,we shall leave the problem how to determine which concepts are vague and which are not.We shall only be concerned with the problem ofﬁnding an adequate model that correctly describes how the notion of membership is used in a given agent’s behavior.73Membership for elementary conceptsIn the original fuzzy logic model,a membership degree function is attributedto each concept,measuring how accurately this concept applies to the objectsat hand.This degree however is not explicitly present in an agent’s mind:this is so for example for young children,for whom notions like real numbersor unit interval are totally meaningless.Nevertheless,given a concept,the agent will be generally able to decide whether two objects have the same ordiﬀerent membership degrees,and which one,in the latter case,has higher degree:for instance,the agent may decide that the concept to-be-a-piece-of-furniture applies more to a car-seat than to a blackboard,without being ableat the same time to attribute a numerical membership degree to any of these items.In other words,the agent associates with each concept f an implicit notion of a membership order.It is this order we now want to build.We shallﬁrst deal with elementary concepts,leaving the case of com-pound concepts in the next section.In order to correctly deﬁne a suitable notion of membership for vague concepts,we start from the widely accepted theory following which each such concept f is given together with aﬁnite auxiliary set∆f which,from the point of view of the agent,includes allthe features that explain or illustrate f,helping diﬀerentiating it from its neighboring concepts.For instance,for the concept to-be-a-bird,the corre-sponding∆f may consist of the concepts to-be-a-vertebrate,to-have-a-beakand to-have-feathers;for the concept to-be-a-tent,it may list the featuresto-be-a-shelter,to-be-made-of-cloth.We interpret∆f as the set of deﬁning features an agent or a group of agents would associate with f.The sets∆fmay be seen as the outputs a dictionary or an encyclopedia would return when given vague concepts as inputs.The elements of∆f are supposed tobe less complex than the root concept f:in the agent’s mind,they constitutean help for the understanding of f.This notion of complexity will be now given a precise meaning by attributing a complexity level c(f)to the set Fof concepts at hand in the following way:•Sharp concepts are given complexity level0.•If∆f consists of sharp concepts,set c(f)=1.•If c(g)has been deﬁned for all concepts g of∆f,set c(f)=1+Max(c(g))g∈∆f We shall make the assumption that this procedure attributes a well-deﬁned complexity level to every element of F.In other words,our theory8only applies to a set F that consists of concepts that either are sharp,or can be recursively deﬁned through sharp concepts.As a matter of fact,most of the elementary concepts one usually deals with have a small complexity level,and we could have made the assumption that the set of concepts at hand solely consists of concepts f of level less than3.However weﬁnd it more convenient to work in a more general framework,as the results are not more diﬃcult to establish.It may be the case that some elements of∆f are more important than others,when considered as a help for deﬁning or illustrating f:for instance, given the concept to-be-a-bird,an agent may think that the feature to-have-wings is more salient than the feature to-be-an-animal,while both features may be part of the same set∆f.Thus,it is necessary to endow each set∆f with a(possibly empty)salience relation that reﬂects the relative importance of its elements as deﬁning features of f.In its most general form,such a relation will be represented by a strict partial order>f.This order has to be taken into account when comparing the f-membership of two items: an object x that falls under the most salient deﬁning features of f will be considered a better instance of f than an object y that only falls under some non-salient deﬁning feature of f.We can now proceed to the construction of the membership preorder rela-tion µf ,which will be deﬁned on the set of objects O,and to the constructionof the membership functionϕf,which will take its values in a totally ordered set(A f,<f).We shall omit the subscripts when there is no ambiguity.We begin with the simplest case of sharp concepts:Deﬁnition1For every elementary sharp concept f,A f is the set{0,1}, andϕf the function:ϕf(x)=1if x falls under f andϕf(x)=0otherwise. The associated membership preorder is deﬁned by x µfy ifϕf(x)≤ϕf(y).The membership preorder and the membership function relative to an arbitrary elementary concept f will be now deﬁned by induction on c(f). This will be done in two steps.3.1The elementary membership orderDeﬁnition2Let f be an elementary concept,and suppose that the totally ordered sets(A g,<g)and the membership functionsϕg have been deﬁned forall elementary concepts g such that c(g)<c(f).The relation µf is thendeﬁned by:9x µf y if for any concept h of∆f such thatϕh(y)<hϕh(x),there exists a concept k of∆f,k>f h,such thatϕk(x)<kϕk(y).The relation µf thus compares the ways objects inherit the deﬁning fea-tures of f and takes into account the relative salience of these features.We will say that a preorder of this type is induced by the(ordered)set∆f.In the particular case where the salience order on∆f is empty,the relation boilsdown to:x µf y if and only ifϕh(x)≤hϕh(y)for all h in∆f,that is if andonly if no deﬁning feature of f applies more to x than to y.The hypothesis that,for k∈∆f,the membership functionsϕk take theirvalue in a totally ordered set guarantees the transitivity of the relation µf .More precisely we have the following result:Lemma1For any elementary concept f,the relation µf is a partial pre-order on O.Proof:We have to prove that µf is a reﬂexive and transitive relation.Reﬂexivity is immediate.For transitivity,suppose that x,y and z are threeobjects such that x µf y and y µfz.We want to show that x µfz.Supposing that there exists a concept h of∆f such thatϕh(z)<ϕh(x),we have to prove the existence of a concept k∈∆f,k more salient than h,such thatϕk(x)<ϕk(z).We make a proof by cases:•Supposeﬁrst thatϕh(x)≤ϕh(y).Then we haveϕh(z)<ϕh(y),and there exists therefore a concept k of∆f,k>f h,such thatϕk(y)<ϕk(z).We can suppose that k is maximal in∆f for this property (∆f is aﬁnite set).Ifϕk(x)≤ϕk(y),we getϕk(x)<ϕk(z)and we are done.Ifϕk(y)<ϕk(x),the hypotheses imply that there exists a concept g in∆f,g>f k such thatϕg(x)<ϕg(y).We cannot have ϕg(z)<ϕg(y),otherwise there would exist a concept l in∆f,l>f g, such thatϕl(y)<ϕl(z),which would contradict the maximality of k.We have thereforeϕg(y)≤ϕg(z)and it follows thatϕg(x)<ϕg(z)as desired.•Suppose now that we haveϕh(y)<ϕh(x).There exists k∈∆f,k>f h, such thatϕk(x)<ϕk(y).Again,we can suppose that k is maximal in∆f for these properties.Ifϕk(y)≤ϕk(z),we getϕk(x)<ϕk(z), as desired.If on the contrary we haveϕk(z)<ϕk(y),there exists a concept g in∆f,g>f k,such thatϕg(y)<ϕg(z).As before,the10maximality of k implies that we necessarily have ϕg (x )≤ϕg (y ).It follows that ϕg (x )<ϕg (z ),and the proof is complete.Let us denote by ≺µf the relation:x ≺µf y iﬀx µf y and not y µf x .Itfollows from the above lemma that ≺µf is a strict partial order on O .Example 1Let f be the concept to-be-a-bird ,and suppose that,from the point of view of an agent,its deﬁning feature set is given by ∆f ={to-be-an-animal ,to-have-two legs ,to-lay-eggs ,to-have-a-beak ,to-have-wings },all of these concepts being considered as sharp concepts for the agent.Suppose also that the salience order is given by:to-lay-eggs >f to-have-two-legs ,to-have-a-beak >f to-lay-eggs and to-have wings >f to-lay-eggsLet r ,m ,t ,b and d respectively stand for a robin,a mouse,a tortoise,a bat and a dragonﬂy,and let us compare their relative birdhood.In order to determine the induced membership order,we ﬁrst build the following array:animaltwo −legs lay −eggs beak wings robinmousetortoisebatdragonﬂyWe readily check that d ≺µf r ,m ≺µf t ,and m ≺µf b .Note that we haveb µf d ,since the concept to-have-two-legs under which the bat falls,contrary to the dragonﬂy,is dominated by the concept to-lay-eggs that applies to the dragonﬂy and not to the bat.On the other hand,we do not have d µf b ,as nothing compensates the fact that the dragonﬂy lays eggs and the bat does not.This yields b ≺µf d .We also remark that the tortoise and the bat are incomparable,that is,we have neither b µf t ,nor t µf b .The strict f -membership order induced on these ﬁve elements is thus given by the following Hasse diagramm:mb d t r &&11We have therefore m≺µf b≺µfd≺µfr and m≺µft≺µfr.We can now precisely translate the notion of membership:an object xwill be considered as falling under f if x is≺µf -maximal in O.We shalldenote by Ext f,the extension of f,the set of all such objects.We close this paragraph with a technical lemma:Lemma2The double inequality x µf y and y µfx holds if and only ifϕh(x)=ϕh(y)for all concepts h of∆f.Proof:Ifϕh(x)=ϕh(y)for all concepts h of∆f,we have clearly x µf y andy µf x.Conversely,suppose that x µfy and y µfx.If we had notϕh(x)=ϕh(y)∀h∈∆f,there would exist a concept h of∆f such thatϕh(x)=ϕh(y), and we could choose h with maximal salience for this property.We wouldhave for instanceϕh(x)<hϕh(y).But since y µf x,there would existk∈∆f,k more salient than h,such thatϕk(y)<kϕk(x),thus contradicting the choice of h.3.2The membership functionIt is clear that the ordering given by the relation µf is not connected:giventwo objects x and y,it may well happen that neither x µf y,nor y µfx.It is nevertheless possible,starting from the strict partial order≺µf ,to build,a membership functionϕf that fairly translates the notion of a degree off-membership.This function will satisfyϕf(x)<ϕf(y)whenever x≺µf y:in a sense,this is the best one can hope(see(12)and her discussion on the impossibility for order relations to correctly represent vagueness).For this purpose,we shall proceed in a way that parallels,though in diﬀerent context, a construction we proposed in(?).Given an object x,we say that x initializes a membership chain of length n if it is possible toﬁnd n objects x1,x2,...,x n with last term x n∈Ext f,such that x≺µf x1≺µfx2≺f...≺µfx n.For instance,any element x∈Ext finitializes a chain of length0,and any object that does not fall under f initializes an membership-chain of strictly positive length l≤|∆f|.In a sense,the length of such a chain measures how distant x is from the set Ext f.Note that,given an object x,the existence and the length of such a chain is determined by the concepts and the objects the agent has at his disposal.Each link of a chain corresponds for this agent to a real(or aﬁctive) given object,together with some given concepts of the universe at hand.12。

证明数据论点的英语作文

The Power of Data in ArgumentationIn the world of argumentation, data holds immense power. It is the cold, hard evidence that can either make or break an argument. Without data, an argument often lackscredibility and persuasiveness. This essay explores the significance of data in argumentation, discussing how itcan strengthen an argument and make it more convincing.First and foremost, data provides a solid foundationfor an argument. When backed by reliable data, an argument becomes more credible and difficult to refute. For example, in a debate about the effectiveness of a new policy, presenting statistical data on its impact on the economy, society, or the environment can significantly strengthenone's position. Data not only adds weight to an argumentbut also helps to establish its legitimacy.Moreover, data can be used to refute opposing arguments. By presenting counter-data, one can effectively challenge the validity of an opponent's claims. For instance, in a discussion about the safety of a particular product, citing statistics on the number of accidents caused by its use can effectively undermine the opponent's argument that it issafe. Data, when used effectively, can turn the tide of an argument in one's favor.Additionally, data can help to clarify complex issues and make them easier to understand. By breaking down complex problems into manageable chunks of data, one can make them more accessible to a wider audience. This, in turn, increases the chances of说服他人接受自己的观点. For instance, in an essay arguing for the need for environmental conservation, presenting data on the rate of deforestation, climate change, and the impact of these issues on human health can help readers understand the urgency of the problem and the need for action.However, it is important to note that not all data is created equal. The credibility of an argument can be compromised if the data presented is incomplete, outdated, or biased. Therefore, it is crucial to ensure that the data used in an argument is reliable, accurate, and representative of the larger population. This involves conducting thorough research, cross-checking sources, and analyzing data critically.In conclusion, data is an essential tool in argumentation. It adds credibility, helps to refute opposing arguments, clarifies complex issues, and makes arguments more convincing. However, to ensure the effectiveness of data in argumentation, it is important to ensure its reliability, accuracy, and representativeness. By doing so, one can turn raw data into powerful ammunition in the battle of ideas.**数据论证的力量**在论证的世界里，数据具有巨大的力量。

《数理统计》课程思政教学的最新探索和实践

Creative Education Studies 创新教育研究, 2023, 11(10), 2986-2990Published Online October 2023 in Hans. https:///journal/ceshttps:///10.12677/ces.2023.1110440《数理统计》课程思政教学的最新探索和实践李龙，朱笑雨*苏州科技大学数学科学学院，江苏苏州收稿日期：2023年8月23日；录用日期：2023年9月21日；发布日期：2023年10月9日摘要《数理统计》是统计专业基础课程，是研究随机现象客观规律性的数学学科，旨在从海量的实际数据中，挖掘具有实用价值的信息。

当前世界处于百年未有之变局，统计学作为第四次工业革命的最重要理论基础之一，正在影响着社会的方方面面。

传统的统计学课程思政内容侧重统计学理论本身，缺乏与当代最前沿的经济、社会、科技的联系，因此需要在传统思政内容的基础上，建设更加符合当代要求的思政课程。

文章从统计学在生命科学、大数据、人工智能、新能源、数字经济等国家发展重要领域的应用为切入点，可以很好地结合社会热点进行思政教育。

关键词《数理统计》，课程思政，第四次工业革命The New Exploration and Practice ofIdeological and Political Teaching in theCourse “Mathematical Statistics”Long Li, Xiaoyu Zhu*School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou JiangsuReceived: Aug. 23rd, 2023; accepted: Sep. 21st, 2023; published: Oct. 9th, 2023Abstract“Mathematical Statistics” is a basic course of statistics, which is a mathematical discipline that stu-dies the objective law of random phenomena, aiming to mine information with practical value from massive actual data. At present, the world is in a century of unprecedented changes, statistics, *通讯作者。

英语哲学思想解读50题

英语哲学思想解读50题1. The statement "All is flux" was proposed by _____.A. PlatoB. AristotleC. HeraclitusD. Socrates答案：C。

本题考查古希腊哲学思想家的观点。

赫拉克利特提出了“万物皆流”的观点。

选项A 柏拉图强调理念论；选项B 亚里士多德注重实体和形式；选项D 苏格拉底主张通过对话和反思来寻求真理。

2. "Know thyself" is a famous saying from _____.A. ThalesB. PythagorasC. DemocritusD. Socrates答案：D。

此题考查古希腊哲学家的名言。

“认识你自己”是苏格拉底的名言。

选项A 泰勒斯主要研究自然哲学；选项B 毕达哥拉斯以数学和神秘主义著称；选项C 德谟克利特提出了原子论。

3. Which philosopher believed that the world is composed of water?A. AnaximenesB. AnaximanderC. ThalesD. Heraclitus答案：C。

本题考查古希腊哲学家对世界构成的看法。

泰勒斯认为世界是由水组成的。

选项A 阿那克西美尼认为是气；选项B 阿那克西曼德认为是无定；选项D 赫拉克利特提出万物皆流。

4. The idea of the "Forms" was put forward by _____.A. PlatoB. AristotleC. EpicurusD. Stoics答案：A。

这道题考查古希腊哲学中的概念。

柏拉图提出了“理念论”，即“形式”。

选项B 亚里士多德对其进行了批判和发展；选项C 伊壁鸠鲁主张快乐主义；选项D 斯多葛学派强调道德和命运。

5. Who claimed that "The unexamined life is not worth living"?A. PlatoB. AristotleC. SocratesD. Epicurus答案：C。

glivenko-cantelli格里文科定理证明

1The Glivenko-Cantelli TheoremLet X i,i=1,...,n be an i.i.d.sequence of random variables with distribu-tion function F on R.The empirical distribution function is the function ofx deﬁned byˆFn(x)=1n1≤i≤nI{X i≤x}.For a given x∈R,we can apply the strong law of large numbers to the sequence I{X i≤x},i=1,...n to assert thatˆFn(x)→F(x)a.s(in order to apply the strong law of large numbers we only need to show that E[|I{X i≤x}|]<∞,which in this case is trivial because|I{X i≤x}|≤1).In this sense,ˆF n(x)is a reasonable estimate of F(x)for a given x∈R. But isˆF n(x)a reasonable estimate of the F(x)when both are viewed as functions of x?The Glivenko-Cantelli Thoerem provides an answer to this question.It asserts the following:Theorem1.1Let X i,i=1,...,n be an i.i.d.sequence of random variables with distribution function F on R.Then,supx∈R|ˆF n(x)−F(x)|→0a.s.(1) This result is perhaps the oldest and most well known result in the very large ﬁeld of empirical process theory,which is at the center of much of modern econometrics.The statistic(1)is an example of a Kolmogorov-Smirnov statistic.We will break the proof up into several steps.Lemma1.1Let F be a(nonrandom)distribution function on R.For each >0there exists aﬁnite partition of the real line of the form−∞=t0< t1<···<t k=∞such that for0≤j≤k−1F(t−j+1)−F(t j)≤ .1Proof:Let >0be given.Let t0=−∞and for j≥0deﬁnet j+1=sup{z:F(z)≤F(t j)+ }.Note that F(t j+1)≥F(t j)+ .To see this,suppose that F(t j+1)<F(t j)+ .Then,by right continuity of F there would existδ>0so that F(t j+1+δ)< F(t j)+ ,which would contradict the deﬁnition of t j+1.Thus,between t j and t j+1,F jumps by at least .Since this can happen at most aﬁnite number of times,the partition is of the desired form,that is−∞=t0< t1<···<t k=∞with k<∞.Moreover,F(t−j+1)≤F(t j)+ .To see this, note that by deﬁnition of t j+1we have F(t j+1−δ)≤F(t j)+ for allδ>0.The desired result thus follows from the deﬁnition of F(t−j+1).Lemma1.2Suppose F n and F are(nonrandom)distribution functions on R such that F n(x)→F(x)and F n(x−)→F(x−)for all x∈R.Thensupx∈R|F n(x)−F(x)|→0.Proof:Let >0be given.We must show that there exists N=N( )such that for n>N and any x∈R|F n(x)−F(x)|< .Let >0be given and consider a partition of the real line intoﬁnitely many pieces of the form−∞=t0<t1···<t k=∞such that for0≤j≤k−1F(t−j+1)−F(t j)≤2.The existence of such a partition is ensured by the previous lemma.For any x∈R,there exists j such that t j≤x<t j+1.For such j,F n(t j)≤F n(x)≤F n(t−j+1)F(t j)≤F(x)≤F(t−j+1),which implies thatF n(t j)−F(t−j+1)≤F n(x)−F(x)≤F n(t−j+1)−F(t j).2Furthermore,F n(t j)−F(t j)+F(t j)−F(t−j+1)≤F n(x)−F(x)F n(t−j+1)−F(t−j+1)+F(t−j+1)−F(t j)≥F n(x)−F(x).By construction of the partition,we have thatF n(t j)−F(t j)−2≤F n(x)−F(x)F n(t−j+1)−F(t−j+1)+2≥F n(x)−F(x).For each j,let N j=N j( )be such that for n>N jF n(t j)−F(t j)>− 2and let M j=M j( )be such that for n>M jF n(t−j )−F(t−j)<2.Let N=max1≤j≤k max{N j,M j}.For n>N and any x∈R,we have that|F n(x)−F(x)|< .The desired result follows.Lemma1.3Suppose F n and F are(nonrandom)distribution functions on R such that F n(x)→F(x)for all x∈Q.Suppose further that F n(x)−F n(x−)→F(x)−F(x−)for all jump points of F.Then,for all x∈R F n(x)→F(x)and F n(x−)→F(x−).Proof:Let x∈R.Weﬁrst show that F n(x)→F(x).Let s,t∈Q such that s<x<t.First suppose x is a continuity point of F.Since F n(s)≤F n(x)≤F n(t)and s,t∈Q,it follows thatF(s)≤lim infn→∞F n(x)≤lim supn→∞F n(x)≤F(t).Since x is a continuity point of F,lim s→x−F(s)=limt→x+F(t)=F(x),3from which the desired result follows.Now suppose x is a jump point of F .Note thatF n (s )+F n (x )−F n (x −)≤F n (x )≤F n (t ).Since s,t ∈Q and x is a jump point of F ,F (s )+F (x )−F (x −)≤lim inf n →∞F n (x )≤lim sup n →∞F n (x )≤F (t ).Sincelim s →x −F (s )=F (x −)lim t →x +F (t )=F (x ),the desired result follows.We now show that F n (x −)→F (x −).First suppose x is a continuity point of F .Since F n (x −)≤F n (x ),lim sup n →F n (x −)≤lim sup n →F n (x )=F (x )=F (x −).For any s ∈Q such that s <x ,we have F n (s )≤F n (x −),which implies thatF (s )≤lim inf n →∞F n (x −).Sincelim s →x −F (s )=F (x −),the desired result follows.Now suppose x is a jump point of F .By as-sumption,F n (x )−F n (x −)→F (x )−F (x −),and,by the above argument,F n (x )→F (x ).The desired result follows.Proof of Theorem 1.1:If we can show that there exists a set N suchthat Pr {N }=0and for all ω∈N (i)ˆFn (x,ω)→F (x )for all x ∈Q and (ii)ˆFn (x,ω)−F n (x −,ω)→F (x )−F (x −)for all jump points of F ,then the result will follow from an application of Lemmas 1.2and 1.3.For each x ∈Q ,let N x be a set such that Pr {N x }=0and for all ω∈N x ,ˆF n (x,ω)→F (x ).Let N 1= x ∈Q .Then,for all ω∈N 1,ˆF n (x,ω)→F (x )by construction.Moreover,since Q is countable,Pr {N 1}=0.4For integer i ≥1,let J i denote the set of jump points of F of size at least 1/i .Note that for each i ,J i is ﬁnite.Next note that the set of all jump points of F can be written as J = 1≤i<∞J i .For each x ∈J ,let M x denotea set such that Pr {M x }=0and for all ω∈M x ,ˆF n (x,ω)−F n (x −,ω)→F (x )−F (x −).Let N 2= x ∈J M x .Since J is countable,Pr {N 2}=0.To complete the proof,let N =N 1∪N 2.By construction,for ω∈N ,(i)and (ii)hold.Moreover,Pr {N }=0.The desired result follows.2The Sample MedianWe now give a brief application of the Glivenko-Cantelli Theorem.Let X i ,i =1,...,n be an i.i.d.sequence of random variables with distribution F .Suppose one is interested in the median of F .Concretely,we will deﬁneMed(F )=inf {x :F (x )≥12}.A natural estimator of Med(F )is the sample analog,Med(ˆFn ).Under what conditions is Med(ˆFn )a reasonable estimate of Med(F )?Let m =Med(F )and suppose that F is well behaved at m in the sense that F (t )>12whenever t >m .Under this condition,we can show usingthe Glivenko-Cantelli Theorem that Med(ˆFn )→Med(F )a.s.We will now prove this result.Suppose F n is a (nonrandom)sequence of distribution functions such thatsup x ∈R |F n (x )−F (x )|→0.Let >0be given.We wish to show that there exists N =N ( )such that for all n >N|Med(F n )−Med(F )|< .Choose δ>0so thatδ<12−F (m − )δ<F (m + )−12,5which in turn implies thatF(m− )<12−δF(m+ )>12+δ.(It might help to draw a picture to see why we should pickδin this way.) Next choose N so that for all n>N,supx∈R|F n(x)−F(x)|<δ.Let m n=Med(F n).For such n,m n>m− ,for if m n≤m− ,thenF(m− )>F n(m− )−δ≥12−δ,which contradicts the choice ofδ.We also have that m n<m+ ,for if m n≥m+ ,thenF(m+ )<F n(m+ )+δ≤12+δ,which again contradicts the choice ofδ.Thus,for n>N,|m n−m|< ,as desired.By the Glivenko-Cantelli Theorem,it follows immediately that Med(ˆF n)→Med(F)a.s.6。

Fractal and Multifractal Time Series

$Fractal and Multifractal Time Series$

2
• Self-aﬃne system: Generalization of a fractal system, where diﬀerent magniﬁcations s and s = sH have to be used for diﬀerent directions in order to obtain a statistically equivalent magniﬁcation. The exponent H is called Hurst exponent. Self-aﬃne time series and time series becoming self-aﬃne upon integration are commonly denoted as fractal using a less strict terminology. • Multifractal system: A system characterised by scaling laws with an inﬁnite number of diﬀerent fractal exponents. The scaling laws must be valid for the same range of the scale parameter. • Crossover: Change point in a scaling law, where one scaling exponent applies for small scale parameters and another scaling exponent applies for large scale parameters. The center of the crossover is denoted by its characteristic scale parameter s× in this article. • Persistence: In a persistent time series, a large value is usually (i. e., with high statistical preference) followed by a large value and a small value is followed by a small value. A fractal scaling law holds at least for a limited range of scales. • Short-term correlations: Correlations that decay suﬃciently fast that they can be described by a characteristic correlation time scale; e. g., exponentially decaying correlations. A crossover to uncorrelated behaviour is observed on larger scales. • Long-term correlations: Correlations that decay suﬃciently slow that a characteristic correlation time scale cannot be deﬁned; e. g., power-law correlations with an exponent between 0 and 1. Power-law scaling is observed on large time scales and asymptotically. The term long-range correlations should be used if the data is not a time series. • Non-stationarities: If the mean or the standard deviation of the data values change with time, the weak deﬁnition of stationarity is violated. The strong deﬁnition of stationarity requires that all moments remain constant, i. e., the distribution density of the values does not change with time. Non-stationarities like monotonous, periodic, or step-like trends are often caused by external eﬀects. In a more general sense, changes in the dynamics of the system also represent non-stationarities.

【翻译笔记】The Tyranny of Metrics

The Tyranny of Metrics翻译笔记1.描述修饰“去形象”主要是指英文喜欢在名词前添加合理的形容词，或者用拟人化形象的动词，以提升语言张力和可读性，在译为中文时如果难以对应可以大胆去掉，特别氏形容词氏大家已知或常识信息的情况下更是如此。

英中切换是形象词一般要去形象化。

Example1原文：Uber riders earn stars for their back-seat behaviour.Social-media posts attract“likes”.Users of dating sites are assigned desirability scores.Apple’siPhones tell their owners how many hours they have spent peering into theirscreens.译文：优步乘客根据乘车表现被评以不同星级；社交媒体上的帖子求“赞”；交友网站的用户被系统评定吸引力指数。

苹果手机记录用户的屏幕使用时间。

分析：译者需要运用“预知”能力，包括对语言的感知，即常见搭配、常用意义，还包括信息的预测，那就要靠平时扩大知识面，搜集生活经验了。

比如back-seat behaviour，和desirability scores这两个词的翻译，虽然可能没有见过同样表达，但是根据生活经验，我们使用优步或滴滴打车，系统会对用户评分，在用车高峰期的时候，评分高的乘客往往被接单的可能性更大，而评分基础就是乘客在用车时的表现。

交友网站用户被打分也是很容易理解的，游戏中也有类似打分制度，游戏中的打分可能代表难度等级，交友网站用户的我们就可以推测代表吸引力、魅力值之类的。

Peer into本意为窥探，凝视，含有情感色彩，但我们常识中苹果手机会告诉我们“屏幕使用时间”，所以这里的翻译不用把peer into的情感色彩表现出来。

Example2原文：That urge is now the premise of one of Apple’s latest products,a watch that keeps tabs on the wearer’s heart rate.译文：正是这种需求促成了苹果最新一代产品的推出——一款可以记录佩戴者心率的手表。

历届诺贝尔经济学奖获得者演讲词

10 Economic Sciences 19691. T HE L URES OF U NSOLVABLE P ROBLEMSDeep in the human nature there is an almost irresistible tendency to concen-trate physical and mental energy on attempts at solving problems that seem to be unsolvable. Indeed, for some kinds of active people only the seemingly un-solvable problems can arouse their interest. Other problems, those which can reasonably be expected to yield a solution by applying some time, energy and money, do not seem to interest them. A whole range of examples illustrating this deep trait of human nature can be mentioned.The mountain climber. The advanced mountain climber is not interested in fairly accessible peaks or fairly accessible routes to peaks. He becomes enthu-siastic only in the case of peaks and routes that have up to now not been con-quered.The Alchemists spent all their time and energy on mixing various kinds of matter in special ways in the hope of producing new kinds of matter. To produce gold was their main concern. Actually they were on the right track in prin-ciple, but the technology of their time was not advanced far enough to assure a success.The alluring symmetry problem in particle physics. Around 1900, when the theory of the atom emerged, the situation was to begin with relatively simple. There were two elementary particles in the picture: The heavy and positively charged PROTON and the light and negatively charged ELECTRON. Subsequently one also had the NEUTRON, the uncharged counterpart of the proton. A normal hydrogen atom, for instance, had a nucleus consisting of one proton, around which circulated (at a distance of 0.5. 10-18 cm) one electron. Here the total electric charge will be equal to 0. A heavy hydrogen atom (deuterium) had a nucleus consisting of one proton and one neutron around which circu-lated one electron. And similarly for the more complicated atoms.This simple picture gave rise to an alluring and highly absorbing problem. The proton was positive and the electron negative. Did there exist a positively charged counterpart of the electron? And a negatively charged counterpart of the proton? More generally: Did there exist a general symmetry in the sense that to any positively charged particle there corresponds a negatively charged counter-part, and vice versa? Philosophically and mathematically and from the view-point of beauty this symmetry would be very satisfactory. But it seemed to be an unsolvable problem to know about this for certain. The unsolvability, however, in this case was only due to the inadequacy of the experimental technology of the time. In the end the symmetry was completely established even experimentally. The first step in this direction was made for the light particles (because here the radiation energy needed experimentally to produce the counterpart, although high, was not as high as in the case of the heavy particles). After the theory of Dirac, the positron, i.e. the positively charged counterpart of the electron, was produced in 1932. And subsequently in 1955 (in the big Berkeley accelerator) the antiproton was produced.The final experimental victory of the symmetry principle is exemplified in the following small summary tableR. A. K. Frisch11Electric charge0-1Note. Incidentally, a layman and statistician may not be quite satisfied with the terminology, because the “anti” concept is not used consistently in connection with the electric charge. Since the antiproton has the opposite charge of the proton, there is nothing to object to the term anti in this connection. The difference between the neutron and the antineutron, however, has nothing to do with the charge. Here it is only a question of a difference in spin (and other properties connected with the spin). Would it be more logical to reserve the terms anti and the corresponding neutr to differences in the electric charge, and use expressions like, for instance counter and the corresponding equi when the essence of the difference is a question of spin (and other properties connected with the spin)? One would then, for in-stance, speak of a counterneutron instead of an antineutron.The population explosion in the world of elementary particles. As research pro-gressed a great variety of new elementary particles came to be known. They were extremely short-lived (perhaps of the order of a microsecond or shorter), which explains that they had not been seen before. Today one is facing a variety of forms and relations in elementary particles which is seemingly as great as the macroscopic differences one could previously observe in forms and relations of pieces of matter at the time when one started to systematize things by considering the proton, the electron and the neutron. Professor Murray Gell-Mann, Nobel prize winner 1969, has made path-breaking work at this higher level of systematization. When will this drive for systematization result in the discovery of something still smaller than the elementary particles?Matter and antimatter. Theoretically one may very precisely consider the existence of the “anti” form of, for instance, a normal hydrogen atom. This anti form would have a nucleus consisting of one antiproton around which circulated one positron. And similarly for all the more complicated atoms. This leads to the theoretical conception of a whole world of antimatter. In theory all this is possible. But to realize this in practice seems again a new and now really unsolvable problem. Indeed, wherever and whenever matter and anti-matter would come in contact, an explosion would occur which would produce an amount of energy several hundred times that of a hydrogen bomb of the same weight. How could possibly antimatter be produced experimentally? And how could antimatter experimentally be kept apart from the normal matter that surrounds us? And how could one possibly find out if antimatter exists in some distant galaxes or metagalaxes? And what reflections would the12 Economic Sciences 1969existence of antimatter entail for the conception of the “creation of the world”, whatever this phrase may mean. These are indeed alluring problems in physics and cosmology which - at least today - seem to be unsolvable problems, and which precisely for this reason occupy some of the finest brains of the world today.Travelling at a speed superior to that of light. It is customary to think that this is impossible. But is it really? It all depends on what we mean by “being in a certain place”. A beam of light takes about two million years to reach from us to the Andromeda nebula. But my thought covers this distance in a few seconds. Perhaps some day some intermediate form of body and mind may permit us to say that we actually can travel faster than light.The astronaut William Anders, one of the three men who around Christmas time 1968 circled the moon in Apollo 8 said in an interview in Oslo (2):“Nothing is impossible . . .it is no use posting Einstein on the wall and say: Speed of light-but not any quicker . . .30 nay 20, years ago we said: Impos-sible to fly higher than 50 000 feet, or to fly faster than three times the speed of sound. Today we do both.”The dream of Stanley Jevons. The English mathematician and economist Stanley Jevons (1835-1882) dreamed of the day when we would be able to quantify at least some of the laws and regularities of economics. Today - since the break-through of econometrics - this is not a dream anymore but a reality. About this I have much more to say in the sequel.Struggle, sweat and tears. This slight modification of the words of Winston Churchill is admirably suited to caracterize a certain aspect of the work of the scientists - and particularly of that kind of scientists who are absorbed in the study of “unsolvable” problems. They pass through ups and downs. Some-times hopeful and optimistic. And sometimes in deep pessimism. Here is where the constant support and consolation of a good wife is of enormous value to the struggling scientist. I understand fully the moving words of the 1968 Nobel prize winner Luis W. Alvarez when he spoke about his wife: “She has provided the warmth and understanding that a scientist needs to tide him over the periods of frustration and despair that seem to be part of our way of life” (3).2. A P HILOSOPHY OF C HAOS. T HE E VOLUTION TOWARDS A M AMMOTH S INGULAR T RANSFORMATIONIn the The Concise Oxford Dictionary (4) - a most excellent book - "philo-sophy"is defined as“love of wisdom or knowledge, especially that which deals with ultimate reality, or with the most general causes and principles of things”.If we take a bird’s eye-view of the range of facts and problems that were touched upon in the previous section, reflections on the “ultimate reality”quite naturally come to our mind.A very general point of view in connection with the “ultimate reality” I developed in lectures at the Institut Henri Poincaré in Paris in 1933. Subse-quently the question was discussed in my Norwegian lectures on statistics (5).R. A. K. Frisch 13The essence of this point of view on “ultimate reality” can be indicated by a very simple example in two variables. The generalization to many variables is obvious. It does not matter whether we consider a given deterministic, em-pirical distribution or its stochastic equivalence. For simplicity consider an empirical distribution.Let x 1 and x 2 be the values of two variables that are directly observed in aseries of observations. Consider a transformation of x 1 and x 2 into a new setof two variables y 1 and y 2. For simplicity let the transformation be linear i.e.The b’s and a’s being constants.(2.2)is the Jacobian of the transformation, as it appears in this linear case.It is quite obvious - and well known by statisticians - that the correlation coefficient in the set (y 1y 2) will be different from-stronger or weaker than-thecorrelation coefficient in the set (x 1x 2) (“spurious correlation”). It all dependson the numerical structure of the transformation.This simple fact I shall now utilize for my reflections on an “ultimate reality”in the sense of a theory of knowledge.It is clear that if the Jacobian (2.2)is singular, something important happens.In this case the distribution of y 1 and y 2 in a (y 1y 2) diagram is at most one-dimensional, and this happens regardless of what the individual observations x 1 and x 2 are - even if the distribution in the (x 1x 2) diagram is a completelychaotic distribution. If the distribution of x 1 and x 2 does not degenerate to apoint but actually shows some spread, and if the transformation determinant is of rank 1, i.e. the determinant value being equal to zero but not all its elements being equal to zero, then all the observations of y 1 and y 2 will lie on a straight linein the (y l y2) diagram. This line will be parallel to the y 1 axis if the first row ofthe determinant consists exclusively of zeroes, and parallel to the y 2 axis if thesecond row of the determinant consists exclusively of zeroes. If the distribution of x 1 and x 2 degenerates to a point, or the transformation determinant is of rankzero (or both) the distribution of y 1 and y 2 degenerates to a point.Disregarding these various less interesting limiting cases, the essence of the situation is that even if the observations x 1 and x 2 are spread all over the (x 1x 2)diagram in any way whatsoever, for instance in a purely chaotic way, the corresponding values of y 1 and y 2 will lie on a straight line in the (y 1y 2) diagramwhen the transformation matrix is of rank 1. If the slope of this straight line is finite and different from zero, it is very tempting to interpret y 1 as the “cause”of y 2 or vice versa. This “cause”,however, is not a manifestation of somethingintrinsic in the distribution of x 1 and x 2, but is only a human figment, a humandevice, due to the special form of the transformation used.What will happen if the transformation is not exactly singular but only14Economic Sciences 1969near to being singular? From the practical viewpoint this is the crucial question. Here we have the following proposition:(2.3)Suppose that the absolute value of the correlation coefficient r x i n(x1x2) is not exactly 1. Precisely stated, suppose that(2.3.1)0 1.This means that ε may be chosen as small as we desire even exactly 0, but it must not be exactly 1. Hence |rX|may be as small as we please even exactly 0, but not exactly 1.Then it is possible to indicate a nonsingular transformation from x1 and x2to the new variables y1 and y2with the following property: However small wechoose the positive, but not 0, number δ, the correlation coefficient rYi n(yl y2) will satisfy the relation(2.3.2) |rY|( 0R. A. K. Frisch 15 techniques. The latter is only an extension of the former. In principle there is no difference between the two. Indeed, science too has a constant craving for regularities. Science considers it a triumph whenever it has been able by some partial transformation here or there, to discover new and stronger regularities. If such partial transformations are piled one upon the other, science will help the biological evolution towards the survival of that kind of man that in the course of the millenniums is more successful in producing regularities. If “the ultimate reality” is chaotic, the sum total of the evolution over time - biological and scientific - would tend in the direction of producing a mammoth singular transformation which would in the end place man in a world of regularities. How can we possibly on a scientific basis exclude the possibility that this is really what has happened? This is a crucial question that con-fronts us when we speak about an “ultimate reality”. Have we created the laws of nature, instead of discovering them? Cf. Lamarck vs. Darwin.What will be the impact of such a point of view? It will, I believe, help us to think in a less conventional way. It will help us to think in a more advanced, more relativistic and less preconceived form. In the long run this may indirectly be helpful in all sciences, also in economics and econometrics.But as far as the concrete day to day work in the foreseeable future is con-cerned, the idea of a chaotic “ultimate reality” may not exert any appreciable influence. Indeed, even if we recognize the possibility that it is evolution of man that in the long run has created the regularities, a pragmatic view for the fore-seeable future would tell us that a continued search for regularities - more or less according to the time honoured methods - would still be “useful” to man.Understanding is not enough, you must have compassion. This search for regularities may well be thought of as the essence of what we traditionally mean by the word “understanding”. This “understanding”is one aspect of man’s activity. Another - and equally important - is a vision of the purpose of the understand-ing. Is the purpose just to produce an intellectually entertaining game for those relatively few who have been fortunate enough through intrinsic abilities and an opportunity of top education to be able to follow this game? I, for one, would be definitely opposed to such a view. I cannot be happy if I can’t believe that in the end the results of our endevaours may be utilized in some way for the betterment of the little man’s fate.I subscribe fully to the words of Abba Pant, former ambassador of India to Norway, subsequently ambassador of India to the United Arab Republic, and later High Commissioner of India to Great Britain:“Understanding is not enough, you must have compassion.” (6).3. A B RIEF S URVEY OF THE D EVELOPMENT OF E CONOMICS IN THE L AST C ENTURY Turning now to the more specifically economic matters, it is inevitable that I should begin by making a brief survey of the development of economics in the last century.In the middle of the 19th century John Stuart Mill (1806-1873) in his famous work “Principles of Economics”said that so far as general principles are concerned the theory of value and price was now completely elaborated.16 Economic Sciences 1969There was nothing more to add, he said, neither for himself nor any other author. To us with our relativistic view on knowledge and the development of science, it is difficult to understand that such a statement could be made. But to the generation that lived at that time these words by Mill appeared to be very close to the truth. In Mill’s “Principles” the ideas of Adam Smith (1723-1790), David Ricardo (1772-l823)and Thomas Robert Malthus (1766-1834) had been knit together into an organic, logically and seemingly complete whole.Subsequent developments have thoroughly denounced Stuart Mill’s words. Two break-throughs have emerged in economic theory since the time of Stuart Mill.The classical theory of value - as we find it streamlined in Stuart Mill - was essentially a theory of production costs based on the thinking of the private entrepreneur. The entrepreneur will think about as follows: “If I could only cut my selling price I would be able to draw the customers to me. This, how-ever, is also the way my competitors think. So, there emerges a sort of gravita-tional force that pulls prices down. The cost of production is so to speak the solid base on to which the prices fall down and remain. Hence the cost of production is “the cause”of prices. This general viewpoint the classical economists applied with great sagacity to a whole range of commodities , to the relation between wages and profits and to the theory ofinternational prices etc.This theory contains, of course, an irrefutable element of truth. But it is too simple to give even a crude presentation of the forces at play. The economic process is an equilibrium affair where both technological and subjective forces. are at play. The subjective element was nearly left out by the classicists.On this point economic theory was completely renewed in the years between 1870 and 1890 when a number of Austrian economists headed by Karl Menger (1840-1921) undertook a systematic study of the human wants and their place in a theory of prices. Similar thoughts were expressed also by the Swiss Léon Walras (1834-1910) and the Englishman Stanley Jevons (1835-l882). This was the first break-through since Stuart Mill.The Englishman Alfred Marshall (1842-1924) subsequently did much to combine the subjective viewpoint and the cost of production viewpoint. This led to what we now usually speak of as the neo-classical theory.Neither the classicists nor the neo-classicists did much to verify their theo-retical results by statistical observations. The reason was partly that the statistics were poor, and partly that neither the classical nor the neo-classical theory was built out with the systematic statistical verification in view. The architec-tural plan of the theory had so to speak not made room for this verification. This fact was criticized by the German historical school under the leadership of Gustav Schmoller (1838-1917) and by the American institutionalists. These schools, however, had an unfortunate and rather naive belief in something like a “theory-free” observation.“Let the facts speak for themselves”. The impact of these schools on the development of economic thought was therefore not very great, at least not directly. Facts that speak for themselves, talk in a very naive language.A. A. K. Frisch17In the first part of the 20th century the picture changed. Partly under the influence of the criticism of the historical school and the institutionalists the theoreticians themselves took up a systematic work of building up the theory in such a way that the theory could be brought in immediate contact with the observational material. One might say that from now on economics moved into that stage where the natural sciences had been for a long time, namely the stage where theory derives its concepts from the observational technique, and in turn theory influences the observational technique.For the first time in history it now seemed that the work on the theoretical front in economics - now to a large extent mathematically formulated - and the work on the outer descriptive front should converge and support each other, giving us a theory that was elaborate enough to retain the concrete observatio-nal material, and at the same time a mass ofobservations that were planned and executed with a view to be filled into the theoretical structure.Of course, there had been forerunners for such a combination of economic theory, mathematics and statistics even earlier. It was represented by such men as Johan Heinrich von Thünen (1783-l850), Augustin Cournot (1801-1877), A. J. Dupuit (1804-1866) and Hermann Heinrich Gossen (1810-1858). But from the first part of the 20ieth century the movement came in for full. This was the beginning of the econometric way of thinking. And this is what I would call the second break-through since Stuart Mill.A crucial point in this connection is the quantification of the economic concepts, i.e. the attempts at making these concepts measurable. There is no need to insist on what quantitative formulation of concepts and relations has meant in the natural sciences. And I would like to state that for more than a generation it has been my deepest conviction that the attempted quantification is equally important in economics.The quantification is important already at the level of partial analysis. Here one has studied the demand for such important commodities as sugar, wheat, coffe, pig iron, American cotton, Egyptian cotton etc.And the quantification is even more important at the global level. Indeed, at the global level the goal of economic theory is to lay bare the way in which the different economic factors act and interact on each other in a highly complex system, and to do this in such a way that the results may be used in practice to carry out in the most effective way specific desiderata in the steering of the economy.As long as economic theory still works on a purely qualitative basis without attempting to measure the numerical importance of the various factors, practically any “conclusion”can be drawn and defended. For instance in a depression some may say: A wage reduction is needed because that will increase the profits of the enterprises and thus stimulate the activity. Others will say: A wage increase is needed because that will stimulate the demand of the consumers and thus stimulate activity. Some may say: A reduction of the interest rate is needed because this will stimulate the creation of new enter-prises. Others may say: An increase of the interest rate is needed because that18Economic Sciences1969will increase the deposits in the banks and thus give the banks increased capacity of lending money.Taken separately each of these advocated measures contains some particle of truth, taken in a very partial sense when we only consider some of the obvious direct effects, without bothering about indirect effects and without comparing the relative strengths of the various effects and countereffects. Just as one would say: If I sit down in a rowing boat and start rowing in the ordinary way, the boat will be driven backwards because of the pressure exerted by my feet in the bottom of the boat.In a global analysis that shall be useful for practical applications in economic policy in the nation as a whole, the gist of the matter is to study the relative strengths of all relevant effects and countereffects, hence the need for quanti-fication of the concepts.This perhaps is the most general and most salient formulation of the need for econometrics. How far we would be able to go in this direction was of course another question. But at least the attempt had to be made if economics were to approach the state of an applied science.It goes without saying that econometrics as thus conceived does not exhaust all the contents of economics. We still need - and shall always need - also broad philosophical discussions, intuitive suggestions of fruitful directions of research, and so on. But this is another story with which I will not be concerned here (7). Let me only say that what econometrics - aided by electronic computers - can do, is only to push forward by leaps and bounds the line of demarcation from where we have to rely on our intuition and sense of smell.4. S OME H ISTORICAL N OTES ON THE F OUNDING OF T HE E CONOMETRIC S OCIETY In the files of the Oslo University Institute of Economics I have located a folder containing letters and copies of letters dating from the years when the plans for an econometric society took shape. Here are interesting ideas and opinions from outstanding people in different parts of the world. Most of these people have now passed away.One of them was my good friend professor Francois Divisia. His letter of 1 September 1926 from his home in Issy les Moulineaux (Seine) was handwritten in his fine characters, 8 pages to the brim with every corner of the paper used. Most of the letter contained discussions on specific scientific questions, but there were also some remarks of an organizational sort. He spoke for instance of his correspondence with professor Irving Fisher of Yale. About this he said: ”Je suppose qu'il s’agit d’une liste destinée àétablir une liason entre les écono-mistes mathématiciens du monde entier”.Whether this was an independent initiative on the part of Fisher in connection with a plan for a society, or it was an outcome of my previous correspondence with Fisher, I have not been able to ascertain, because the files are missing. Divisia continues:“Dans la politique, je ne suis pas très partisan des organismes internationaux . . .mais dans les domaines desinteresses comme celui de la science, j’en suis au contraire partisan sans restriction”.Answering Divisia in a letter of 4 September 1926 I said inter alia: “JeR. A. K. Frisch19 saisis avec enthousiasme l’idee d’une liste ou d’un autre moyen de communication entre les économistes mathematiciens du monde entier. J’ai eu moi-même l’idée de tâcher de réaliser une association avec un périodique consacré à ces questions. Il est vrai que les périodiques ordinaires tels que la Revue d’économie politique ou l’Economic Journal, etc. acceptent occasionnellement des memoires mathematiques, mais toujours est-il que l’auteur d’un tel memoire se trouve duns l’obligation de restreindre autant que possible l’emploi de symboles mathematiques et le raisonnement par demonstration mathematique.Je connais déjà plusieurs economistes-mathématiciens dans differents pays, et j'ai pensé érire un jour ou l’autre une lettre à chacun d’eux pour avoir leur opinion sur la possiblité d’un périodique, (que dites-vous d’une “Econometrica”?, la soeur du”Biometrika”.) Maintenant je serai heureux d’avoir votre opinion d’abord. Si vous pensez que cela vaut la peine on pourra peut-être commencer par former un cercle restreint qui s’adressera plus tard au public. Dans les années à venir j’aurai probablement l’occasion de voyager souvent en Amérique et en Europe, alors j'aurai l’occasion de faire la connaissance des économistes qui pourront s’intéresser à ce projet, et j’aurai l’occasion de faire un peu de propagande. Peut-être pourra-t-on obtenir l’appui d’une des grandes fondations américaines pour la publication du périodique.Voici une liste de quelque personnes que je connais par correspondance comme étant très intéressées au sujet de l’économie pure: Jaime Algarra, Professeur d’éc. pol. UniversitéBarcelone, L. von Bortkievicz, Professeur de Stat. Univ. Berlin, E. Bouvier, Prof. de S C. fin. Univ. Lyon, K. Goldziher, Prof. Techn. Hochschule, Budapest, K. G. Hagström, Actuaire, Stockholm, Charles Jordan, Docteur és S C., Budapest, Edv. Mackeprang, Dr. polit., Copenhague, W. M. Persons, Prof. de Stat. Harvard Univ. Cambridge. Mass. U.S.A., E. Slutsky, Moscou, A. A. Young, Prof. d’éc. polit., Harvard Univ. Cam-bridge. Mass. U.S.A., P. Rédiadis. Contreamiral, Athènes.”I mentioned also a number of others, among whom were: Anderson, Prof. Ecole Supérieure de Commerce, Varna, Bulgarie, Graziani, Prof. d’éc. pol. Univ. Napoli, Italie, Huber, Dir. de la Stat.gén. de la France, Paris, Ricci, Prof. Univ. Roma, Gustavo del Vecchio R. Univ. Commerciale, Trieste.In a letter of 22 September 1926 Divisia answered inter alia: “Je suis, vous le savez, tout à fait d’accord avec vous sur l’utilité d’une Association Internationale d’Éco-nomie pure et j'aime beaucoup le titre d’"Econometrica" auquel vous avez songé pour un périodique. Toutefois, avant de passer aux realisations, je pense qu’il est indispensable de réunir tout d’abord un certain nombre d’adhésions. .. . je me demande s’il ne serait pas aussi possible et opportun de s’aboucher à une organisation existente comme l’lnstitut international de statistique. . . .Enfin, d’ores et déjà, tout mon concours vous est acquis.”In a letter of 1 November 1926 I wrote to Divisia: “Mon départ pour l’Amérique a été ajourné de quelques mois. J’en ai profité pour écrire aux personnes suivantes: Bortkievicz, Université de Berlin, A. L. Bowley, London School of Economics, Charles Jordan,Université de Budapest, Eugen Slutsky, Moscou, pour avoir leur opinion sur l’utilité et la possibilité de réaliser d’abord un cercle restreint et plus turd peut-être une association formelle . . .J’ai trouvé que je n’ai pas pû expliquer la chose d’une meilleure fagon qu’en copiant certains passages de votre dernière lettre . . .C’est peut-être là une petite indiscretion dont je me suis rendu coupable.”The same day 1 November 1926 I wrote to the four persons in question. In。

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D R A F T LECTURE NOTES Discrete Mathematics Maintainer:Anthony A.Aaby An Open Source Text DRAFT Version:α0.1Last Edited:August 18,2004Copyright c 2004by Anthony AabyWalla Walla College204S.College Ave.College Place,WA99324E-mail:aabyan@Original Author:Anthony AabyThis work is licensed under the Creative Commons Attribution License.To view a copy of this license,visit /licenses/by/2.0/or send a letter to Creative Commons,559Nathan Abbott Way,Stanford,California 94305,USA.This book is distributed in the hope it will be useful,but without any warranty; without even the implied warranty of merchantability orﬁtness for a particular purpose.No explicit permission is required from the author for reproduction of this book in any medium,physical or electronic.The author solicits collaboration with others on the elaboration and extension of the material in this ers are invited to suggest and contribute material for inclusion in future versions provided it is oﬀered under compatible copyright provisions.The most current version of this text and L A T E Xsource is available at /~aabyan/Math/index.html.Dedication:WWC Computer Science Students4Contents1Sets,Relations,and Functions91.1Informal Set Theory (9)1.2Relations (12)1.3Functions (13)1.4References (14)2Paradox152.1Antinomies of intuitive set theory (15)2.2Zeno’s paradoxes (17)3The Basics of Counting193.1Counting Arguments (19)3.2The Pigeonhole Principle (19)3.3Permutations and combinations (20)3.4Solving recurence relations (20)3.5References (21)4Numbers and the Loss of Innocence234.1How many kinds of numbers? (23)4.2Pythagoras,the Pythagoreans,&Pure Mathematics (24)4.3How big is big? (25)4.4Why we will never catch up (26)5The Inﬁnite295.1The inﬁnite (29)6Cardinality and Countability3156.1The Cardinal Numbers (31)6.2Countability (31)7The Indescribable357.1Is the universe indescribable? (35)7.2Formal Languages (36)7.3Grammars (37)8Proof Methods in Logic398.1Preliminaries (39)8.2The Axiomatic Method (41)8.2.1Classical logic (41)8.2.2Hilbert’s Axiomatization (43)8.3Hilbert Style Proofs (44)8.4Natural Deduction (45)8.5The Analytic Properties (47)8.6The Method of Analytic Tableaux (49)8.7Sequent Systems(Gentzen) (52)9Many-sorted Algebra579.1Historical Perspectives and Further Reading (60)9.2Exercises (60)10Graphs and Trees6310.1Vertex,vertices (63)10.2Graphs (64)10.2.1Paths and cycles (64)10.3Trees and Forests (66)10.4Traversal strategies (67)11Discrete Probability6911.1Deﬁnition of probability (69)11.2Complementary events (70)11.3Conditional probability (70)11.4Independent events (70)11.5Bayes’theorem (70)611.6Random variables (71)11.7Expectation (71)Bibliography7378List of Figures7.1Alphabet and Language (37)8.1Formulas of Logic (42)8.2Natural Deduction Inference Rules (46)8.3Analytic Subformula Classiﬁcation (48)8.4Block Tableau Construction (50)8.5Block Tableau Inference Rules (50)8.6Tableau for¬[(p∨q)→(p∧q)] (51)8.7Tableau for∀x.[P(x)→Q(x)]→[∀x.P(x)→∀x.Q(x)] (52)8.8Analytic Sequent Inference Rules (53)9.1Algebraic Deﬁnition of Peano Arithmetic (58)9.2Algebraic deﬁnition of an Integer Stack (59)9Chapter1Sets,Relations,and Functions1.1Informal Set Theory(Set theory is)Theﬁnest product of mathematical genius and oneof the supreme achievements of purely intellectual human activity.-David HilbertBag A bag is an unordered collection of elements.It is also called a multiset and may include duplicates.Set A set is an unordered collection of distinct elements selected from a domain of discourse or the universe of values,U.S,X,A,B,...are sets.a,b,x, y,...are elements.Set deﬁnition/description Sets may be deﬁned by extension:speciﬁcation by explicit listing of members A={x0,...,x n}.A set of one element is called a singleton set.Sets may be deﬁned by intension/comprehension: speciﬁcation by a membership condition or rule for inclusion in the set.A={x|P(x)}={x:P(x)}.P(x)is usually a logical expression.As no restrictions are placed on the condition or rule this method is called the unrestricted principle of comprehension or abstraction.Some Sets∅={x|x=x}The empty set is the set that has no elements.U The set of all the elements in the universe of discourse.N={0,1,2,...}The set of natural numbers.Z={...−1,0,1,...}The set of integers.11CHAPTER1.SETS,RELATIONS,AND FUNCTIONSA=∼A={x|x/∈A}=U\A The complement of a set A is a set consisting of those elements not found in the set A.Cross product-A×B={(a,b)|a∈A and b∈B}The cross product of a pair of sets is the set of ordered pairs of elements from each set.Note: sets are unordered while pairs are ordered thus{a,b}={b,a}while the tuple(a,b)=(b,a).Subset:A⊆B if x∈A then x∈B A set A is a subset of a set B if every element of A is an element of B.Proper subset-A⊂B={x|if x∈A then x∈B and A=B}A subset A of a set B is a proper subset if the sets are not equal.DRAFT COPY August18,200412RMAL SET THEORYCHAPTER1.SETS,RELATIONS,AND FUNCTIONS1.3.FUNCTIONSCHAPTER1.SETS,RELATIONS,AND FUNCTIONSChapter2ParadoxAntinomy a contradiction between two apparently equally valid principles or between inferences correctly drawn from such principlesParadox a self-contradictory statement that atﬁrst seems true.2.1Antinomies of intuitive set theoryThe paradoxes in intuitive set theory are actually antinomies and are the result of the use of the unrestricted principle of comprehension/abstraction(deﬁning a set A={x:P(x)}where no restriction is placed on P(x).The most famous is Russell’s paradox.Russell(1901)and Zermelo:Let A={x|x/∈x}.Is A∈A?Boththe assumption that A is a member of A and A is not a member ofA lead to a contradiction(If R={x|x/∈x}then R∈R iﬀR/∈R).Two popular forms of this paradox are:•Is there is a bibliography that lists all bibliographies that don’tlist themselves.•In a village,there is a barber(a man)who shaves all those menwho do not shave themselves.Who shaves the barber?Logical antinomies•Burali-Forti(1897):Is there a set of all ordinal numbers?May have been discovered by Cantor in1885.•Cantor(1899):If there is a set of all sets,its cardinality must be the greatest possible cardinal yet the cardinality of the power set of the set17CHAPTER2.PARADOX2.2.ZENO’S PARADOXESCHAPTER2.PARADOXChapter3The Basics of Counting3.1Counting ArgumentsA set A isﬁnite if there is some n∈N such that there is a bijection from the set {0,1,2,...,n−1}to the set A.The number n is called the cardinality of A and we say“A has n elements,”or“n is the cardinal number of A.”The cardinality of A is denoted by|A|.3.2The Pigeonhole PrincipleThe fundamental rule of counting:The Pigeonhole Principle.The following are equivalent1.If m pigeons are put into m pigeonholes,there is an empty hole iﬀthere’sa hole with more than one pigeon.2.If n>m pigeons are put into m pigeonholes,there’s a hole with morethan one pigeon.3.Let|A|denote the number of elements in aﬁnite set A.For twoﬁnite setsA and B,there exists a1-1correspondence f:A→B iﬀ|A|=|B|. Theorem3.1N can be placed in1-1correspondence with any inﬁnite subset of itself.Proof:by the natural ordering of the elements of N.Theorem3.2|I|=|N|Proof:evens count positives and odds count negatives.Theorem3.3|N|=|Q|Place Q in tabular form and count along successive diagonals.21CHAPTER3.THE BASICS OF COUNTING(n−r)!Theorem3.9Let r,n∈N and r≤n.The number of combinations of n things taken r at a time is nr =n!3.5.REFERENCESCHAPTER3.THE BASICS OF COUNTINGChapter4Numbers and the Loss of InnocenceThere are several ideas that are introduced in this section that are covered in more detail in later sections:1.Relationship between language an new ideas.2.Emergence of pure mathematics.3.Consequences ofﬁxed world views.4.Big numbers as prelude to a discussion of inﬁnity5.Big numbers as a prelude to a discussion of the limits of computation. 4.1How many kinds of numbers?In our examination of informal set theory,we saw an example of how informal language could lead to paradox.The solution oﬀered was more precision and careful use of the nguage may also lead to new ideas as in the following example.In this example,our language is that of equations.Diﬀerent types of numbers are required as solutions for slightly diﬀerent forms of the equation.•Natural Numbers are solutions to equations of the form:x+a=b where a≤b,e.g.,x+3=7•Negative numbers are solutions to equations of the form:x+a=b wherea>b,e.g.,x+5=325CHAPTER4.NUMBERS AND THE LOSS OF INNOCENCE2is the length of the diagonal of a unit square,andπis the ratio of the length of the diameter of a circle to the length of its circumference.•Imaginary numbers are solutions to equations of the form;x2=−1All these numbers are solutions to polynomial equations with integer coeﬃcients and are collectively called algebraic numbers.Numbers which are not algebraic are called transcendental numbers.Among the transcendental numbers are pi and e.4.2Pythagoras,the Pythagoreans,&Pure Math-ematicsPythagoras and the PythagoreansThe material here has been stolen from else where.Pure mathematicsPure mathematics-numbers detached from reality-the irrationals,and non-constructive(indirect)proofs.Zeno of EleaGreek philosopher,born at Elea,about490B.C.At his birthplace Xenophanes and Parmenides had established the metaphysical school of philosophy known as the Eleatic School.The chief doctrine of the school was the oneness and immutability of reality and the distrust of sense-knowledge which appears to testify to the existence of multiplicity and change.Zeno’s contribution to the literature of the school consisted of a treatise,now lost,in which,according to Plato,he argued indirectly against the reality of motion and the existence of the manifold.There were,it seems,several discourses,in each of which he DRAFT COPY August18,2004264.3.HOW BIG IS BIG?CHAPTER4.NUMBERS AND THE LOSS OF INNOCENCE4.4.WHY WE WILL NEVER CATCH UP.Running Time Func-tion Example:n=256(instructions)1microsec/instruction 1×10−6sec/instructionConstant time O(1)check the timea log log n+b0.000003secLog N time O(log n)an+b0.0025secN Log N time O(n log n)an2+bn+c0.065secPolynomial time O(n k)matrix multiplication ak n+... 3.67x1061centuries(k=2) Still computable Ackermann’s function Computable functionsNP-non-deterministic polyno-mial complexityP-deterministic polynomialcomplexity(tractable prob-lems)29DRAFT COPY August18,2004CHAPTER4.NUMBERS AND THE LOSS OF INNOCENCEChapter5The Inﬁnite5.1The inﬁniteIts inﬁnite all the way up and down!•Small numbers:the inﬁnitesimals,non-standard arithmetic and non-standard analysis•<ahref="../Math/Cardinality.html">Counting and the deﬁnition ofinﬁnity;cardinality of N,I,Q,&R and transﬁnite arithmetic-Georg Can-tor•[0,1]and[−∞,+∞],R and R n•℘(N)and inﬁnite binary strings•When will it ever end?-Hierarchy of inﬁnitiesℵ0,ℵ1,...•Paradoxs–One way inﬁnite,bounded and inﬁnite,unbounded and inﬁnite–<ahref="Paradox.html">Zeno’s paradox and inﬁnite algorithms–Gabriel’s horn–The Axiom of Choice–Banach-Tarski paradox–Spaceﬁlling curves-fractals in general31CHAPTER5.THE INFINITEChapter6Cardinality and Countability6.1The Cardinal NumbersIf a set A isﬁnite,there is a nonnegative integer,denoted#A or|A|,which is the number of elements in A.That number is one of theﬁnite cardinal numbers. To do arithmetic with cardinal numbers,you use facts aboutﬁnite sets and the number of elements in them,such as the following:•If A and B can be put into one-to-one correspondence,then#A=#B, and conversely.•If A is contained in B,then|A|≤|B|.•If A is disjoint from B and C is their union,then|C|=|A|+|B|.•If A and B are sets,and C=A×B is the set of all ordered pairs of elements,theﬁrst from A and the second from B,then|C|=|A|×|B|.•If C is the set of all subsets of A,i.e.,C=℘(A),then|C|=2|A|.6.2CountabilityIf the set is inﬁnite,the corresponding cardinal number is not one of theﬁnite cardinal numbers,so it is called a transﬁnite(or inﬁnite)cardinal number.The smallest inﬁnite cardinal number isℵ0=|{0,1,2,...}|.Sets having this cardinal number are called countably inﬁnite sets,or just countable sets,because they can be put into one-to-one correspondence with the positive integers,or counting numbers33CHAPTER6.CARDINALITY AND COUNTABILITY6.2.COUNTABILITY02i∞b...where each of the b’s are0or1.The1’s indicate that the corre-sponding natural number is in the set.The binary sequence of allzeros corresponds to the empty set.The binary sequence of all onescorresponds to N.Theorem6.7|A|<|℘(A)|Proof:Since A⊂℘(A)(every element of A is in℘(A)),|A|≤|℘(A)|.Assume that there is a1to1correspondence f between A and℘(A).Let B={x|x∈A,x∈f(x)}-x is not a member of the set towhich it corresponds.Let y∈A be such that f(y)=B.If y∈Bthen by the deﬁnition of B,y∈B.If y∈B then by the deﬁnitionof B,y∈B.A contradiction∴Therefore,|A|=|℘(A)|and we haveestablished the theorem.Theorem6.8|N|<|℘(N)|.Examples:There are inﬁnite sequences such as•1/3=0.33333...•1/1+1/2+...+1/n+...•All x P(x)=P x0∧P x1∧...Arithmetic of the inﬁnite cardinals•ℵ0+n=n+ℵ0=ℵ0•ℵ0+ℵ0=ℵ0•ℵ0∗n=n∗ℵ0=ℵ0(n>0)•ℵ0∗ℵ0=ℵ0•ℵn0=ℵ0(n>0Subtraction and division are not deﬁnable operations in this arithmetic.The Associative Laws of Addition and Multiplication hold,and the Commutative 35DRAFT COPY August18,2004CHAPTER6.CARDINALITY AND COUNTABILITYChapter7The IndescribableLanguage and structures•How big is the English Language?-Alphabet=Σ,Words⊂Σ*,Sentences ⊂Σ∗∗,Texts⊂Σ∗∗∗–|Σ|=n,|Σ∗|=ℵ0,Σ*=Σ**=Σ***•How big is the universe?Constants=Σ,Strings=Σ∗,Relations=2|Σ∗|•Theorem:An inﬁnite universe is not completely describable.Proof:Fora givenﬁnite alphabet(Σ,|Sigma|=n),there are at most,countablyinﬁnite many descriptions(|Σ∗|=ℵ0).For a given inﬁnite set of constants (N,|N|=ℵ0),there are uncountably many relations(|℘(N)=ℵ1≤|℘∪i∈N N i)|.Therefore,some relation in N cannot be described,ℵ0<ℵ1.Q.E.D.7.1Is the universe indescribable?How big is the English Language?-Alphabet=Σ;,Words⊂Σ∗,Sentences ⊂Σ∗∗,Texts⊂Σ∗∗∗•|Σ|=n,•|Σ∗|=ℵ0,•|Σ∗|=|Σ∗∗|=|Σ∗∗∗|How big is the universe?Is itﬁnite or inﬁnite?Can it be described in terms of a possibly inﬁnite set of constants(=Σ)and a set of relations(=℘(Σ∗))on those constants?37CHAPTER7.THE INDESCRIBABLE7.3.GRAMMARS Σan alphabet.Σis a nonempty,ﬁnite set of symbols.Λhe empty string.Λis a string with no symbols at all.L a language L over an alphabetΣis a collection of strings of ele-ments ofΣΣ∗The set of all possibleﬁnite strings of elements ofΣis denotedbyΣ∗.Λis an element ofΣ∗and L is a subset ofΣ∗.Figure7.1:Alphabet and LanguageCHAPTER7.THE INDESCRIBABLEChapter8Proof Methods in Logic8.1PreliminariesLetΣbe a set of symbols andΣ*be the set of all strings ofﬁnite length composed of symbols inΣincluding the empty string.A language L is a subset ofΣ*.Alternately,let G= Σ,P,S be a grammar whereΣis a set of symbols, P a set of grammar rules,and S the symbol for sentences in the language.The notation L(G)designates the language deﬁned by the grammar G.The set of strings in L/L(G)are called sentences or formulas.Three sets of formulas are distinguished,axioms(A),theorems(T),and formu-las(F).In monotonic logic systems the relationship among them is:A⊂T⊂F=L⊂Σ*If the set of theorems is the same as the set of formulas(T=F),then the system is of little interest and in logic is said to be contradictory.Inference rules I are functions from sets of formulas to formulas(I:℘(L)→L for each I∈I).The set of theorems are constructed from the set of axioms by the application of rules of inference.A proof is a sequence of statements,each of which is an axiom,a previously proved theorem,or is derived from previous statements in the sequence by means of a rule of inference.The notation U⊢T is used to indicate that there is a proof of T from the set of formulas U.The task of determining whether or not some arbitrary formula A is a member of the set of theorems is called theorem proving.There are several styles of proofs.The semi-formal style of proof common in mathematics papers and texts is a paragraph style.Formal proofs are presented in several formats.The following are the most common.•Hilbert style proofs•Natural Deduction41CHAPTER8.PROOF METHODS IN LOGIC8.2.THE AXIOMATIC METHODCHAPTER8.PROOF METHODS IN LOGICThe set of atomic formulas,P,is deﬁned byP={P i j t k...t k+i−1|t l∈C,i,j,k,l∈N}with f∈P where C={F i j t k...t k+i−1|t k∈C,i,j,k∈N}is a set of terms,{P0j|j∈N}is a set of propositional constants,and{F0j|j∈N}is a set of individual constants.The set of formulas,F,is deﬁned byF::=P|→FF|2F|∀x.[F]x twhere V={x i|i∈N}is a set of individual variables,t∈C,x∈V, and textual substitution,[F]t x,is a part of the meta language and designates the formula that results from replacing each occurrence of t with x.Additional operators and inﬁx notation:(A→B)≡→AB¬A≡(A→f)(A∨B)≡(¬A→B)(A∧B)≡¬(A→¬B)f≡(A∧¬A)(A↔B)≡((A→B)∧(B→A))3A≡¬2¬A∃x.A≡¬∀x.¬AFigure8.1:Formulas of Logic8.2.THE AXIOMATIC METHODCHAPTER8.PROOF METHODS IN LOGIC8.4.NATURAL DEDUCTION Hilbert Style Proof FormatQ By Modus Ponens explanation explanationA→B By Contrapositive AssumptionexplanationBut A holds because explanationP∧Q→R By Deduction Assumption Assumption explanationP By Contradiction Assumption explanationP By Contradiction Assumption explanationR By Case Analysis explanation explanation explanationP↔R By Mutual implication explanation explanation∀n.P By Inductionexplanation(Base step) Assumption(Induction hypothesis) explanation(Induction step)8.4Natural DeductionNatural deduction was invented independently by S.Jaskowski in1934and G.Gentzen in1935.It is an approach to proof using rules that are designed to mirror human patterns of reasoning.There are no logical axioms,only inference rules.For each logical connective,there are two kinds of inference rules,an introduction rule and an elimination rule.•Each introduction rule answers the question,under what conditions can the connective be introduced.•Each elimination rule answers the question,underheat conditions can the connective be eliminated.The natural deduction rules of inference are listed in Figure8.2.47DRAFT COPY August18,2004CHAPTER8.PROOF METHODS IN LOGICIntroductionRules¬¬A AA,B A∧B∨A∨B AA⊢B A,A→B∀x.∀x.P(x)[P(x)]c x for any c∈CP(c)∃x.P(x)8.5.THE ANALYTIC PROPERTIESCHAPTER8.PROOF METHODS IN LOGIC。

高中英语哲学思想单选题50题

高中英语哲学思想单选题50题1. Which of the following statements best represents the idea of Plato's Theory of Forms?A. The physical world is the ultimate reality.B. Ideas are more real than the physical objects.C. Sensory experiences are the only source of knowledge.D. Everything is constantly changing and unpredictable.答案：B。

柏拉图的理念论认为理念（形式）比具体的物质世界更真实，A 选项说物质世界是终极现实，与柏拉图的观点相悖；C 选项感官经验是唯一知识来源并非柏拉图的观点；D 选项一切都在不断变化且不可预测不符合柏拉图的理念论。

2. In Aristotelian philosophy, the concept of "entelechy" refers to:A. The potentiality of a thing to become something else.B. The final cause that guides the development of a thing.C. The randomness in the evolution of all beings.D. The complete absence of purpose in nature.答案：B。

亚里士多德哲学中的“隐德来希”指的是引导事物发展的最终原因，A 选项指的是事物成为其他东西的可能性；C 选项说的是所有生物进化的随机性不符合；D 选项自然界完全没有目的也不正确。

3. According to Stoicism, which of the following is most important for a person to achieve inner peace?A. Pursuing pleasure and material wealth.B. Controlling one's emotions and accepting fate.C. Always striving to change the external world.D. Focusing on personal achievements and recognition.答案：B。

信息计量学概论

3 科技报告
– Report（报告书） – technical note （技术札记） – memorandum （备忘录） – paper （论文） – bulletin （通报） – technical translation （技术译丛） – special publication （特种出版物）； – primary report （初步报告） – progress report （进展报告） – interim report （期中/临时报告） – final report （最终报告）等。
增进和深化信息学理论研究方面旳应用；图书馆管理中旳应用；在信息分析和预测中旳应用；信息检索方面旳应用；在科学评价方面旳应用；
在其他社会学科中旳应用。
2.2.1 信息计量学旳研究对象
根据巴克兰（美国）旳解释，信息计量学旳研究对象比文件计量学和科学计量学旳研究对象范围广得多。主要涉及：
（7）学位论文（8）产品资料（9）技术档案（10）科技报纸（11）光盘数据（12）网络数据
38
文件信息计量----10类文件信息源
1 科技图书
– 专著 – 论文集 – 教材 – 百科全书 – 字（词/辞）典 – 手册等。
2 科技期刊
– acta（学报） – journal（杂志） – annual（纪事） – bulletin（通报） – transaction（汇刊） – proceeding（会刊） – review（评论） – progress / advance（进展）等。
3 引文分析法
– 引文数量分析（时序、著者、国别、语种等） – 引文网研究（耦合、同引、链引等） – 引文主题有关性分析
信息计量学
第一章绪论

人大在职硕专业英语考试

一．选择题(每题1分，共10分)1. Categorical data can be further divided into nominal and ___ data.A. quantitativeB. scaleC. qualitativeD. ordinal2. For a sample with the data {10,8,20,25,12}, the sample mean is ____A. 14B. 15C. 16D. 173. Let X be Binomial(4,, then P(X=0) is ____A. 1/8B. 1/4C. 1/16D. 1/24. Which number is the square of some integerA. 6B. 7C. 8D. 95. The range for the Pearson correlation coefficient is____A. [0,1]B. [-1,1]C. (-1,1)D. [-1,0]6. Which of the following is NOT used to a measure of central tendencyA. meanB. medianC. modeD. variance7. Which of the following is a method of statistical inferenceA. meanB. frequency tableC. box-plotD. t-test8. Let X,Y be the two random variable, which of the following is TRUEA. E(X-Y)=EX-EYB. E(XY)=EXEYC. E(X/Y)=EX/EYD. Var(X+Y)=Var(X)+Var(Y)9. What does the abbreviation . stands forA. that is C. for exampleC. such thatD. becausedoes the abbreviation . stands forA. that is C. for exampleC. such thatD. because二．将下面的统计术语翻译成中文或者英文(每题1分，共20分)1.描述性统计2.点估计3.正态分布4. 最大似然估计5. 总体6. 假设检验7. 实验设计8. 方差分析9. 中心极限定理10. 大数定律11. Probability12. Statistics13. Unbiased estimate14. Multiple regression15. Bernoulli distribution16. Time series17. Multicollinearity18. Density function19. Histogram20. Random variable三．将下面的句子翻译成英文(每题4分，共20分)1.统计学是一门收集，整理，分析和解释数据的学科2.X1, X2… X n是一列独立同分布的随机变量。

有关统计的英语作文短语

有关统计的英语作文短语Title: The Essence of Statistics: A Journey throughData Analysis.In the realm of information and knowledge, statistics stands as a beacon, guiding us through the vast sea of data. It is the language of numbers, the translator of complex phenomena into meaningful representations. This essaydelves into the intricacies of statistics, unpacking its value, applications, and the challenges it poses in today's data-driven world.The Foundation of Statistics: Data Collection and Representation.Statistics begins with data, the raw material of analysis. Data collection is a crucial step, ensuring that the information gathered is accurate, representative, and relevant. From surveys and experiments to observational studies and administrative records, the methods used tocollect data vary depending on the research objectives. Once collected, data is then represented using numerical summaries, graphs, or charts, making it easier to identify patterns, trends, and outliers.Descriptive Statistics: Unlocking the Basics.Descriptive statistics is the first step in the analytical journey. It involves summarizing and describing the collected data using measures like mean, median, mode, and range. These measures provide a snapshot of the data's central tendency, dispersion, and shape. Furthermore, graphical representations like histograms, box plots, and scatter plots aid in visualizing the distribution and relationships within the data.Inferential Statistics: Making Sense of the Unknown.Where descriptive statistics provides a snapshot of the observed data, inferential statistics takes a leap of faith into the unknown. It allows researchers to make predictions or generalizations about a population based on a sample.Techniques like hypothesis testing and confidence intervals are the cornerstones of inferential statistics, helping us determine whether a claim about a population is supportedby the evidence.The Challenges of Statistics: The Dangers of Misinterpretation.However, the power of statistics also lies in its potential for misuse. Statistical fallacies, such ascherry-picking data, overfitting, and the misrepresentation of confidence intervals, can lead to misleading conclusions. It is, therefore, crucial to have a sound understanding of statistical principles and ethical guidelines to ensurethat data analysis is accurate and reliable.Applications of Statistics: The Breadth of Its Reach.The applications of statistics are vast and diverse. It finds its way into almost every field, from social science and medicine to business and technology. In research, statistics helps us identify patterns in data, testhypotheses, and make predictions. In business, it aids in market analysis, product testing, and decision-making. In public health, statistics is used to track disease outbreaks, evaluate treatment effectivenesss, and plan resource allocation.The Future of Statistics: The Convergence of Technologies.As technology continues to evolve, statistics is poised to play an even more crucial role. The convergence of big data, machine learning, and artificial intelligence is reshaping the landscape of data analysis. The ability to process and analyze vast amounts of data in real-time, coupled with the predictive powers of AI, is opening up new frontiers in research and innovation.In conclusion, statistics is not just a tool or a technique; it is a way of thinking, a framework for understanding the world through numbers. It helps us make sense of complexity, extract knowledge from data, and make informed decisions. In a world increasingly driven by data,the importance of statistics cannot be overstated. As we embark on this journey through data analysis, it is crucial to remember that the power of statistics lies not just in the numbers but in the questions we ask and the insights we gain from them.。

证明数据论点的英语作文

证明数据论点的英语作文The Importance of Data in Supporting Arguments.In the world of学术讨论and research, data holds the key to validating arguments and building robust theories. The significance of data in academic writing cannot be overstated, as it provides a solid foundation for claims and enhances the credibility of the author's assertions. This essay aims to demonstrate the importance of data in supporting arguments by examining its role in学术讨论and research, and discussing the various types of data that can be used to back up claims.Firstly, it is crucial to understand that data is the backbone of any argument. It provides empirical evidence that supports or contradicts a particular claim. In学术讨论, data is gathered through various methods such as experiments, surveys, observations, and case studies. This information is then analyzed and interpreted to draw conclusions or support arguments.One of the main reasons why data is so important in学术讨论is that it helps to establish objectivity.Subjective opinions and personal biases can easily cloud an argument, but data provides an objective measure that can be verified and replicated by others. This objectivity is crucial in ensuring that the findings of academic research are reliable and trustworthy.Moreover, data allows for precise measurement and quantification of phenomena. In many fields, such as science, economics, and social sciences, quantitative data is essential for making accurate predictions and understanding complex systems. By breaking down information into measurable units, researchers can identify patterns, trends, and relationships that might not be apparent otherwise.Qualitative data, on the other hand, provides insights into the underlying reasons and motivations behind observed phenomena. It helps to understand the perspectives, experiences, and beliefs of individuals or groups, whichare crucial for understanding social and cultural phenomena. Qualitative data can be collected through interviews, focus groups, observations, and document analysis, among other methods.In addition to establishing objectivity and providing precise measurements, data also enhances the credibility of arguments. When an argument is supported by solid data, itis more likely to be accepted and believed by others. Thisis because data provides evidence that can be checked and verified, which increases the confidence in the validity of the argument.Furthermore, data can be used to refute opposing arguments. By presenting counter-evidence or showing inconsistencies in the opposing view, data can be used to discredit opposing claims and strengthen the position ofthe argument. This is a crucial aspect of academic debate, as it helps to clarify the truth and promote the progressof knowledge.In conclusion, data plays a crucial role in supportingarguments in学术讨论and research. It provides an objective measure that ensures objectivity, precision, and credibility. By gathering and analyzing data, researchers can build robust theories and make reliable predictionsthat contribute to the advancement of knowledge. Therefore, it is essential for academics and researchers to prioritize data collection and analysis in their work, as it is the foundation upon which all credible arguments are built.。

高三英语统计学分析单选题30题

高三英语统计学分析单选题30题1.The purpose of statistics is to collect, analyze and present ___.A.datarmationC.numbersD.results答案：A。

本题主要考查统计学中“统计的目的是收集、分析和呈现什么”。

选项A“data”（数据）符合统计学的定义，统计就是对数据进行处理。

选项B“information”（信息）比较宽泛，统计学主要针对具体的数据。

选项C“numbers”（数字）只是数据的一种表现形式，不全面。

选项D“results”（结果）不准确，统计的目的不是单纯呈现结果，而是通过数据来呈现。

2.In statistics, a sample is a subset of ___.A.the populationB.a groupC.peopleD.items答案：A。

在统计学中，样本是总体的一部分。

选项A“the population” 总体）正确。

选项B“a group” 一组）不确切。

选项C“people” 人）太局限。

选项D“items” 物品）也不准确。

3.Statistics helps us make inferences about a population based on ___.A.assumptionsB.samplesC.guessesD.estimates答案：B。

统计学帮助我们基于样本对总体进行推断。

选项B“samples”（样本）正确。

选项A“assumptions”（假设）不准确。

选项C“guesses”（猜测）不科学。

选项D“estimates”（估计）只是其中一方面，不全面。

4.The mean is a measure of ___.A.central tendencyB.variabilityC.distributionD.skewness答案：A。

均值是集中趋势的一种度量。

论数这个概念

论数这个概念Numbers are an essential part of our lives. We use them to quantify, measure, count, and analyze various aspects of the world around us. Numbers provide a universal language that allows us to communicate information accurately and efficiently. They play a crucial role in fields such as mathematics, science, economics, and technology, shaping the way we understand and interact with the world.数字是我们生活中的一个重要组成部分。

我们使用它们来量化、测量、计数和分析我们周围世界的各个方面。

数字提供了一种普遍语言，使我们能够准确高效地传达信息。

它们在数学、科学、经济和技术等领域发挥着至关重要的作用，塑造着我们理解和与世界互动的方式。

In addition to their practical applications, numbers also have deep symbolic and cultural significance. They can represent concepts such as infinity, unity, perfection, and eternity. Different cultures have assigned symbolic meanings to specific numbers, influencing rituals, beliefs, and traditions. For example, the number seven is often associated with luck or spirituality in many cultures, while thenumber thirteen is considered unlucky in others. These symbolic meanings add layers of meaning and complexity to our understanding of numbers.除了实际应用之外，数字还具有深刻的象征和文化意义。

Detailed Balance and Intermediate Statistics

a r X i v :q u a n t -p h /0308084v 2 15 D e c 2003Detailed Balance and Intermediate StatisticsR.Acharyaa and P.Narayana Swamyb aProfessor Emeritus Physics,Arizona State University,Tempe AZ 85287b Professor Emeritus Physics,Southern Illinois University,Edwardsville IL 62026Abstract We present a theory of particles,obeying intermediate statistics (“anyons”),interpolating between Bosons and Fermions,based on the princi-ple of Detailed Balance.It is demonstrated that the scattering probabilities of identical particles can be expressed in terms of the basic numbers,which arise naturally and logically in this theory.A transcendental equation determining the distribution function of anyons is obtained in terms of the statistics pa-rameter,whose limiting values 0and 1correspond to Bosons and Fermions respectively.The distribution function is determined as a power series involv-ing the Boltzmann factor and the statistics parameter and we also express the distribution function as an inﬁnite continued fraction.The last form enables one to develop approximate forms for the distribution function,with the ﬁrst approximant agreeing with our earlier investigation.Electronic address:a Raghunath.acharya@,b pswamy@August 2003PACS 05.30.-d,05.90.+m,05.30.PrTypeset using REVT E XI.INTRODUCTIONWe formulate a theory of particles obeying intermediate statistics,interpolating between Bosons and Fermions,which might be called anyons.Our formulation will be based on two assumptions:1)The exchange symmetry or permutation of the coordinates of the particles in the many particle wave function is accompanied by multiplication by a complex number f,thus generalizing the symmetric or antisymmetric wave functions and2)The principle of Detailed Balance:if n1,n2represent the average occupation numbers of states labelled by 1and2,then the number of transitionsﬂowing from1to2must equal thatﬂowing from2 to1at equilibrium.The particles described by this theory may or may not be the same as the anyons obeying intermediate or fractional statistics discussed in the literature.The objects named anyons [1–8]carry both an electric charge and a magneticﬂux.They have attracted a great deal of attention and have been the subject of intense investigation in the literature.The anyons arise from the special circumstance in2+1space-time dimensions,where the permutation group is the braid group and the Chern-Simons theory provides a natural realization of the anyons[9,10].There has been a great deal of discussion in the literature on the thermo-statistics of anyons[11].Since the real world is strictly in a3+1dimensional space,anyons may not be real particles:they could be quasi-particles playing important roles in condensed matter phenomena.More recently,the subject of generalized statistics has been investigated in one dimension in the context of many kinds of statistics[12].In contrast to the theory of anyons familiar in the literature,our present approach is not limited to two space dimensions and is valid in the real world of3+1dimensional space-time.It is important to point out that this is due to the fact that we do not invoke the spin-statistics theorem and do not require ordinary spin to be interpreted in two dimensional space.On one hand,the theory based only on Detailed Balance does not have the features of the full-ﬂedged theory of anyons which takes advantage of the braid group in two dimensions. On the other hand,in this formulation we investigate the idea of interpolating statistics only in the context of the statistical mechanics of a gas in equilibrium,without the constraints imposed by quantumﬁeld theory.Here we shall use the name anyons,for convenience,to refer to particles obeying intermediate statistics,with the understanding that the present work needs to be developed further by incorporating additional assumptions before the connection with true anyons described in the literature could be established.The subject of anyons has been well investigated in the literature,especially in the context of quantumﬁeld theory and the braid group[2,5,7,8].Interesting results have also been derived to describe the thermostatistics of anyons,such as determining the virial coeﬃcients[11,13].However,the theoretical basis of the statistical mechanics of anyons has not really been established.For instance,the distribution function for the anyons has not been determined in an exact form.We had used an ansatz for the distribution function in our earlier work in order to derive many of the thermodynamic properties[11].More recently,Chaturvedi and Srinivasan[14]have done a comparative study of the diﬀerent interpolations between Bose-Einstein(BE)and Fermi-Dirac(FD)statistics available in theliterature,in the context of the most general interpolations,including Haldane statistics and Gentile statistics,with the conclusion that the distribution function introduced by us[11] has some desirable features.Consequently it is worthwhile to investigate the basic theoretical structure of anyons from the point of view of statistical mechanics and investigate the distribution function of the anyons if possible.This is the goal of this paper.Not only shall we determine the distribution function of the particles obeying the interpolation statistics,we shall also formulate a theory which leads to this determination without resorting to the restriction to a two dimensional space.Our formulation will be based on the quantum theory of many particles permitting a generalized interpolating exchange symmetry statistics with no other assumptions.Remarkably,this formulation leads to an exact theory which requires the employment of basic numbers.In Sec.II we study the scattering probability of many particle states obeying interpolating statistics.We establish the fact that the basic numbers arise naturally and logically in this theory.We introduce the method of Detailed Balance in Sec.III and derive a transcendental equation for the distribution function in terms of the Boltzmann factor and the statistics determining parameter.The distribution function of anyons is studied in Sec.IV where we show that a closed form solution is not possible.We present the exact solution for the dis-tribution function as an inﬁnite series as well as in the form of an inﬁnite continued fraction which is amenable to approximations.Sec.V contains a brief summary and conclusions.Let us introduce an ensemble of particles,the anyons,which obey a generalized statistics, interpolating between(BE)and(FD)statistics.We begin with the framework for building the wave functions of these anyons by a generalized procedure of f-symmetrizing in such a way that it will reduce,in appropriate limits,to the standard procedure of symmetrizing for Bosons and anti-symmetrizing for Fermions.The operation of permutation or exchange of the coordinates of the many particle wave function results in multiplication by the complex number f,the exchange symmetry parameter,so thatPΨn(···,q i,···,q j,···)=fΨn(···,q j,···,q i,···).(1) Since the Hamiltonian has the property P−1HP=H,it follows that H(PΨn)=E n(PΨn) and thus PΨn is an eigenfunction of the Hamiltonian with the same eigenvalue asΨn.Thus fΨn is proportional toΨn and consequently we may take|f|2Ψn=Ψn.Hence the general exchange symmetry that would lead to an intermediate statistics could be implemented by the complex number f with the property|f|2=1.We shall accordingly choose the exchange symmetry to be implemented by f=e iπα,whereαis the statistics determining parameter, 0≤α≤1,so that f∗=f−1.The limits f→1,−1correspond respectively to the BE and FD statistics representing Bosons and Fermions.This procedure for incorporating exchange symmetry among anyons is justiﬁed[7]by the special property of rotations in two dimensions.Here we may treat it as an assumption or an ansatz,not restricted to two space dimensions but valid in any number of dimensions.II.MANY PARTICLES AND QUANTUM PROBABILITIES Following Feynman[15],we consider the two particle scattering amplitude deﬁned by the product a1b2,where a1= 1|a describes the scattering of particle a into state1and b2= 2|b describes the process b→2.We shall take1and2to be the same state at the end in order to deal with identical particles.The exchange symmetry has to do with the process corresponding to a→2,b→1which is indistinguishable from the direct process and the amplitude for this process would be f a2b1due to the exchange factor f.The total probability amplitude is the sum of the direct and exchange processes.Employing the abbreviation 1|a = 2|a =a,weﬁnd the probability of this two particle scattering process involving non-identical particles to bep(2)non=(1+|f|2)|a|2|b|2=2(|a|2|b|2),(2)since|f|2=1.This probability is the same as for ordinary Bosons[15].However,for identical particles,we need to take account of interference between the two processes and that makes a great deal of diﬀerence.We obtain the probability in this case to bep(2)identical=(1+|f|2+f+f−1)|a|2|b|2,(3)since f∗=f−1and this probability depends on the statistics determining parameterα.In the limit f→1,this would reduce to the case of Bosons and it would be twice as much as in Eq.(2)for the non-identical particles.For arbitrary f the probability for the process involving identical particles relative to that for non-identical particles is given by1P(2)identical=(1+2f−1+2f−2+f−3)(1+2f+2f2+f3).(5)6We can reduce this to the form1P(3)=.(6)6 6+7(f+f−1)+4(f2+f−2+1)+(f3+f−3+f+f−1)Recognizing the pattern here,we observe that the right hand side in the expressions for the probabilities in Eqs.(4),(5),(6)contain basic numbers[16],with the base f.They are indeed expressed succinctly in terms of the basic numbers deﬁned by[n]f=f n−f−nsinπα,(9) will be found most useful.In terms of the basic numbers,the probabilities may now be expressed conveniently and succinctly asP(2)=16(6+7[2]+4[3]+[4]),P(4)=1sin tnk=1sin(2k−1)t= sin ntWe shall alsoﬁnd the following result quite useful:[n−1][n]=[2]+[4]+[6]+···[2(n−1)].(13) In other words,[2][3]=[2]+[4],[3][4]=[2]+[4]+[6],etc.This is proved by using the identityn−1k=0sin ky=sin(n−1)y2cscy2(2+[2]),P(3)=14!(2+[2])(2+2[2]+[3])(2+2[2]+2[3]+[4]).(15)In this manner,generalizing to n particles,we are led to the probability for the n-anyon state:P(n)=1P(n),(17)which can be determined from Eq.(16).This enhancement factor provides the essential step in the method of Detailed Balance.This brings us to the second assumption upon which our theory of interpolating statistics rests:if n1,n2represent the average occupation numbers of states1and2respectively,then the number of transitionsﬂowing from1to2must equal thatﬂowing from2to1at equilibrium.This is the principle of Detailed Balance.We should stress that this principle is characteristic of thermodynamic equilibrium and may be regarded as a consequence of the second law of thermodynamics[18].Indeed the principle of Detailed Balance is valid when thermodynamic equilibrium prevails or the validity of microscopic reversibility in the language of statistical physics.This notion is based on the reversibility of the microscopic equations of motion,or on the Hermitian nature of the scattering Hamiltonian[19].The principle of Detailed Balance can thus be stated asn1F(n2)eβE1=n2F(n1)eβE2,(18)where the population of each level is governed by the Boltzmann factor and F(n)is the enhancement factor.This yieldsnz−1eβE−1.(20)where z=eβµis the fugacity of the gas.Let us now proceed in the same manner for arbitrary f.The enhancement factor for arbitrary f is given by Eqs.(16)and(17)which reduces to the simple formF(n)=P(n+1)n+1(2+2[2]+2[3]+···2[n]+[n+1]).(21)To proceed further,we employ the resultnk=0[k]=2cos(πα/2)[n/2][(n+1)/2],(22)which is easy to prove by using identities involving sums of trigonometric functions.This gives us the following result for F(n),after some algebra:F(n)=1n+1{[(n+1)/2]cos(πα/2)}2.(25)We observe that this reproduces the expected result F(n)→n+1in the Bose limit.Upon now invoking the Detailed Balance we obtain the important result1n(n+1){[(n+1)/2]cos(πα/2)}2.(26)This can be rewritten in the following convenient form in order to deal with the distribution function:eβ(E−µ)=1sin2πα/2.(27)Solving this equation should,in principle,lead to the distribution function for the anyons in this formulation.At this point before dealing with the distribution function,we need to study the nature of the intermediate statistics as an interpolation between the BE and FD limits.Speciﬁcally we need to study the limits corresponding to BE and FD statistics.We have already observed that F(n)→n+1in the Bose limit,f→1.Indeed,it is readily veriﬁed that Eq.(26)reproduces the correct BE statistical distribution.The case of Fermi limit,α→1,f→−1,is,however,somewhat complicated.We have:[n]→n,−n for odd and even occupation numbers respectively.Evaluating the limit in Eq.(26),the enhancement factor may be put in the form,lim α→1F(n)=1n+1{1,0,1,0,1,···},(29)and weﬁnd that F(n)has repeating values1/(n+1)and0in the limitα→1,f→−1. This is thus a special and interesting feature of this theory.Thus for arbitrary values of the parameter f,including the Fermi limit f→−1,the theory of intermediate statistics requires the existence of generalized fermions beyond the exclusion principle.IV.THE DISTRIBUTION FUNCTIONWe begin by rewriting Eq.(27)in the formeβ(E−µ)=1sin2x.(30)Here x=πα/2in terms of the statistics determining parameter.The object is to determine the average occupation number n,the distribution function,so that we can compare with the standard BE or FD distribution and understand the nature of the interpolating statistics. We can expand the right hand side in a power serieseβ(E−µ)=1a0=csc2x(x sin2x−sin2x)a1=csc2x(x2cos2x−x sin2x+sin2x)a2=csc2x −x2cos2x−sin2x+(x−2x3/3)sin2xa3=csc2x (x2−x4/3)cos2x+sin2x−(x−2x3/3)sin2x ,(32) and so on.The coeﬃcients to any desired order can be obtained by using Mathematica.It is clear that a0→1,while a n→0for n≥1in the Bose limit,which is consistent with the BE distribution as described in Sec.III.In the case of the Fermi limit,a0→−1and it leads to the generalized fermions.We can rewrite the above as1g n2−a2gn4− (33)where g=eβ(E−µ)−a0.This series can be reverted to express n as a series in powers of1/g, thusn=1/g+B/g2+C/g3+D/g4+E/g5+F/g6···,(34) whereB=a1/g,C=(2a21/g2+a2/g),D=(5a1a2/g2+a3/g+5a31/g3),E=E=6a1a3/g2+3a22/g2+14a41/g4+a4/g+21a21a2/g3,F=7a1a4/g2+7a2a3/g2+84a31a2/g4+a5/g+28a21a3/g3+28a1a22/g3+42a51/g5,(35) and so on.Upon rearrangement,we can rewrite the above form for the distribution function as the following series:n(E)=1/g+a1/g3+a2/g4+(2a21+a3)/g5+(5a1a2+a4)/g6+(5a31+6a1a3+3a22+a5)/g7+ (36)We observe that this is a power series in1/g but the term with1/g2is absent.We can rewrite Eq.(36)in the formn(E)=1/g+∞k=3αk/g k,(37)whereα3=a1,α4=a2,2a21+a3=α5,5a1a2+a4=α6,(38) etc.We can now express this in the form of a continued fraction.We invoke the standard algorithm which can be introduced as follows[21].If a continued fraction is of the formB nc1+b2then the successive convergents(approximants)are obtained by the recurrence formulaeB n=c n B n−1+b n B n−2,C n=c n C n−1+b n C n−2,B−1=1,C−1=0.(40) Thus in order to put the series in Eq.(37)in the continued fraction form,we proceed with theﬁrst convergent and set c0=B0/C0=ing the recurrence relations,Eq.(40), we determine B0=0,C0=1,b1=1,c1=g,B1=1,C1=g.This determines theﬁrst convergent as n=1/g.Next,the recurrence relations lead to b2=−α3g,c2=g2+α3,B2= g2+α3,C2=g3which determines the second convergent to ben=1g2+α3.(41)Continuing on with successive convergents in this manner,we obtain the desired inﬁnite continued fraction form for the distribution function as followsn(E)=1g2+α3−α4gα4g+α5−···(42)This is the distribution function in the exact theory,albeit not in a closed form.Approxi-mations can be implemented in order to investigate the thermostatistical properties of the anyons.Theﬁrst approximant,n(E)=12,the exact form of the distributionfunction isn(E)|α=1g+a0ga0g−(a0−π3/48)+···,(44) where a0(1V.SUMMARYThe thermodynamic distribution function of anyons in two space dimensions,obeying in-terpolating statistics is not known in the literature and is an open question.We have not only found an answer,thus determining the distribution function,we have also demonstrated that the theory describing interpolating statistics has several remarkable and interesting features. Our investigation leads to a generalized deﬁnition of permutation symmetry in arbitrary di-mensions and not restricted to two space dimensions.We have shown that the theory of permutation symmetry that would describe particles obeying interpolating statistics is suc-cinctly formulated in the language of basic numbers.These basic numbers arise naturally and automatically in this formulation but do not explicitly invoke any deformed oscillator algebra.Our theory is based on the principle of Detailed Balancing which is a consequence of the Second law of thermodynamics.Furthermore,this formulation leads to the determi-nation of the exact distribution function,without having to introduce any approximation. In this manner,we have formulated the theory which leads to a transcendental equation for the distribution function of anyons in terms of the statistics determining parameter and the Boltzman factor containing the energy of the state.We obtain a solution of this equation which we express as a power series as well as in the form of a continued fraction.We show that theﬁrst approximation of this theory reproduces a form of the distribution function introduced by us in an earlier investigation.An important feature of our formulation consists of the fact that the basic numbers arise naturally and automatically in this theory,speciﬁcally the symmetric formulation of the basic numbers.The basic numbers arise automatically in our theory but they are not part of an oscillator algebra,nor do we introduce the construction of Fock space for the particles obeying the intermediate statistics.The theory reduces to BE statistics in the Bose limit, f→1,α→0.The Fermi limit,f→−1,α→1leads to generalized fermions beyond the exclusion principle and thus correspond to inﬁnite dimensional representation and they reduce to the familiar FD statistics only when n is restricted to0,1by hand.Although the algebra of q-oscillators appears in the literature on the subject of anyons [7,8,20],it is also known that q-oscillators may have nothing to do with anyons since the for-mer exist in arbitrary space-time dimensions.In our formulation,we do not use q-oscillator algebra,neither do we use Fock states.We do not make use of the Hamiltonian other than to recognize that the Hamiltonian should be of a form that permits Detailed Balance.The Hamiltonian consists of only the kinetic energy terms and is the free Hamiltonian[5]in the anyon gauge and the anyon wave functions satisfy“twisted”boundary conditions.It is interesting that theﬁrst approximant(ﬁrst convergent)of our solution corresponds to the approximate form for the distribution function1n(E)=approximation.It should be stressed that while the form above is a point of agreement for an approximate theory,the form of a in the present formulation as in Eq.(32),as a function of the statistics determining parameter,is very diﬀerent from that of[11].Finally we obtain the exact form of the distribution function for the caseα=1REFERENCES[1]J.Leinnas and J.Myrheim,Nuovo Cimento B37,(1977)1.[2]F.Wilczek(ed.)Fractional Statistics and Anyon Superconductivity,World Scientiﬁc(1991),Singapore.[3]S.Forte,Rev.Mod.Phys.64(1992)193.[4]S.S.Chern et al,(ed.)Physics and Mathematics of Anyons,World Scientiﬁc(1991),Singapore.[5]A.Lerda,Anyons,Springer(1992),Berlin.[6]A.Khare,Fractional Statistics and Quantum Theory,World Scientiﬁc Pub.(1998)Sin-gapore.[7]M.Frau et al,arXiv:hep-th/9407161v1,July1994.[8]L.Frappat et al,Phys.Lett.B369(1996)313.[9]F.Wilczek,Phys.Rev.Lett,48,(1982)114;Phys.Rev.Lett,49,(1982)957.[10]R.Jackiw and S.Templeton,Phys.Rev.,D23,(1981)2291;C.R.Hagen,Ann.Phys.,157,(1984)342.[11]R.Acharya and P.Narayana Swamy,J.Phys.A;math.Gen.A27,(1994)7247;Seealso other references therein.[12]A.Polychronakos,Les Houches Lectures,Summer1998,<arXiv;hep-th/9902157>,February1999.[13]D.Arovas et al,Nuc.Phys.B251(1985),117;See ref.([11])for a full list of references.[14]S.Chaturvedi and V.Srinivasan,Physica A246(1997),576.[15]R.P.Feynman,R.B.Leighton and M.Sands,The Feynman Lectures on Physics,Addison-Wesley Pub.Co.(1965),Reading MA.[16]H.Exton,q-Hypergeometric functions and applications,Ellis Horwood Ltd.(1983)Chich-ester.[17]I.S.Gradshteyn and I.M.Ryzhik,Table of Integrals,Series and products,Academic press(1980)New York.[18]S-K Ma,Statistical Mechanics,World Scientiﬁc Publishing(1985)Singapore;L.E.Reichl,A Modern Course in Statistical Physics,second edition,John Wiley&sons Inc.(1998),New York;ndsberg,editor,Problems in thermodynamics and statis-tical physics,Pion Limited(1971),London.[19]D.C.Mattis,Statistical Mechanics Made Simple,World Scientiﬁc Publishing,(2003)Singapore.[20]M.Chaichian,R.G.Felipe and C.Montonen,J.Phys.A:Math.Gen.A26(1993)4017–4034.[21]G.E.Andrews,R.Askey and R.Roy,Special Functions,Cambridge University Press,(2001)New York.13。

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“On the existence of statistics intermediate between those ofFermi-Dirac and Bose-Einstein“J. Dunning-Davies,Department of Physics,University of Hull,Hull HU6 7RX,England.J.Dunning-Davies@Abstract.Once again the possibility of the existence of particle statistics intermediate between those of Fermi-Dirac and Bose-Einstein surfaces. Here attention is drawn to the fact that some fifteen years ago it was shown that such so-called ‘intermediate’ statistics correspond to no physical process and the stationary probability distributions of intermediate statistics are not compatible with any mechanism which allows a variation between Fermi-Dirac and Bose-Einstein statistics.“What is between Fermi-Dirac and Bose-Einstein statistics?” This is, at first sight, a seemingly interesting question and maybe that is why it raises its head every few years or so. Whenever this happens, it becomes apparent that people haven’t searched out all the background literature. In 1995, Viefers, Ravndal and Haugset [1] asked this same question and even referred to the ‘fundamental discovery of Leinaas and Myrheim in 1977 of the possibility for intermediate quantum statistics for identical particles in two dimensions interpolating between standard Fermi-Dirac and Bose-Einstein statistics’ [2]. This ignored the fact that such intermediate statistics, with no restriction to two dimensions, had been proposed by Gentile [3] in 1940. As is mentioned in an article of 1989 [4], between 1940 and 1963, several articles appeared which examined both the possibility of systems obeying such intermediate statistics and the actual properties of such systems. However, as was pointed out in1989 [4], the arguments in these various articles seemed to go against the grain of certain limit theorems in probability theory; theorems which assert that, as the number of indistinguishable particles increases without limit while the number of energy cells remains finite, the probability distribution should converge to a normal distribution [5]. However, many authors appeared to feel that it is in precisely this case that the differences between intermediate and other forms of statistic would be manifested. The limit theorems of probability imply that, in the limit, the statistics should become classical!It has been shown [4] that Bose-Einstein statistics arises only in the limit as d = ∞, where d represents the number of energy levels. All values of d between one and infinity were shown to give rise to intermediate statistics which cannot be described by stationary probability distributions that tend to the binomial and negative binomial distributions as d → 1 and d → ∞ respectively [6]. Hence, the stationary probability distributions of so-called intermediate statistics are not compatible with any mechanism which allows a variation between Fermi-Dirac and Bose-Einstein statistics. Also, it was realised that no purely thermodynamic argument may be given either in favour of, or against, the existence of intermediate statistics. Therefore, a statistical argument was employed by showing that the stationary probability distribution may only satisfy the recursion relation obtained from the time independent master equation when it coincides with the binomial, negative binomial or Poisson distributions - corresponding to Fermi-Dirac, Bose-Einstein or classical statistics respectively. It seems surprising to realise that this was the first time mention had been made of the Poisson case in studies on intermediate statistics. Classical statistics was shown to emerge as the number of energy cells increases without limit independently of the size of the occupation numbers; it may never depend on the size of the particle number since any conclusions drawn from the generating function and its binomial expansion, in which the particle number merely appears as an index, must lead to the same conclusions. The entire argument appeared to lay to rest the possibility of the existence of particles with spins other than those with semi-integral or integral values.Note also that it was the basic tenet of the book “Statistical Physics: a Probabilistic Approach” [7] that probability may be used in the development of a statistical basis for thermodynamics. If this is so, it means that any form of physical statistics must be governed by probability distributions, just as fermions and bosons are governed by the negative binomial and binomial distributions respectively. It should be noted that, athigh temperatures, these two distributions merge into the Poisson distribution which governs classical, or Boltzmann, particles. However, there is no distribution between the negative binomial and binomial. Hence, it must be concluded that so-called ‘intermediate’ statistics correspond to no physical process. It should be realised also that the argument is independent of the dimensionality of the system and of any conditions that might have followed from quantum mechanics as Wilczek [8] would have one believe. As far as dimensionality is concerned, it only affects the expression for the density of states, having no effect upon the statistics. The probability distributions govern the statistics of collections of non-interacting particles and so attention is restricted to the negative binomial, binomial and Poisson distributions which govern fermions, bosons and classical particles respectively.Papers seem to be appearing with monotonous regularity extolling the virtues of these ‘intermediate’ statistics. Some [9] seem to refer back to the article by Haldane [10] as a basic source. The results obtained mirror those referred to earlier and obtained many years ago in a wide variety of articles; they assume that the occupation number of the energy levels may take any value between one and infinity, perform well-known mathematical manipulations and then make deductions pertaining to physics. These articles are essentially mathematical in nature; physics and statistical theory are not to the fore. One exception is provided by Landsberg [11] whose final conclusion is that there are only two genuine physical cases - those described by Fermi-Dirac and Bose-Einstein statistics. This conclusion agrees with the more detailed examination of this problem by Lavenda and Dunning-Davies [4], an article which existed, incidentally, before even that of Haldane.There may well be physical situations involving apparently non-interacting particles which are not readily explained by Fermi-Dirac or Bose-Einstein statistics but the correct explanations for these particular situations cannot lie with intermediate statistics, at least not until the objections already raised against such statistics [4] are first refuted. Indeed, for the situations falling into this category, it might be worth investigating the possibility that some interaction between the particles is playing a more important role than currently thought. After all, there is no actual physical situation where the particles involved are totally non-interacting. The ideal gases merely provide a convenient, tractable model which may, or may not, accurately reflect physical reality. In some situations, it is the physical model which should be examined critically. Maybe those which continue to spawn discussions of so-called ‘intermediate statistics’ provide examples where this is so.References.[1] S.Viefers, F. Ravndal & T. Haugset; Am. J. Phys. 63 (1995) 369[2] J. M. Leinaas & J. Myrheim; Nuovo Cimento B 37 (1977) 1[3] G. Gentile; Nuovo Cimento 19 (1940) 493[4] B. H. Lavenda & J. Dunning-Davies; J. Math. Phys. 30 (1989) 1117[5] R. Von Mises; Mathematical Theory of Probability and Statistics (Academic Press, New York, 1964)[6] B. H. Lavenda; Int. J. Theor. Phys. 27 (1988) 1371[7] B. H. Lavenda; Statistical Physics: A Probabilistic Approach,(Wiley, New York, 1991)[8] F. Wilczek; Scientific American, 264 (1991) 24[9] C. Wolf; Hadronic J. 20 (1997) 657K. Byczuk, et al. arXiv:cond-mat/0403735[10] F. D. M. Haldane, Phys. Rev. Lett. 67 (1991) 937[11] P. T. Landsberg; Mol. Phys. 6 (1963) 341。