decompositions of

APOS: A Constructivist Theory of Learningin Undergraduate Mathematics Education ResearchEd Dubinsky, Georgia State University, USAandMichael A. McDonald, Occidental College, USAThe work reported in this paper is based on the principle that research in mathematics education is strengthened in several ways when based on a theoretical perspective. Development of a theory or model in mathematics education should be, in our view, part of an attempt to understand how mathematics can be learned and what an educational program can do to help in this learning. We do not think that a theory of learning is a statement of truth and although it may or may not be an approximation to what is really happening when an individual tries to learn one or another concept in mathematics, this is not our focus. Rather we concentrate on how a theory of learning mathematics can help us understand the learning process by providing explanations of phenomena that we can observe in students who are trying to construct their understandings of mathematical concepts and by suggesting directions for pedagogy that can help in this learning process.Models and theories in mathematics education can•support prediction,•have explanatory power,•be applicable to a broad range of phenomena,•help organize one’s thinking about complex, interrelated phenomena,•serve as a tool for analyzing data, and•provide a language for communication of ideas about learning that go beyond superficial descriptions.We would like to offer these six features, the first three of which are given by Alan Schoenfeld in “Toward a theory of teaching-in-context,” Issues in Education, both as ways in which a theory can contribute to research and as criteria for evaluating a theory.In this paper, we describe one such perspective, APOS Theory, in the context of undergraduate mathematics education. We explain the extent to which it has the above characteristics, discuss the role that this theory plays in a research and curriculum development program and how such a program can contribute to the development of the theory, describe briefly how working with this particular theory has provided a vehicle for building a community of researchers in undergraduate mathematics education, and indicate the use of APOS Theory in specific research studies, both by researchers who are developing it as well as others not connected with its development. We provide, in connection with this paper, an annotated bibliography of research reports which involve this theory.APOS TheoryThe theory we present begins with the hypothesis that mathematical knowledge consists in an individual’s tendency to deal with perceived mathematical problem situations by constructing mental actions, processes, and objects and organizing them in schemas to make sense of the situations and solve the problems. In reference to these mental constructions we call it APOS Theory. The ideas arise from our attempts to extend to the level of collegiate mathematics learning the work of J. Piaget on reflective abstraction in children’s learning. APOS Theory is discussed in detail in Asiala, et. al. (1996). We will argue that this theoretical perspective possesses, at least to some extent, the characteristics listed above and, moreover, has been very useful in attempting to understand students’learning of a broad range of topics in calculus, abstract algebra, statistics, discrete mathematics, and other areas of undergraduate mathematics. Here is a brief summary of the essential components of the theory.An action is a transformation of objects perceived by the individual as essentially external and as requiring, either explicitly or from memory, step-by-step instructions on how to perform the operation. For example, an individual with an action conception of left coset would be restricted to working with a concrete group such as Z20 and he or she could construct subgroups, such asH={0,4,8,12,16} by forming the multiples of 4. Then the individual could write the left coset of 5 as the set 5+H={1,5,9,13,17} consisting of the elements of Z20which have remainders of 1 when divided by 4.When an action is repeated and the individual reflects upon it, he or she can make an internal mental construction called a process which the individual can think of as performing the same kind of action, but no longer with the need of external stimuli. An individual can think of performing a process without actually doing it, and therefore can think about reversing it and composing it with other processes. An individual cannot use the action conception of left coset described above very effectively for groups such as S4, the group of permutations of four objects and the subgroup H corresponding to the 8 rigid motions of a square, and not at all for groups S n for large values of n. In such cases, the individual must think of the left coset of a permutation p as the set of all products ph, where h is an element of H. Thinking about forming this set is a process conception of coset.An object is constructed from a process when the individual becomes aware of the process as a totality and realizes that transformations can act on it. For example, an individual understands cosets as objects when he or she can think about the number of cosets of a particular subgroup, can imagine comparing two cosets for equality or for their cardinalities, or can apply a binary operation to the set of all cosets of a subgroup.Finally, a schema for a certain mathematical concept is an individual’s collection of actions, processes, objects, and other schemas which are linked by some general principles to form a framework in the individual’s mind that may be brought to bear upon a problem situation involving that concept. This framework must be coherent in the sense that it gives, explicitly or implicitly, means of determining which phenomena are in the scope of the schema and which are not. Because this theory considers that all mathematical entities can be represented in terms of actions, processes, objects, and schemas, the idea of schema is very similar to the concept image which Tall and Vinner introduce in“Concept image and concept definition in mathematics with particular reference to limits and continuity,” Educational Studies in Mathematics, 12, 151-169 (1981). Our requirement of coherence, however, distinguishes the two notions.The four components, action, process, object, and schema have been presented here in a hierarchical, ordered list. This is a useful way of talking about these constructions and, in some sense, each conception in the list must be constructed before the next step is possible. In reality, however, when an individual is developing her or his understanding of a concept, the constructions are notactually made in such a linear manner. With an action conception of function, for example, an individual may be limited to thinking about formulas involving letters which can be manipulated or replaced by numbers and with which calculations can be done. We think of this notion as preceding a process conception, in which a function is thought of as an input-output machine. What actually happens, however, is that an individual will begin by being restricted to certain specific kinds of formulas, reflect on calculations and start thinking about a process, go back to an action interpretation, perhaps with more sophisticated formulas, further develop a process conception and so on. In other words, the construction of these various conceptions of a particular mathematical idea is more of a dialectic than a linear sequence.APOS Theory can be used directly in the analysis of data by a researcher. In very fine grained analyses, the researcher can compare the success or failure of students on a mathematical task with the specific mental constructions they may or may not have made. If there appear two students who agree in their performance up to a very specific mathematical point and then one student can take a further step while the other cannot, the researcher tries to explain the difference by pointing to mental constructions of actions, processes, objects and/or schemas that the former student appears to have made but the other has not. The theory then makes testable predictions that if a particular collection of actions, processes, objects and schemas are constructed in a certain manner by a student, then this individual will likely be successful using certain mathematical concepts and in certain problem situations. Detailed descriptions, referred to as genetic decompositions, of schemas in terms of these mental constructions are a way of organizing hypotheses about how learning mathematical concepts can take place. These descriptions also provide a language for talking about such hypotheses.Development of APOS TheoryAPOS Theory arose out of an attempt to understand the mechanism of reflective abstraction, introduced by Piaget to describe the development of logical thinking in children, and extend this idea to more advanced mathematical concepts (Dubinsky, 1991a). This work has been carried on by a small group of researchers called a Research in Undergraduate Mathematics Education Community (RUMEC) who have been collaborating on specific research projects using APOS Theory within abroader research and curriculum development framework. The framework consists of essentially three components: a theoretical analysis of a certain mathematical concept, the development and implementation of instructional treatments (using several non-standard pedagogical strategies such as cooperative learning and constructing mathematical concepts on a computer) based on this theoretical analysis, and the collection and analysis of data to test and refine both the initial theoretical analysis and the instruction. This cycle is repeated as often as necessary to understand the epistemology of the concept and to obtain effective pedagogical strategies for helping students learn it.The theoretical analysis is based initially on the general APOS theory and the researcher’s understanding of the mathematical concept in question. After one or more repetitions of the cycle and revisions, it is also based on the fine-grained analyses described above of data obtained from students who are trying to learn or who have learned the concept. The theoretical analysis proposes, in the form of a genetic decomposition, a set of mental constructions that a student might make in order to understand the mathematical concept being studied. Thus, in the case of the concept of cosets as described above, the analysis proposes that the student should work with very explicit examples to construct an action conception of coset; then he or she can interiorize these actions to form processes in which a (left) coset gH of an element g of a group G is imagined as being formed by the process of iterating through the elements h of H, forming the products gh, and collecting them in a set called gH; and finally, as a result of applying actions and processes to examples of cosets, the student encapsulates the process of coset formation to think of cosets as objects. For a more detailed description of the application of this approach to cosets and related concepts, see Asiala, Dubinsky, et. al. (1997).Pedagogy is then designed to help the students make these mental constructions and relate them to the mathematical concept of coset. In our work, we have used cooperative learning and implementing mathematical concepts on the computer in a programming language which supports many mathematical constructs in a syntax very similar to standard mathematical notation. Thus students, working in groups, will express simple examples of cosets on the computer as follows.Z20 := {0..19};op := |(x,y) -> x+y (mod 20)|;H := {0,4,8,12,16};5H := {1,5,9,13,17};To interiorize the actions represented by this computer code, the students will construct more complicated examples of cosets, such as those appearing in groups of symmetries.Sn := {[a,b,c,d] : a,b,c,d in {1,2,3,4} | #{a,b,c,d} = 4};op := |(p,q) -> [p(q(i)) : i in [1..4]]|;H := {[1,2,3,4], [2,1,3,4], [3,4,1,2], [4,3,2,1]};p := [4,3,2,1];pH := {p .op q : q in H};The last step, to encapsulate this process conception of cosets to think of them as objects, can be very difficult for many students. Computer activities to help them may include forming the set of all cosets of a subgroup, counting them, and picking two cosets to compare their cardinalities and find their intersections. These actions are done with code such as the following.SnModH := {{p .op q : q in H} : p in Sn};#SnModH;L := arb(SnModH); K := arb(SnModH); #L = #K; L inter K;Finally, the students write a computer program that converts the binary operation op from an operation on elements of the group to subsets of the group. This structure allows them to construct a binary operation (coset product) on the set of all cosets of a subgroup and begin to investigate quotient groups.It is important to note that in this pedagogical approach, almost all of the programs are written by the students. One hypothesis that the research investigates is that, whether completely successful or not, the task of writing appropriate code leads students to make the mental constructions of actions, processes, objects, and schemas proposed by the theory. The computer work is accompanied by classroom discussions that give the students an opportunity to reflect on what they have done in the computer lab and relate them to mathematical concepts and their properties and relationships. Once the concepts are in place in their minds, the students are assigned (in class, homework and examinations) many standard exercises and problems related to cosets.After the students have been through such an instructional treatment, quantitative and qualitative instruments are designed to determine the mental concepts they may have constructed and the mathematics they may have learned. The theoretical analysis points to questions researchers may ask in the process of data analysis and the results of this data analysis indicates both the extent to which the instruction has been effective and possible revisions in the genetic decomposition.This way of doing research and curriculum development simultaneously emphasizes both theory and applications to teaching practice.Refining the theoryAs noted above, the theory helps us analyze data and our attempt to use the theory to explain the data can lead to changes in the theory. These changes can be of two kinds. Usually, the genetic decomposition in the original theoretical analysis is revised and refined as a result of the data. In rare cases, it may be necessary to enhance the overall theory. An important example of such a revision is the incorporation of the triad concept of Piaget and Garcia (1989) which is leading to a better understanding of the construction of schemas. This enhancement to the theory was introduced in Clark, et. al. (1997) where they report on students’ understanding of the chain rule, and is being further elaborated upon in three current studies: sequences of numbers (Mathews, et. al., in preparation); the chain rule and its relation to composition of functions (Cottrill, 1999); and the relations between the graph of a function and properties of its first and second derivatives (Baker, et. al., submitted). In each of these studies, the understanding of schemas as described above was not adequate to provide a satisfactory explanation of the data and the introduction of the triad helped to elaborate a deeper understanding of schemas and provide better explanations of the data.The triad mechanism consists in three stages, referred to as Intra, Inter, and Trans, in the development of the connections an individual can make between particular constructs within the schema, as well as the coherence of these connections. The Intra stage of schema development is characterized by a focus on individual actions, processes, and objects in isolation from other cognitive items of a similar nature. For example, in the function concept, an individual at the Intra level, would tend to focus on a single function and the various activities that he or she could perform with it. TheInter stage is characterized by the construction of relationships and transformations among these cognitive entities. At this stage, an individual may begin to group items together and even call them by the same name. In the case of functions, the individual might think about adding functions, composing them, etc. and even begin to think of all of these individual operations as instances of the same sort of activity: transformation of functions. Finally, at the Trans stage the individual constructs an implicit or explicit underlying structure through which the relationships developed in the Inter stage are understood and which gives the schema a coherence by which the individual can decide what is in the scope of the schema and what is not. For example, an individual at the Trans stage for the function concept could construct various systems of transformations of functions such as rings of functions, infinite dimensional vector spaces of functions, together with the operations included in such mathematical structures.Applying the APOS TheoryIncluded with this paper is an annotated bibliography of research related to APOS Theory, its ongoing development and its use in specific research studies. This research concerns mathematical concepts such as: functions; various topics in abstract algebra including binary operations, groups, subgroups, cosets, normality and quotient groups; topics in discrete mathematics such as mathematical induction, permutations, symmetries, existential and universal quantifiers; topics in calculus including limits, the chain rule, graphical understanding of the derivative and infinite sequences of numbers; topics in statistics such as mean, standard deviation and the central limit theorem; elementary number theory topics such as place value in base n numbers, divisibility, multiples and conversion of numbers from one base to another; and fractions. In most of this work, the context for the studies are collegiate level mathematics topics and undergraduate students. In the case of the number theory studies, the researchers examine the understanding of pre-college mathematics concepts by college students preparing to be teachers. Finally, some studies such as that of fractions, show that the APOS Theory, developed for “advanced” mathematical thinking, is also a useful tool in studying students’understanding of more basic mathematical concepts.The totality of this body of work, much of it done by RUMEC members involved in developing the theory, but an increasing amount done by individual researchers having no connection with RUMEC or the construction of the theory, suggests that APOS Theory is a tool that can be used objectively to explain student difficulties with a broad range of mathematical concepts and to suggest ways that students can learn these concepts. APOS Theory can point us towards pedagogical strategies that lead to marked improvement in student learning of complex or abstract mathematical concepts and students’ use of these concepts to prove theorems, provide examples, and solve problems. Data supporting this assertion can be found in the papers listed in the bibliography.Using the APOS Theory to develop a community of researchersAt this stage in the development of research in undergraduate mathematics education, there is neither a sufficiently large number of researchers nor enough graduate school programs to train new researchers. Other approaches, such as experienced and novice researchers working together in teams on specific research problems, need to be employed at least on a temporary basis. RUMEC is one example of a research community that has utilized this approach in training new researchers.In addition, a specific theory can be used to unify and focus the work of such groups. The initial group of researchers in RUMEC, about 30 total, made a decision to focus their research work around the APOS Theory. This was not for the purpose of establishing dogma or creating a closed research community, but rather it was a decision based on current interests and needs of the group of researchers.RUMEC was formed by a combination of established and beginning researchers in mathematics education. Thus one important role of RUMEC was the mentoring of these new researchers. Having a single theoretical perspective in which the work of RUMEC was initially grounded was beneficial for those just beginning in this area. At the meetings of RUMEC, discussions could focus not only on the details of the individual projects as they developed, but also on the general theory underlying all of the work. In addition, the group’s general interest in this theory and frequent discussions about it in the context of active research projects has led to growth in the theory itself. This was the case, for example, in the development of the triad as a tool for understanding schemas.As the work of this group matures, individuals are beginning to use other theoretical perspectives and other modes of doing research.SummaryIn this paper, we have mentioned six ways in which a theory can contribute to research and we suggest that this list can be used as criteria for evaluating a theory. We have described how one such perspective, APOS Theory is being used, in an organized way, by members of RUMEC and others to conduct research and develop curriculum. We have shown how observing students’ success in making or not making mental constructions proposed by the theory and using such observations to analyze data can organize our thinking about learning mathematical concepts, provide explanations of student difficulties and predict success or failure in understanding a mathematical concept. There is a wide range of mathematical concepts to which APOS Theory can and has been applied and this theory is used as a language for communication of ideas about learning. We have also seen how the theory is grounded in data, and has been used as a vehicle for building a community of researchers. Yet its use is not restricted to members of that community. Finally, we provide an annotated bibliography which presents further details about this theory and its use in research in undergraduate mathematics education.An Annotated Bibliography of workswhich develop or utilize APOS TheoryI. Arnon. Teaching fractions in elementary school using the software “Fractions as Equivalence Classes” of the Centre for Educational Technology, The Ninth Annual Conference for Computers in Education, The Israeli Organization for Computers in Education, Book of Abstracts, Tel-Aviv, Israel, p. 48, 1992. (In Hebrew).I. Arnon, R. Nirenburg and M. Sukenik. Teaching decimal numbers using concrete objects, The Second Conference of the Association for the Advancement of the Mathematical Education in Israel, Book of Abstracts, Jerusalem, Israel, p. 19, 1995. (In Hebrew).I. Arnon. Refining the use of concrete objects for teaching mathematics to children at the age of concrete operations, The Third Conference of the Association for the Advancement of the Mathematical Education in Israel, Book of Abstracts, Jerusalem, Israel, p. 69, 1996. (In Hebrew).I. Arnon. In the mind’s eye: How children develop mathematical concepts – extending Piaget's theory. Doctoral dissertation, School of Education, Haifa University, 1998a.I. Arnon. Similar stages in the developments of the concept of rational number and the concept of decimal number, and possible relations between their developments, The Fifth Conference of the Association for the Advancement of the Mathematical Education in Israel, Book of Abstracts. Be’er-Tuvia, Israel, p. 42, 1998b. (In Hebrew).The studies by Arnon and her colleagues listed above deal with the development ofmathematical concepts by elementary school children. Having created a framework thatcombines APOS theory, Nesher’s theory on Learning Systems, and Yerushalmy’s ideas ofmulti-representation, she investigates the introduction of mathematical concepts as concreteactions versus their introduction as concrete objects. She establishes developmental paths for certain fraction-concepts. She finds that students to whom the fractions were introduced asconcrete actions progressed better along these paths than students to whom the fractions were introduced as concrete objects. In addition, the findings establish the following stage in thedevelopment of concrete actions into abstract objects: after abandoning the concrete materials, and before achieving abstract levels, children perform the concrete actions in their imagination.This corresponds to the interiorization of APOS theory.M. Artigue, Enseñanza y aprendizaje del análisis elemental: ¿qué se puede aprender de las investigaciones didácticas y los cambios curriculares? Revista Latinoamericana de Investigación en Matiemática Educativa, 1, 1, 40-55, 1998.In the first part of this paper, the author discusses a number of student difficulties and tries toexplain them using various theories of learning including APOS Theory. Students’unwillingness to accept that 0.999… is equal to 1 is explained, for example, by interpreting the former as a process, the latter as an object so that the two cannot be seen as equal until thestudent is able to encapsulate the process which is a general difficulty. In the second part of the paper, the author discusses the measures that have been taken in France during the 20thCentury to overcome these difficulties.M. Asiala, A. Brown, D. DeVries, E. Dubinsky, D. Mathews and K. Thomas. A framework for research and curriculum development in undergraduate mathematics education, Research in Collegiate Mathematics Education II, CBMS Issues in Mathematics Education, 6, 1-32, 1996.The authors detail a research framework with three components and give examples of itsapplication. The framework utilizes qualitative methods for research and is based on a veryspecific theoretical perspective that was developed through attempts to understand the ideas of Piaget concerning reflective abstraction and reconstruct them in the context of college levelmathematics. For the first component, the theoretical analysis, the authors present the APOStheory. For the second component, the authors describe specific instructional treatments,including the ACE teaching cycle (activities, class discussion, and exercises), cooperativelearning, and the use of the programming language ISETL. The final component consists ofdata collection and analysis.M. Asiala, A. Brown, J. Kleiman and D. Mathews. The development of students’ understanding of permutations and symmetries, International Journal of Computers for Mathematical Learning, 3, 13-43, 1998.The authors examine how abstract algebra students might come to understand permutations of a finite set and symmetries of a regular polygon. They give initial theoretical analyses of what it could mean to understand permutations and symmetries, expressed in terms of APOS. Theydescribe an instructional approach designed to help foster the formation of mental constructions postulated by the theoretical analysis, and discuss the results of interviews and performance on examinations. These results suggest that the pedagogical approach was reasonably effective in helping students develop strong conceptions of permutations and symmetries. Based on thedata collected as part of this study, the authors propose revised epistemological analyses ofpermutations and symmetries and give pedagogical suggestions.M. Asiala, J. Cottrill, E. Dubinsky and K. Schwingendorf. The development of student’s graphical understanding of the derivative, Journal of Mathematical Behavior, 16(4), 399-431, 1997.In this study the authors explore calculus students’ graphical understanding of a function and its derivative. An initial theoretical analysis of the cognitive constructions that might be necessary for this understanding is given in terms of APOS. An instructional treatment designed to help foster the formation of these mental constructions is described, and results of interviews,conducted after the implementation of the instructional treatment, are discussed. Based on the data collected as part of this study, a revised epistemological analysis for the graphicalunderstanding of the derivative is proposed. Comparative data also suggest that students who had the instructional treatment based on the theoretical analysis may have more success indeveloping a graphical understanding of a function and its derivative than students fromtraditional courses.M. Asiala, E. Dubinsky, D. Mathews, S. Morics and A. Oktac. Student understanding of cosets, normality and quotient groups, Journal of Mathematical Behavior,16(3), 241-309, 1997.Using an initial epistemological analysis from Dubinsky, Dautermann, Leron and Zazkis(1994), the authors determine the extent to which the APOS perspective explains students’mental constructions of the concepts of cosets, normality and quotient groups, evaluate the。

（ＥＣ）、碳酸丙烯酯（ＰＣ）、二乙基碳酸酯（ＤＥＣ）、二甲基碳酸酯（ＤＭＣ）。

溶剂均采用精馏结合分子筛吸附的方法提纯至纯度（质量分数）＞－９９．９５％（气相色谱仪为ＧＣ－１４Ｃ，日本岛津）。

电解液的配制及电池的装配均在充满高纯氩气的手套箱中进行【畎Ｈ２０）＜Ｉ１０－６】，ＬｉＰＦ６浓度为１ｍｏＦＬ。

电解液中水和酸（Ｉ－ＩＦ）含量均低于２０ｘ１０－６，分别用ＫａｒｌＦｉｓｈｅｒ水分测定仪ＫＦ８３１和ＫａｒｌＦｉｓｈｅｒ电位滴定仪７９８ＧＰＴＴｉｔｒｉｎｏ（瑞士万通）测定。

１．２电池性能测试电池电化学性能测试用ＣＲ２０３２扣式半电池，以ＭＣＭＢ电极片为正极，金属锂片作为负极，电解液为不含与含（质量分数）２％ＶＣ的ｌｍｏｉ／ＬＬｉＰＦｃ，／（ＥＣ：ＰＣ：ＤＥＣ：ＤＭＣ）＝（３：１：４：２１。

组装成半电池后，在室温下用Ｌａｎｄ多通道充放电测试系统在０．０１～２．０Ｖ电压区间进行恒电流充放电实验。

金属锂片作为对电极和参比电极，ＭＣＭＢ电极为工作电极，在多通道测试系统（ＳｏｌａｒｔｒｏｎＡｎａｌｙｔｉｃａｌ１４８０Ｍｕｌｔｉｓｔａｔ，英国）上对电池进行循环伏安测试，扫描速率为０．２ｍＶ／ｓ。

１．３表面性质分析用ＪＳＭ．６５１０（ＪＥＯＬ）扫描电镜观察石墨电极表面的形貌，加速电压为１５ｋＶ。

用ＡＴＲ－ＩＲＮｉｃｏｌｅｔｉＳｌ０（ＴｈｅｒｍｏＦｉｓｈ．ｅｒ）傅里叶变换红外光谱研究电极表面的ＳＥＩ膜成分，扫描范围６８０～２０００ｃｍ—ｉ。

２结果与讨论２．１石墨电极的循环伏安行为图ｌ为石墨电极在不含ＶＣ（ａ）与含ＶＣ添加剂（ｂ）的电解液体系中的前三次循环伏安图。

从图ｌ（ａ）可以看出，石墨电极在不含ＶＣ电解液的循环伏安图谱中具有以下特征：（１）在第一次负向扫描过程中，在电极电位为１．５Ｖ左右出现一个还原峰，对应于电极表面吸附的溶剂分子的还原，在１．１３Ｖ左右出现一个还原峰，对应于电解液本体中溶剂组分的还原分解，并形成同体电解质相界面膜（ＳＥＩ）；随着电位的不断降低，电解液的阴极电流逐渐增大，对应于锂离子向石墨电极嵌入，生成锂碳嵌合物的量不断增加；（２）正向扫描时，在０．４５Ｖ左右出现一个氧化峰，对应于锂碳嵌合物发生阳极氧化，锂离子从石墨电极中脱出。

基于VMD最优分解次数的研究发表时间：2019-08-15T15:58:02.740Z 来源：《科技新时代》2019年6期作者：夏雄飞[导读] 本文通过对VMD分解信号后的若干个IMF模态分量进行希尔伯特变换求得该信号的平均瞬时频率分量，并根据平均瞬时频率分量确定最佳的信号分解次数。

民航湖南空管分局，421000，摘要：非平稳、非线性信号的常用处理方法有小波变换、短时傅里叶变换、魏格纳分布等，然而这些方法都有一定的限制。

最近EMD 分解算法和VMD算法对于非平稳、非线性信号具有非常好分解效果。

本文通过对比VMD分解算法和EMD分解算法，发现VMD分解算法能够很好地抑制杂波信号干扰并能够在一定程度上抑制模态混叠现象，能够很好地避免EMD分解算法带来的缺陷。

其次VMD分解算法更容易表达信号波动的大趋势，而EMD分解则不容易观察到该特性。

最后针对VMD算法的分解次数是有人为设定的，因此如何选择最佳分解层数，对于信号分解是否彻底也具有至关重要的作用，本文通过计算VMD算法分解后的IMF模态分量的平均瞬时频率，确定VMD分解算法的最佳分解次数。

关键字：非线性信号；非平稳信号；EMD分解；VMD分解；Abstract: Commonly used methods for non-stationary and nonlinear signals are wavelet transform, short-time Fourier transform, and Wigner distribution. However, these methods have certain limitations. Recently, the EMD decomposition algorithm and the VMD algorithm have very good decomposition effects for non-stationary and nonlinear signals. By comparing the VMD decomposition algorithm and the EMD decomposition algorithm, it is found that the VMD decomposition algorithm can suppress the clutter signal interference and suppress the modal aliasing phenomenon to a certain extent, which can avoid the defects caused by the EMD decomposition algorithm. Secondly, the VMD decomposition algorithm is more likely to express the general trend of signal fluctuation, and EMD decomposition is not easy to observe this characteristic. Finally, the number of decompositions for the VMD algorithm is artificially set. Therefore, how to choose the optimal decomposition layer is also crucial for the complete decomposition of the signal. This paper calculates the average of the IMF modal components after the decomposition of the VMD algorithm. The instantaneous frequency determines the optimal number of decompositions of the VMD decomposition algorithm.Keywords: nonlinear signal; non-stationary signal; EMD decomposition; VMD algorithm;1. 引言传统时频分析方法由于是将信号以规定的频率成分做线性变化处理取得了比较好的效果。

2.1 Show that (1,-1),(1,2) and (2,1) are linearly dependent.Solution: we have (1,-1)+(1,2) -(2,1)=0, so (1,-1),(1,2) and (2,1) are linearly dependent.2.2 Suppose V is a vector space with basis vectors 01and ,and A is a linear operator from V to V such that 0110and A A ==. Give a matrix representation for A, with respect to the input basis 0,1,and the output basis 0,1.Find input and output basis which give rise to a different matrix representation of A.Solution: 0001A =+,1001A =+,so 0110A ⎛⎫⎪⎝⎭=.BecauseV is a vector space with basis vectors 01and ,we can get another basis of V arevectors))0101and +-,)))010101A A +=+=+,))))01011001A A -=-=-=,so now 1001A ⎛⎫ ⎪-⎝⎭=2.3 Suppose A is a linear operator from vector space V to vector space W, and B is a linear operator from vector space W to vector space X. Let ,j k w and x v i be bases for the vector space V ,W, and X, respectively. Show that the matrix representation for linear transformation BA is the matrix product of the matrix representatitions for B and A, with respect to the appeopriate bases.Solution: ,i jij j kj k jkA v AB B x ωω==∑∑i ji j ji j ji kj k ki k jjjkkBA v B A A B A B x BA x ωω====∑∑∑∑∑So the matrix representation for linear transformation BA is the matrix product of the matrix representatitions for B and A.2.4 Show that the identity operator on a vector space V has a matrix representation which is one along the diagonal and zero everywhere else, if the matrix representation is taken with respect to the same input and output bases. This matrix is known as the identity matrix.Solution: let v i be the basis for the vector space V, if the matrix representation is taken with respecttothesameinputandoutputbases,wehave121100...00...0i i nA v v v v v v v v i i i -+=+++++++=,so 100 (00)10 (00)01...0:::0000...1A ⎛⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭=2.5 Verify that (. , .) just defined is an inner product on nC Solution:let12,,,......i nv ωωωω are vectors innC, we have(1)(),,i i iiiv i v λωλω⎛⎫=∑ ⎪⎝⎭∑, (2)()()****,,,*i i i i i i ii i v v vv v ωωωωω⎛⎫=== ⎪⎝⎭∑∑∑so()()*,,v vωω=. (3) ()*,0,i iiv v v v=≥∑ only if 0v =, (),0v v =So (. , .) just defined is an inner product on n C2.6 Show that any inner product (. , .) is conjugate-linear in the first argument,()*,,i i i i i iv v λωλω⎛⎫= ⎪⎝⎭∑∑ Solution:()()()*****,,,,,i i i i i iiii ii i iiiv v v v vλωλωλωλωλω⎛⎫⎛⎫==== ⎪⎪⎝⎭⎝⎭∑∑∑∑∑2.7 Verify that ()1,1ω≡ and ()1,1v ≡- are orthogonal. What are the normalized forms ofthese vectors.Solution: because ((1,1),(1,-1))=1-1=0,()1,1ω≡ and ()1,1v ≡- are orthogonal.v ω==))1,11,1- 2.8 Prove that the Gram-Schmidt procedure produces an orthonormal basis for V .Solution:()121211212112111212121211212112121,,v v v v v v v v v v v v v v v v ωωωωωωωωωωωωωωωωω---====---121111212110v v v ωωωωωωω-==-, let2.9 The Pauli matrices can be considered as operators with respect to an orthonormal basis 0 and 1 for a two-dimensional Hilbert space.Express each of the Pauli operators in the outer product notation. Solution:10010010000,1,01,10,0,110100100001⎛⎫⎛⎫⎛⎫⎛⎫⎛⎫⎛⎫====== ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭⎝⎭⎝⎭⎝⎭So00011,0110,0110,0011x y z i i σσσσ=+=+=-+=-2.10 Suppose i v is an orthonormal basis for an inner product space V . What is the matrix representation for the operator j k v v , with respect to the i v basis.Solution:,,,j i i k i i j k i iii i g i g iiiii gv m v v n v v v m v nm n v v ====∑∑∑∑∑2.11 Find the eigenvectors, eigenvalues, and diagonal representations of the Pauli matrices X,Y ,Z.Solution: X: eigenvalues:1 and -1, eigenvectors : and ⎛⎪ ⎪⎝⎭X ⎛ ⎛ =- ⎪⎝ ⎪⎝⎭Y: eigenvalues:1 and -1, eigenvectors: andY =- Z: eigenvalues:1 and -1, eigenvectors : 01and 0011Z ==-2.12 Prove that the matrix 1011⎛⎫⎪⎝⎭is not diagonalizable. Solution: for matrix 1011⎛⎫⎪⎝⎭, there is only one eigenvalue 1λ=, the eigenvectors is 01⎛⎫⎪⎝⎭,because ()01001111⎛⎫⎛⎫≠ ⎪ ⎪⎝⎭⎝⎭, the matrix 1011⎛⎫ ⎪⎝⎭is not diagonalizable2.13 Ifω and v are two vectors, show that ()†v v ω=Solution:()(),,vv v v v ωωωωωω==, according to (2.32), we have()†v v ωω=.2.14 Show that the adjoint operation is anti-linear, †*i i i i i i a A a A +⎛⎫= ⎪⎝⎭∑∑Solution:()11221122,......i i n n n n i v a A v a A a A a A a v A a A a v A ωωωωω⎛⎫⎛⎫=+++=+++ ⎪ ⎪⎝⎭⎝⎭∑ ()()()*1122,,...,,n n i i i a A v a A v a A v a A v ωωωω++++⎛⎫=+++= ⎪⎝⎭∑So, according to (2.32), †*i i i i i i a A a A +⎛⎫= ⎪⎝⎭∑∑2.15 Show that ()A A ++=Solution: ()()(),,v Av A A v A v ωωωω+++===, so ()AA ++=2.16 Show that any projector P satisfies the equation 2P P = Solution: 11122...ni P in n ===+++∑,()()21122 (1)122...P n n n n=++++++11112222...n n n n =++1122....n n P =++=2.17 Show that a normal matrix is Hermitian if and only if it has real eigenvalues. Solution:2.18 Show that all eigenvalues of a unitary matrix have modulus I , that is, can be written in theform i e θfor some real θ.Solution: let the eigenvector of a unitary matrix is α, so U αλα=, U αλαλα==,Because U αα=, we have 1λ=2.19 Show that the Pauli matrices are Hermitian and unitary. Solution: X **010*********X X XX ⎛⎫⎛⎫⎛⎫===⎪⎪ ⎪⎝⎭⎝⎭⎝⎭, *X X = Y **00100001i i Y Y YY i i --⎛⎫⎛⎫⎛⎫=== ⎪⎪ ⎪⎝⎭⎝⎭⎝⎭, *Y Y =Z ***101010,010101Z Z ZZ Z Z ⎛⎫⎛⎫⎛⎫====⎪⎪ ⎪--⎝⎭⎝⎭⎝⎭So the Pauli matrices are Hermitian and unitary.2.20 Suppose 'A and ''A are matrix representation of an operator A on a vector space V withrespect to different orthonormal bases, i j v and ω. Then the element of 'A and ''A are'''ij i j ij i j A v A v andA A ω==. Characterize the relationship between 'A and ''ASolution:2.21 Repeat the proof of the spectral decomposition in Box 2.2 for the case when M is Hermitian, simplifying the proof wherever possible. Solution:2.22 Prove that two eigenvectors of a Hermitian operator with different eigenvalues are necessarily orthogonal.Solution: 12,letU U αλαβλβ==, we have*2U βλβ+++=, *21U U βαλβλα+++=()*2110λλβα+-=,***12112212211,1,1,1,,1because λλλλλλλλλλ====≠≠()*2110λλ-≠, 0βα+= ,so two eigenvectors of a Hermitian operator with differenteigenvalues are necessarily orthogonal.2.23 Show that the eigenvalues of a projector P are all either 0 or 1.Solution: 22P PP P P ααλαλαλα====, 2P P =, we have 2P P ααλα== So 2λαλα=, ()20λλα-=, the eigenvalues of a projector P are all either 0 or 1.2.24 Show that a positive operator is necessarily Hermitian.(Hint: Show that an arbitrary operator A can be written A=B+Ic where B and C are Hermitian.)Solution: let operator A is positive, so A is 合同 with I. **A C IC C C == where C is ainvertible matrix. **A C C A ==, so a positive operator is necessarily Hermitian2.25 Show that for any operator A, A A +is positive.Solution: ()0AX A AX AX AX +++=≥, so A A +is positive.2.26 Let (=0+1ψ23and ψψ⊗⊗ explicitly, both in terms of tensorproducts like 01, and using the Kronecker product. Solution: ((()20+10+1000110112ψ⊗==+++()(3000110110+1ψ⊗=+++⊗=2.27 Calculate the matrix representation of the tensor products of the Pauli operators (a) X and Z; (b) I and X; (c) X and I. Is the tensor product commutative?Solution: (a) 001001100001100110000100X Z ⎡⎤⎢⎥-⎡⎤⎡⎤⎢⎥⊗=⊗=⎢⎥⎢⎥⎢⎥-⎣⎦⎣⎦⎢⎥-⎣⎦(b) 010010011000011000010010I X ⎡⎤⎢⎥⎡⎤⎡⎤⎢⎥⊗=⊗=⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎢⎥⎣⎦(c) 00100110000110011000010X I ⎡⎤⎢⎥⎡⎤⎡⎤⎢⎥⊗=⊗=⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎢⎥⎣⎦the tensor product is not commutative. 2.28 Show that the transpose , complex conjugation, and adjoint operations distribute over thetensor product, ()()()***;;TT T A B A B A B A B A B A B +++⊗=⊗⊗=⊗⊗=⊗Solution:()*******1112111121*******21222**21222******1212............=::::::::......n n n n m m mn m m mn A B A B A B A B A B A B A B A B A B A B A B A B A B A B A BA B A B A B A B A B ⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⊗==⊗⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦()T T 1112111211TT T 2122212222TT T 1212............::::::::......TTn m Tn T T m m m mn n n mn A B A B A B A B A B A B A B A B A B A B A B A B A B A B A BA B A B A B A B A B ⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⊗===⊗⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦So ()A B A B +++⊗=⊗2.29 Show that the tensor product of two unitary operators is unitary. Solution :let U,V are two unitary operators.()()()()m n mn U V U V U V U V U U V V I I I +++++⊗⊗=⊗⊗=⊗=⊗=So the tensor product of two unitary operators is unitary.2.30 Show that the tensor product of two Hermitian operators is Hermitian. Solution: let A,B are two Hermitian operators.()A B A B A B +++⊗=⊗=⊗ so the tensor product of two Hermitian operators is Hermitian.2.31 Show that the tensor product of two positive operators is positive. Solution: let A,B are two positive operators.()0T T T X A B X X AX X BX ⊗=⊗≥,so he tensor product of two positive operators ispositive.2.32 Show that the tensor product of two projectors is a projector. Solution: let A,B are two projectors()()22A B A B A B A B ⊗⊗=⊗=⊗,so the tensor product of two projectors is a projector2.33TheHadamardoperatorononequbitmaybewrittenas()()010011H ⎤=++-⎦. Show explicitly that the Hadamard transform on nqubits, nH ⊗, may be writtenas (),1x ynx yHx y ∙⊗=-. Write out an explicit matrixrepresentation for 2H ⊗.Solution: ()()(),0100111x yx yH x y ∙⎡⎤=++-=-⎣⎦Let(),1x yk x yH x y∙⊗=-,()()()1,1010011x yk k x yH H H x y ∙⊗+⊗⎡⎤=⊗=-++-⎣⎦2.34 Find the square root and logarithm of the matrix 4334⎡⎤⎢⎥⎣⎦Solution: the eigenvalues of matrix 4334⎡⎤⎢⎥⎣⎦are 121,7λλ== For11λ=, the eigenvector is 11-⎡⎤⎢⎥⎣⎦, for 27λ=, the eigenvector is 11⎡⎤⎢⎥⎣⎦So 1111P -⎡⎤=⎢⎥⎣⎦, 11221122P -⎡⎤-⎢⎥=⎢⎥⎢⎥⎢⎥⎣⎦114311101111112273411071111222222⎡⎤-⎢⎥--⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤==-+⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦⎢⎥⎢⎥⎣⎦2.35 let v be any real , three-dimensional unit vector and θ a real number. Prove that()()()exp cos sin i v I i v θθσθθσ∙=+∙ where 31i i i v v σσ=∙=∑Solution:2.36 Show that the Pauli matrices except for I have trace zero. Solution: tr[X]=0+0=0, tr[Y]=0+0=0, tr[Z]=1-1=0.2.37 If A and B are two linear operators show that tr(AB)=tr(BA). Solution: 112211111()...m mm m nii ii ni in ji ij i i i i j Tr AB A B AB A B A B ======+++=∑∑∑∑∑112211111()...n n n m ni i i i mi im ji ij i i i i j Tr BA B A B A B A A B ======+++=∑∑∑∑∑, so ()()Tr AB Tr BA =2.38 If A and B are two linear operators, show that ()()()Tr A B Tr A Tr B +=+ And if z is a arbitrary complex number show that ()()Tr zA zTr A = Solution: ()iii Tr A A≡∑ ()iii Tr B B≡∑()()()()ii ii ii ii iiiTr A Tr B A B A B Tr A B +=+=+=+∑∑∑()()ii ii iiTr zA zA z A zTr A ≡==∑∑2.39 The set V L of linear operators on Hilbert space V is obviously a vector space —the sum of two linear operators is a linear operator , zA is a linear operator is A is a linear operator and z is a complex number, and there is a zero element 0 . An important additional result is that the vector space V L can be given a natural inner product structure ,turning it into a Hilbert space. (1) Show that the function (),., on V V L L ⨯ defined by(),()A B tr A B +≡ is an innerproduct function. This inner product is known as the Hilbert —Schmidt or trace inner product. (2) If V has d dimensions show that V L has dimension 2d(3) Find an orthonormal basis of Hermitian metrices for the Hilbert space V L . Solution: (1) (),()(),iiiiiiiiA B tr A B tr A B A B λλλλ++⎛⎫≡== ⎪⎝⎭∑∑∑∑ (),()A B tr A B +≡, ()(),()(),B A tr B A tr A B A B ++++===()***1122111,()...0n n ni i i i im im i i i A A tr AA A A A A A A +=====+++≥∑∑∑If and only if 0A =, (),A A =0, so (),()A B tr A B +≡ is an inner product function.(2) (3)2.40 Verify the commutation relations [][][],2;,2;,2X Y iZ Y Z iX Z X iY ===Solution: []01000110,2210001001i i X Y XY YX i iZ i i --⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤=-=-==⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥-⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦[]01010001,2200101010i i Y Z YZ ZY i iX i i --⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤=-=-==⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥--⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦[]1001011002,20110100120Z X ZX XZ iY ⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤=-=-==⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥---⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦2.41 Verify the anti-commutation relations {},0i j σσ= where i j ≠ are both chosen from the set 1,2,3. Also verify that (i=0,1,2,3)2i I σ=Solution:{}12010001,0100010i i i i σσ--⎡⎤⎡⎤⎡⎤⎡⎤=+=⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦⎣⎦{}23010100,0001010i i i i σσ--⎡⎤⎡⎤⎡⎤⎡⎤=+=⎢⎥⎢⎥⎢⎥⎢⎥--⎣⎦⎣⎦⎣⎦⎣⎦{}3110010110,001101001σσ⎡⎤⎡⎤⎡⎤⎡⎤=+=⎢⎥⎢⎥⎢⎥⎢⎥--⎣⎦⎣⎦⎣⎦⎣⎦2221230101001010,,1010000101i i I I I i i σσσ--⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤======⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥--⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦2.42 Verify that []{},,2A B A B AB +=Solution:[]{},,22A B A B AB BA AB BA AB +-++==2.43 Show that for ,1,2,3j k =,31j k jk jkl l l I i σσδεσ==+∑Solution: when j=k ,j k jk I I σσδ==; when j k ≠ 31j k jkl l l i σσεσ==∑So for ,1,2,3j k =, 31j k jk jkl l l I i σσδεσ==+∑2.44 Suppose []{},0,,0A B A B ==, and A is invertible. Show that B must be 0. Solution: []{},0,,0A B AB BA A B AB BA =-==+= so 20,0AB AB == Because A is invertible, 0,0A AB B -==2.45 Show that [],,A B B A +++⎡⎤=⎣⎦Solution: [](),,A B AB BA B A A B B A ++++++++⎡⎤=-=-=⎣⎦2.46 Show that [][],,A B B A =-Solution: []()[],,A B AB BA BA AB B A =-=--=-2.47 Suppose A and B are Hermitian. Show that [],i A B is Hermitian. Solution:[]()()[],,,i A B iAB iBA iB A iA B i A B B A i A B i A B ++++++++++++⎡⎤=-=-+=-==⎣⎦2.48 What is the polar decomposition of a positive matrix P? Of a unitary matrix U? Of a Hermitian matrix , H?Solution: if P is a positive matrix, the polar decomposition of P is P=IP If U is a unitary matrix, the polar decomposition of U is U=UI If H is a Hermitian matrix, the polar decomposition of H is2.49 Express the polar decomposition of a normal matrix in the outer product representation. Solution:2.50 Find the left and right polar decompositions of the matrix 1011⎡⎤⎢⎥⎣⎦. Solution: let A= 1011⎡⎤⎢⎥⎣⎦, 101010111121A A +⎡⎤⎡⎤⎡⎤==⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦,the eigenvalues of A A + is 1λ=,the eigenvector of λ is 01⎡⎤⎢⎥⎣⎦ 约当标准型：1010,22010P P -⎡⎤⎡⎤⎢⎥==⎢⎥⎢⎥⎣⎦⎣⎦ 11001110221200110P A AP -+⎡⎤⎡⎤⎡⎤⎡⎤⎢⎥==⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦⎣⎦110121101P P -⎡⎤⎡⎤⎢⎥==⎢⎥⎢⎥⎣⎦⎣⎦=JA= 1011⎡⎤⎢⎥⎣⎦1011I ⎡⎤=⎢⎥⎣⎦2.51 Verify that the Hadamard gate H is unitary.Solution: 111111,111111H H H H HH I +++⎤⎡⎤=====⎥⎢⎥---⎦⎣⎦So the Hadamard gate H is unitary.2.52 Verify that 2H I =Solution: 2H H H I +==2.53 What are the eigenvalues and eigenvectors of H? Solution: the eigenvalues of H are121,1λλ==-For 11λ=,the eigenvector is 11⎡⎤⎥⎦, for 21λ=-, the eigenvector is11⎡⎤⎢⎥⎣⎦2.54SupposeAandBarecommutingHermitianoperators.Provethat()()()exp exp exp A B A B =+.Solution: ()()2211exp exp ......2!2!A B I A A I B B ⎛⎫⎛⎫=++++++ ⎪⎪⎝⎭⎝⎭()()()2232231133...2!3!I A B A AB BA B A A B AB B =+++++++++++ ()()()()2311...exp 2!3!I A B A B A B A B =+++++++=+2.55 Prove that ()12,U t t defined in equation (2.91) is unitary.Solution: ()()()()21211212,exp ,,exp iH t t iH t t U t t U t t h h +---⎡⎤⎡⎤==⎢⎥⎢⎥⎣⎦⎣⎦()()()()()()212121211212,,exp exp exp iH t t iH t t iH t t iH t t U t t U t t Ih h h h +------⎡⎤⎡⎤⎡⎤==+=⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦(()21iH t t h- and()21iH t t h-- are commuting )So ()12,U t t defined in equation (2.91) is unitary.2.56 Use the spectral decomposition to show that ()log K i U ≡- is Hermitian for any unitaryU, and thus exp()U iK = for some Hermitian K. Solution:2.57 Suppose {}{}l m L and M are two sets of measurement operators. Show that a measurement defined by the measurement operators{}l L followed by a measurement defined by themeasurement operators {}m M is physically equivalent to a single measurement defined by measurement operators {}lm N with the representation lm m l N M L = Solution: Suppose the state of the quantum system is ϕ, after operators {}lm N measure, thestate of the system isIf a measurement defined by the measurement operators {}l L followed by a measurement defined by the measurement operators {}m M , after the measurement by operators {}l L , thestate become{}m M ,the state become==So we can see that a measurement defined by the measurement operators {}l L followed by a measurement defined by the measurement operators {}m M is physically equivalent to a single measurement defined by measurement operators {}lm N with the representation lm m l N M L =2.58 Suppose we prepare a quantum system in an eigenstateϕ of some observable M, withcorresponding eigenvalue m .What is the average observed value of M , and the standard deviation?(system state ?)Solution: the average observed value of M is2.59 Suppose we have qubit in the state 0, and we measure the observable X. What is theaverage value of X? What is the standard deviation of X?Solution: X⎛⎛=-⎪⎝⎪⎝⎭It gives the result +1 with probability1002=It gives the result -1 with probability1002⎛⎛=⎪⎝⎪⎝⎭the average value of X is 0. the standard deviation of X is 1.2.60 Show that vσ∙has eigenvalues 1±, and that the projectors onto the corresponding eigenspaces are given by ()2P I vσ±=±∙Solution: 312123v v ivvv iv vσ-⎡⎤∙=⎢⎥+-⎣⎦，321213v iv vviv v vλλσλ---∙==--+So 222222221231230,1,1,1v v v v v vλλλ---=++===±,,0mP P P IandP P P P P P P P+-+++---+-=+====∑v P Pσ+-∙=-So the projectors onto the corresponding eigenspaces are given by ()2P I vσ±=±∙2.61 Calculate the probability of obtaining the result +1 for a measurement of vσ∙, given that the state prior to measurement is 0.What is the state of the system after the measurement if +1 is obtained?Solution: after the measurement if +1 is obtained, the state of the system is312012I vvv ivσ+∙+⎡⎤=⎥+⎦2.65 Express the states ((0101+-in a basis in which they are not the same up to a relative phase shift.Solutiong: 1211,11v v ⎤⎤==⎥⎥-⎦⎦((1212010,010v v v v +=+-=+they are not the same up to a relative phase shift.2.66 Show that the average value of the observable 12X Z for a two qubit system measured in the state(0011+ is zero.Solution: 10100000,11,0011000011⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥==+=⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦ ,120010000110000100X Z ⎡⎤⎢⎥-⎢⎥=⎢⎥⎢⎥-⎣⎦the average value of the observable 12X Zis12X Z ϕϕ]0010100010100101000001001⎡⎤⎡⎤⎢⎥⎢⎥-⎢⎢⎥==⎢⎢⎥⎢⎢⎥-⎣⎦⎣⎦2.68 Prove thata b ψ≠ for all single qubit state a and b .Solution: 11111222212210,,01a b a b a b let a and b a b a b a b a b ψ⎡⎤⎡⎤⎢⎥⎢⎥⎡⎤⎡⎤⎢⎥⎢⎥====⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎢⎥⎢⎥⎣⎦⎣⎦11122122,0a b a b f a b then a b a b ψ⎧=⎪=⎪=⎨=⎪⎪=⎩, it is impossible, so a b ψ≠ for all single qubit state a and b .2.69 Verify that the Bell basis forms an orthonormal basis for the two qubit state space.Solution:00011001,0110ββ⎡⎤⎡⎤⎢⎥⎢⎥⎥⎥====⎥⎥⎥⎥⎣⎦⎣⎦10011001,0110ββ⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥====⎢⎥⎢⎥-⎢⎥⎢⎥-⎣⎦⎣⎦So the Bell basis forms an orthonormal basis for the two qubit state space.2.71 Let ρbe a density operator. Show that ()21trρ≤, with equality if and only if ρis a pure state.Solution: 22i j i j ji j i j ji j j i i j j j jρλλλλλ====∑∑∑∑()2221j jj jtr tr j jρλλ⎛⎫==≤⎪⎝⎭∑∑. If ρis a pure state, λ=1, ()21trρ=2.72 The Bloch sphere picture for pure states of a single qubit was introduced in Section 1.2. This description has an important generalization to mixed states as follows:(1) Show that an arbitrary density matrix for a mixed state qubit may be written as()2I rρσ=+∙, where r is a real three-dimensional vector such that 1r≤.This vector is known as the Bloch vector for the state ρ(2) What is the Bloch vector representation for the state 2Iρ=?(3) Show that a state ρis pure if and only if 1r=(4) Show that for pure states the description of the Bloch vector we have given coincides with that in Section 1.2.Solution: (1) 312123r r irrr ir rσ-⎡⎤∙=⎢⎥+-⎣⎦()3123121222122r r irI rrr irσ+-⎡⎤⎢⎥+∙=⎢⎥-+⎢⎥⎢⎥⎣⎦Because 1r≤,we have 2221231r r r++≤,3122221233312111220014222r r irr r r randrr ir+----+=≥≥-+,so()312312122122r r irI rrr irσ+-⎡⎤⎢⎥+∙=⎢⎥-+⎢⎥⎢⎥⎣⎦is a self-adjoint non-negative operator with unit trace.So an arbitrary density matrix for a mixed state qubit may be written as ()2I rρσ=+∙.(2) the Bloch vector representation for the state2I ρ= is r=0(3) ()222312322223123212*4412*4r r r r I r r r r r ρσ+⎡⎤+⎢⎥⎢⎥==-⎢⎥⎢⎥⎣+∙+⎦++++()222123212tr r r r ρ+++=,if and only if 1r =,()21tr ρ=So a state ρ is pure if and only if 1r =. (4)2.73 Let ρ be a density operator and supposei ψ is some linearly independent set of purestates spanning the support of ρ.(The support of a Hermitian operator A is the vector space spanned by the eigenvectors of A with non-zero eigenvalues.)Show that there is a unique probability distribution ()i p such thati i i ip ρψψ=∑, and moreover, the probabilitiesi p are given by 11i i ip ψρψ-=, where1ρ- is defined to be the inverse of ρ, when ρis considered as an operator acting only on the support of ρ.(This definition removes the problem that ρ may not have an inverse.) Solution:2.74 Suppose a composite of systems A and B is in the state a b , where a is a pure state of system A,and b is a pure state of system B.Show that the reduced density operator of system A alone is a pure state. Solution:()()()()(),AB A AB B B B a b a b tr tr a b a b tr a a b b a aρρρ====⊗=2.75 For each of the four Bell states, find the reduced density operator for each qubit. Solution: for Bell state(0011+,00001100001111112ρ+++==()()()()()1111100001100001111112tr tr tr tr tr ρ+++=000010100101111122I+++==()2tr ρ=。

磷石膏低温分解研究及进展谢龙贵马丽萍*陈宇航戴取秀崔夏许文娟（昆明理工大学环境科学与工程学院，云南昆明650093）摘要：磷石膏是磷化工中产生量最大的工业废渣，其大量堆存导致严重的环境污染，同时也造成土地和硫资源的大量浪费。

将其用于制硫酸联产水泥能很好的解决磷石膏的堆存问题，然而，磷石膏热分解的高能耗成为了该项技术推广应用的瓶颈，因此，不少学者对磷石膏的低温分解做了一定的研究，研究表明，反应气氛和外加助剂对磷石膏分解温度的降低具有显著贡献。

关键词：磷石膏；低温分解；研究进展The Research Progress of Low-temperature Decomposition ofPhosphogypsumXIE Longgui, MA Liping*,CHEN Y uhang, DAI Quxiu, CUI Xia, XU Wenjuan(Faculty of Environmental Science and Engineering ,Kunming University of Science and Technology，Kunming 650093,China)Abstract: Phosphogypsum is the largest industrial solid waste of phosphorous chemical industry. Its store up in quantity causes serious environmental pollution, moreover, great amount of land and sulfur are wasted. To use phosphogypsum to produce sulphuric acid and cement can decrease its storage volume at a large extent, however, the decomposition of phosphogypsum under high temperature becomes the limits to popularize this technology. Lots of scholars have done a great amount of work to lower the decomposition temperature, the results showed that the reaction atmosphere and some additives do a significant contribution to lower the decomposition temperature.Keywords: Phosphogypsum, Low-temperature decomposition, Research progress基金项目：国家高技术发展计划“863”项目（2007AA06Z321）。

英语可数与不可数名词研究一、引言在英语教学中，可数名词与不可数名词一直是让老师尴尬，学生痛苦的部分。

每次提及bread，chocolate，paper，water，money等不可数名词时，老师说的最多的就是：记下来，这个词不可数，没有复数，不能加s，做主语时，谓语动词用单数。

但当学生问及原因时，却总也不能做出很好的回答。

根据英语传统语法，如：张道真（1995），薄冰（1998），章振邦（1999），Quirk（1985）等学者以名词能否计数的特征来区分可数名词与不可数名词，这也确实给学习者带来了帮助但还是有其缺陷。

因为这种区分只能解决人们较熟悉的典型的可数名词和不可数名词但并不能解决很多的例外。

比如说bread 这个词，国内学生很难理解为什么它是不可数的。

与传统观念不同，美国语言学家Langacker（1987）从认知的视角出发系统地探讨了名词数的问题他依据有界性（Boundedness），同质性（Homogeneity），分割性（Contractibility）以及叠加性（Replicability）区分可数名词和不可数名词。

例如，car作为一个典型的可数名词，占有一定的空间，具有明确边界；car由引擎车轮方向盘座位车门等部件构成，内部不同质；car不可切分，拆卸后的car就不再是car；而car与car的叠加其结果为两个car。

与此相反，不可数名词water没有明确边界；内部同质，无论切分还是叠加，其结果依旧是water。

名词可数性的这种认知解释确实让人耳目一新，这种提法也看似可以解决所有的问题。

但它一方面过于抽象，另一方面也有其不可触及之处。

比如还是同样的问题，bread 一词是有界的，可分割的，但它却是不可数名词。

本文基于以上学者对可数与不可数名词区分的探讨，提出了一种可有效区分可数名词与不可数名词的通俗易懂的方法——分割法，一种顺应学生汉语思维方式来诠释可数名词与否的方法，以期让中学英语教师摆脱那种必须讲又讲不通的尴尬处境，让学生对英语学习更加充满激情，学起来更加轻松愉快。

关于信息超图一些基本概念的注记于德玉;吉日木图【摘要】随着信息科学的发展,超图在信息科学中有着非常广泛的应用,例如网络工程、数据库理论、聚类、分子超图的识别和分子超图的结构等等.在《The Information Hypergraph Theory》一书中,信息超图的基本概念被引入,这一概念来源于信息科学的数据库理论.本文对信息超图的基本概念进行解读和补充,并得到了几个新的结果.【期刊名称】《内蒙古民族大学学报（自然科学版）》【年(卷),期】2017(032)002【总页数】3页(P98-100)【关键词】孤立点;耳朵;无圈;伴随超图;补图【作者】于德玉;吉日木图【作者单位】内蒙古民族大学数学学院,内蒙古通辽028043;内蒙古民族大学离散数学研究所,内蒙古通辽028043;内蒙古民族大学数学学院,内蒙古通辽028043;内蒙古民族大学离散数学研究所,内蒙古通辽028043【正文语种】中文【中图分类】O157.5超图是有限集合的子集系统，是离散数学中最一般的结构.早期的定理有Sperner 定理和Ramsey定理等.Duchet〔1〕概括了超图的历史原由:20世纪，公理化趋势的影响和离散数学带来的快速增长的经济利益促使科学家去发展一种有限集合族的系统的组合方法.20世纪60年代，Berge在《Graphs and Hypergraphs》中首次提出“超图”一词，并建立了无向超图理论，用拟阵来研究超图在运筹学中的应用.Erdös和Hajnal〔2〕奠定了超图的理论基础.作为普通图的推广，超图在树、圈、覆盖和染色等方面已取得一些重要结果〔3，4〕.关系数据模型是科学家研究数据库时需要解决的一个重要问题，而关系数据设计的关键是设计一个数据库图式，使信息不损失.上世纪80年代，信息学家在研究信息理论时，发现属性集R上的一个数据库图式就是R上的一个超图.进而引进了无圈超图的概念.这与传统超图的有关定义不同，称之为“信息超图”，并证明其在计算机科学和信息科学中十分有用〔5，6〕，本文基于超图的研究，给出了超图的孤立点、耳朵和有圈之间的关系，还给出了一致无圈超图的伴随超图是无圈的必要条件以及一致超图的伴随超图和补图的关系.设X是有限集合表示X的所有k-元子集构成的集合，2X表示X的所有子集构成的集合.定义2.1〔7〕是一个有限集合是V的一个子集合族，即ε⊆2V.如果和，我们称H=(V,ε)是V上的一个超图，或简称ε是一个在V上的超图，集合ε称为边集.称为ε的阶，称为ε的规模.如果ε的每条边e，都有|e|=k，则称ε是一个k-一致超图.如果k=2，则ε是一个通常的图.如果ε是由V的所有k-元子集构成的集合，则称ε是一个完全k-一致超图，记为定义2.2〔7〕一个超图ε称为不可约超图，或简单超图，如果它的任意两条边都互不包含，即∀e,e′∈ε, e≠e′，都有e\e′≠∅.下面除特别说明，所说的超图都是简单超图.定义2.3〔7〕设ε是V上的一个超图，一个顶点u∈V，如果u仅属于ε的一条边，则u被称为ε的一个孤立点.定义2.4〔7〕一条边e∈ε，如果存在另外一条边e′∈ε使得e\e′中每个顶点都是孤立点，则e称为ε的一个耳朵.特别地，如果e的所有顶点都是孤立点，则e也是一个耳朵.例如:ε={a bc,cde,efa,ace}，b，d，f都是ε的孤立点，abc，cde，efa都是ε的耳朵.定义2.5〔7〕令ε是V上的一个超图，称为ε的伴随超图，如果={V \e:e∈ε}.定义2.6〔13〕设ε是V上的一个k-一致超图，称εc为ε的补超图，如果超图ε的伴随超图和补超图概念不同.例如，则.很显然定义2.7〔12〕一个k-一致超图H是一个l-圈，如果存在H的顶点的一个循环序列使得H的每条边由k个连续的顶点组成，且每一对连续的边都恰相交于l个顶点(1 ≤l≤k-1).如果l=k-1，则H被称为一个紧圈，如果l=1，则H被称为一个松圈.下面除特别说明，所说的圈都是紧圈.引理2.8〔7〕对一个n阶m条边的约简无圈超图，有m≤n-max |e|+1.引理2.9〔7〕对一个最大n阶m条边的k-一致无圈超图，有m=n-k+1.引理2.10〔7〕一个不少于两条边的无圈超图至少有两个耳朵.无圈公理〔7〕一个超图ε可以通过重复使用下面两种运算化为空集:(a1)若x是孤立顶点，去掉x.(a2)若ei⊆ej,i≠j，去掉ei.一个等价的表述:ε可以通过重复移去耳朵变为空集.满足无圈公理的超图被称为无圈超图，否则称为有圈的.一个无圈超图的部分超图可能是有圈的.例如:ε={a bc,cde,efa,ace} 是无圈的，而是ε的部分超图，但ε′是有圈的.超图的这种性质是普通图所没有的.定理3.1若超图ε有耳朵，则ε一定有孤立点.证明由超图耳朵定义，设超图的一个耳朵，则存在另外一条边e′∈ε使得e\e′中每个顶点都是孤立点，所以ε有孤立点.注意到此定理的逆定理不一定成立.反例:设.显然ε有一个孤立点a，但是ε没有耳朵.定理3.2若超图ε无孤立点，则超图ε一定有圈.证明由定理3.1，超图ε没有孤立点，则ε没有耳朵，故由超图的无圈公理，ε不能变为空集，从而超图ε是有圈的.定理3.3设ε是一个最大的n阶k-一致无圈超图.若ε的伴随超图是一个无圈超图，则n≤2k.证明由伴随超图的定义有，n阶k-一致超图ε的伴随超图是n阶(n -k)-一致超图，且伴随超图的边数 ||与原超图的边数|ε|相等，即||= |ε|.又若是无圈的，由引理 2.8有，由引理2.9可知，最大的n阶k-一致无圈超图ε的边数|ε|=n-k+1.从而n-k+1≤k+1，即n≤2k.注意到此定理的逆定理不一定成立.反例:设，显然其伴随超图.容易验证ε是一个最大的n阶k-一致无圈超图，且满足n≤2k.但由于没有孤立点，由定理3.2，所以有圈.定理3.4设ε是n阶k-一致超图，其伴随超图和补图εc有如下关系:（1）当n≠2k时，和εc没有包含关系.（2）当n=2k时，ε⋂=∅的充分必要条件是⊆εc.证明（1）由n阶k-一致超图补图和伴随超图的定义可知，超图ε的补图εc是n 阶k-一致超图，伴随超图是n阶(n -k)-一致超图.由于n≠2k，所以n-k≠k，即伴随超图不是k-一致超图，所以和εc没有包含关系.（2）当n=2k时，ε是n阶k-一致超图，所以和εc均为n阶k-一致超图.必要性.反证法.假设⊄εc，则存在一条边e，使得e∈且e∉εc，即e∈ε，所以ε⋂≠∅.这与ε⋂=∅矛盾.充分性.反证法.假设ε⋂≠∅，设存在一条边e，使得e∈ε且e∈，由补图的定义可知，e∈且e∉εc.所以⊄εc，这与⊆εc矛盾.当n=2k时，ε⋂=∅的充分必要条件是⊆εc.〔1〕P Duchet.Hypergraphs，Handbook of combinatorics〔M〕.Beckeley:The MIT Press，1995.381-423.〔2〕Erd s P，A.Hajnal.On the chromatic number of graphs and Set systems〔J〕.Acta Math Acda sci Hungar，1966（17）:61-99.〔3〕贝尔热，卜月华.超图有限集的组合学〔M〕.南京:东南大学出版社，2002. 〔4〕B Bollobábinatorics〔M〕.Cambridge:Cambridge University Press，1986.〔5〕Beer C，Fagin R，Maier D et al.On the Desirability of Acyclic Database Schemes〔J〕.Journal of the Acm，1983，30（3）: 479-513. 〔6〕T T Lee.partII information structures of database schemes〔J〕.IEEE Transactions on Software Engineering，1987，13（10）: 1062-1072. 〔7〕王建方.超图的理论基础〔M〕.北京:高等教育出版社，2006.〔8〕吴望名.图论及其应用〔M〕.北京：科学出版社，1984.〔9〕吉日木图.图的标号及超图的分解问题的研究〔D〕.大连:大连理工大学，2006.〔10〕沙特朗，张萍.图论导引〔M〕.北京：人民邮电出版社，2007.〔11〕王建方，闫桂英.超图的圈结构〔J〕.科学通报，2001，（19）:1585-1589. 〔12〕Daniela Kuhn，Deryk Osthus.Decompositions of complete uniform hypergraphs into Hamilton Berge cycles〔J〕.Journal of Combinatorial Theory，2014，126（1）:128-135.〔13〕Chang An，S Miao.The Laplacian of the Complement of a k-uniform hypergraph〔J〕.Journal of Mathematical Study，2000，33（3）:251-260.。

The management of the construction industry is exponentially increasing in complexity since it has to deal with highly fragmented, complex and unique combinations of business relations, communications and processes. Each phase in the construction cycle requires effective communication/sharing of information/knowledge as well as coordination among project participants and stakeholders, thus leading to timing and technical content transfer enhancements and/or problems as well as related efficiencyand/or profitability loss. Recent advances in the Information and Communication Technology (ICT) are considered essential and promising in improving sharing/exchange of project information as well as communication among construction industry stakeholders whilst reducing associated costs and time.One facet of such advances in ICT is evident from the proactive indulgement of the construction supply chain in collaborative extranets. Extranets are able to capture the supply chain communication practices and provide a controlled communication hub as well as document management databases with all ICT enhanced features. In parallel, the construction industry has recognized the importance of sharing and exchanging project information across the supply chain through project integrated databases. However, such advancements in the ICT utilisation in construction industry were unable to develop a system that exchanges/shares project information, at the element level, over a well defined matrix of communication that can also integrate into the business processes of the members of supply chain.The objective of this study is to highlight the needs to bridge this gap through the development of an augmented process model which will enable integrated databases to support collaborative extranets at the Tender stage. This paper will discuss the features, limitation and nature of the process and information models which can be integrated into the core business processes of the construction supplychain members. This is basically achieved by investigating the communication processes and information exchanged across the supply chain during the construction tendering phase.Article Outline1. Introduction2. Research methodology3. Construction tendering stage4. ICT application in construction tendering4.1. Web-collaborative extranets4.2. Project integrated databases5. Process based thinking, role and its application in CI6. Information modelling, its role and application in CI7. Case studies validation8. The need for an augmented process model9. ConclusionsReferencesFRAMEWORK FOR INVESTMENT DECISION-MAKING UNDER RISK AND UNCERTAINTY FOR INFRASTRUCTURE ASSET MANAGEMENT Review Article Research in Transportation EconomicsResources, supplier investment, product launch advantages, and first productperformance Original Research ArticleJournal of Operations ManagementEvaluation of a computer integration strategy in a science teacher's professional development program Original Research ArticleStudies In Educational EvaluationThe present study investigates the relationship between change in kinaesthetic ability and change in openness to experience among Dance Movement Therapy (DMT) trainees, compared with Art Therapy and Social Science students. A field study was conducted using a quasi-experimental pre–post control group design. Participants were 62 graduate students. A standard measure of openness to experience was used and kinaesthetic ability was evaluated using a table of movement dimensions based on Laban Movement Analysis (LMA). Results suggested an increase among the DMT cohort in all domains of kinaesthetic ability as well as in openness to experience. This increase in kinaesthetic ability was related to increases in openness to experience. Therapeutic elements and movement experience in DMT training may thus promote an increase in both kinaesthetic ability and openness to experience.Safety in construction – a comprehensive description of the characteristics of high safety standards in construction work, from the combined perspective of supervisors and experienced workers Original Research ArticleJournal of Safety ResearchAnalysis of export demand for Ghana's timber products: A multivariateco-integration approach Original Research ArticleJournal of Forest EconomicsArticle OutlineIntroductionAuthentic MovementThe Chace approachLaban Movement AnalysisOpenness to experience and movementMethodParticipantsInstrumentsMovement assessmentOpenness to experienceProcedureResultsChange in kinaesthetic abilityChange in openness to experienceThe relationship between change in kinaesthetic ability and change in openness to experience among the DMT groupDiscussionLimitations and suggestions for further researchImplications, contributions and conclusionsReferencesThe Impact of Organizational Integration and Product Development Proficiency on Market Success Original Research ArticleIndustrial Marketing ManagementProposal of Polish Guidelines for Conducting Financial Analysis and Their Comparison to Existing Guidance on Budget Impact in Other Countries Original Research ArticleValue in HealthThis study analyzed the factors that affect the export demand for Ghana's timber products. Export demand functions and error correction models for sawnwood, plywood, and veneer using data from 1961 to 2006 were estimated. Six categories of explanatory variables were hypothesized to determine export demand: world price of wood products, income of importing countries, Ghana's external debt, exchangerates, time-related variables, and policy changes (log export ban, reduction in annual allowable cut and the imposition of export levy on air-dried sawnwood). All variables except external debt were determined from augmented Dickey–Fuller unit root tests to be integrated of order one. The Johansen multivariate cointegration test showed that there was only one cointegration relationship in each timber product data. Exchange rates and income were significant determinants of exported timber products and had the theoretically expected positive signs. The three policy initiatives significantly reduced the exports of sawnwood, but increased the exports of plywood and veneer. Price was moderately elastic for sawnwood and plywood and had the expected negative signs in both cases, while it was positive and inelastic for veneer. The error-correction coefficients show that 68% of shocks to veneer exported is corrected in the following year, while only approximately 20% and 19% of this are corrected for sawnwood and plywood, respectively. Sawnwood and plywood face stiff competition in the international market and this has revenue and tax policy implications for Ghana's forestry sector. Policies that encourage domestic processing and restrictions on both legal and illegal harvesting would work to ensure greater value-added benefits to, and sustainable forest management in, Ghana.Flyrock phenomena and area security in blasting-related accidentsSafety ScienceArticle OutlineIntroductionOverview of Ghana's forest products export sectorTrends in forest products exportsRegulatory and policy environmentPrevious studiesMethodsTheoretical export demand functionDataEconometric methodsResultsUnit roots and cointegration testsLong-run coefficients and short-run dynamicsDiscussionConclusions and policy implicationsReferencesISO 9000 certification and construction project performance: The Malaysian experience Original Research ArticleInternational Journal of Project ManagementThis study provides evidence on the role of financial development in accounting for economic growth in low- and middle-income countries classified by geographic regions. To document the relationship between financial development and economic growth, we estimate both panel regressions and variance decompositions of annual GDP per capita growth rates to examine what proxy measures of financial development are most important in accounting for economic growth over time and how much they contribute to explaining economic growth across geographic regions and income groups. We find a positive relationship between financial development and economic growth in developing countries. Moreover, short-term multivariate analysis provides mixed results: a two-way causality relationship between finance and growth for most regions and one-way causality from growth to finance for the two poorest regions. Furthermore, other variables from the real sector such as trade and government expenditure play an important role in explaining economic growth. Therefore, it seems that awell-functioning financial system is a necessary but not sufficient condition to reach steady economic growth in developing countries.Article Outline1. Introduction2. Literature review3. Data and proxy measures3.1. Structuring the panel dataset3.2. Proxy measures for financial development and economic growth4. Panel estimations and multivariate time-series methodology4.1. Panel estimations with convergent term4.2. Multivariate time-series models5. Empirical results5.1. Summary statistics of proxy measures5.2. Analysis of panel regressions5.3. Analysis of VAR results by geographic regions and income groups5.3.1. East Asia & Pacific (low- and middle-income countries)5.3.2. Europe & Central Asia (low- and middle-income countries)5.3.3. Latin America & Caribbean (low- and middle-income countries)5.3.4. Middle East & North Africa (low- and middle-income countries)5.3.5. South Asia (low- and middle-income countries)5.3.6. Sub-Saharan Africa (low- and middle-income countries)5.3.7. High-income OECD countries5.3.8. High-income non-OECD countries6. ConclusionsAppendix A. Time-series averages of variables by country (1980–2007)ReferencesThe pragmatics of clinical hypermedia: experiences from 5 years of participatory design in the MEDEA project Original Research ArticleComputer Methods and Programs in BiomedicineResearch Highlights►ISO 9000 certified companies have enhanced project performance. ► Exceptions relate to the areas of partnerships and managing resources. ► ISO 9000 moderates the relationship between practices and project success.The financial role of merchants in the development of U.S. manufacturing, 1815–1860Original Research ArticleExplorations in Economic HistoryCasino management: Exploring gender-based differences in perceptions of managerial work Original Research ArticleInternational Journal of Hospitality Management。

How to Split a Flow ?Tzvika Hartman ∗,Avinatan Hassidim ∗,Haim Kaplan ∗†,Danny Raz ∗‡,Michal Segalov ∗∗Google,Inc.Israel R&D Center †Tel Aviv University ‡Technion,Israel{tzvika,avinatan,haimk,razdan,msegalov }@Abstract —Many practically deployed flow algorithms produce the output as a set of values associated with the network links.However,to actually deploy a flow in a network we often need to represent it as a set of paths between the source and destination nodes.In this paper we consider the problem of decomposing a flow into a small number of paths.We show that there is some fixed constant β>1such that it is NP-hard to find a decomposition in which the number of paths is larger than the optimal by a factor of at most β.Furthermore,this holds even if arcs are associated only with three different flow values.We also show that straightforward greedy algorithms for the problem can produce much larger decompositions than the optimal one,on certain well tailored inputs.On the positive side we present a new approximation algorithm that decomposes all but an -fraction of the flow into at most O (1/ 2)times the smallest possible number of paths.We compare the decompositions produced by these algorithms on real production networks and on synthetically generated data.Our results indicate that the dependency of the decomposition size on the fraction of flow covered is exponential.Hence,covering the last few percent of the flow may be costly,so if the application allows,it may be a good idea to decompose most but not all the flow.The experiments also reveal the fact that while for realistic data the greedy approach works very well,our novel algorithm which has a provable worst case guarantee,typically produces only slightly larger decompositions.I.I NTRODUCTIONOften when tackling network design and routing prob-lems,we obtain a flow (or a multicommodity flow)between a source-sink pair (or pairs)by running a maximum flow algorithm,a minimum cost flow algorithm,or solving a linear program that captures the problem.1These algorithms represent and output a flow by associating a flow value with each arc.The value of each arc is at most the arc capacity,and in each node the total incoming flow equals the total outgoing flow,except for the source and the sink.See Figure 1for an example.To deploy a flow in a network,say a traffic engineered MPLS network [7],[15],or an open-flow network [13],we often need to represent the flow as the union of paths from the source to the sink,such that each path is coupled with the amount of flow that it carries.In general,there may be many ways to represent a flow as a union of source-sink paths (see Figure 1).While all these representations route the same flow1Linearprogramming or even convex programming is typically the methodof choice for more complicated multicommodity flow and network designproblems.Fig.1.A flow.We can decompose this flow into 4flow-paths e,f,g,h of value 1,a,b,g,h of value 1,a,b,i of value 1,and a,b,c,d of value 2.A smaller decomposition consists of the paths e,f,i of value 1,a,b,g,h of value 2,a,b,c,d of value 2.they may differ substantially in the number of paths used,their latency,and in other parameters.Our focus in this paper is on algorithms that decompose a flow into a minimum number of paths.(If we are given a multicommodity flow,we just apply the algorithm to each flow separately.)Minimizing the number of paths is desirable for many reasons.When engineering traffic in a network each path consumes entries in tables of the routers that it goes through.Each path also requires periodic software maintenance to check its available bandwidth and other quality parameters [16],[4].Therefore,reducing the number of paths saves resources and makes network management more efficient,both for automatic software and network operators.Evidently,the number of paths is not the only important quality criteria of the decomposition.Other parameters are also important.Among them are the maximum and average latency of the paths,and the maximum number of paths going through a single link or a single node.Optimizing more than one parameter is not naturally well defined,and thereby more complicated.We focus on the number of paths as our main objective criterion,but we do address other criteria in our experimental study.Flow algorithms use the notion of residual graph which contains all nonsaturated arcs on which we can push more flow.Initially the residual is identical to the original graph.An augmenting path based algorithm finds a directed path from the source to the sink in the residual graph (called an augmenting path ),pushes the maximum possible flow along it and updates the residual graph.It stops when there is no directed path from the source to the sink in the residual graph.Different algorithms choose different augmenting paths,and their running time is affected by the number of paths they use.Why not take the paths produced by such an algorithm asour decomposition?There are two problems with this naive approach.First,the augmenting paths used by these algorithms are paths in the residual graph which contains arcs that are not in the flow,therefore these paths may not be contained in the original flow.See Figure 2for such an example.Second,the number of these augmenting paths may be large,since even the fastest augmenting path algorithms use a quadratic (in the size of the graph)number ofpaths.p 1212stsFig.2.In this flow f the capacities of all the arcs are 1.The dashed edges represent long paths.Assume that we run an augmenting path maximum flow algorithm,like,for example the algorithm of Edmonds and Karp on G (f ).The first augmenting path can be a,b,c .(Augmenting path algorithm augment flow on shortest residual paths.)After pushing one unit of flow on a,b,c we obtain the residual graph at the bottom of the figure.The next augmenting path is d,e,b ,f,g which uses the arc b which is opposite to b and is not in the original flow.The last augmenting path would be the concatenation of p 1,b and p 2.A.Our ResultsWe consider two natural greedy algorithms for the problem,also studied by [18].One picks the widest flow-path,removes it and recurses,we call it greedy width .The other picks a shortest (by latency)path,removes it and recurse,we call it greedy length .We show that in the worst case these greedy algorithms produce a large decomposition compared to the optimal one.Specifically,the number of paths which they produce could be as large as Ω(√m )OP T where m is the number of arcs and OP T is the size of the optimal decomposition.Furthermore,this bad performance occurs even if we want to decompose only a constant fraction of the flow.We give a new algorithm similar to the greedy algorithms mentioned above,and prove that for any it decomposes(1− )fraction of the flow into at most O (OP T2)paths.In particular,for an appropriate choice of parameters,it can decompose 1/3of the flow into at most OP T paths (see Section V).We consider two versions of this algorithm,one which we call the bicriteria width algorithm which is similar to the greedy width algorithm and the other which we call the bicriteria length algorithm which is similar to the greedy length algorithm.It is known that the problem of decomposing a flow into the minimum number of paths is strongly NP-hard by a reduction from 3-partition [18].But notice that the 3-partition problem is easy if all the flow values on the arcs are powers of 2,or if there are constantly many different values.So the reduction of[18]does not indicate what is the complexity of our problem in these cases,which are common in real networks.We show that even when there are only three different flow values on the arcs (and even if these values are only 1,2,or 4)then the problem is NP-hard.Furthermore,our reduction also shows that it is hard to approximate the problem better than some fixed constant,i.e.there is some fixed constant βsuch that no algorithm can guarantee a decomposition with less than β·OP T paths in polynomial time unless P =NP .In contrast,we show that if there are only two flow values x and y ,such that x divides y ,then there is a simple algorithm that finds an optimal decomposition.We implemented the greedy algorithms and our new ap-proximation algorithms and compared their performance on data extracted from Google’s backbone network and on syn-thetically generated layered networks,representing intra data center networks.For the Google network,we used a dataset consisting of a few hundred demands observed at a particular time period.For each such a demand,we generated a fractional multicommodity flow using an LP solver.We decomposed various fractions of each of the flows using these algorithms and compared the size of the decomposition they produce and the average and maximum latency of the paths.Our findings are as follows:1)The greedy width algorithm produces the most compact decompositions on the data we used,with the bicriteria width algorithm lagging behind by a few percent.This indicates that flows on which the greedy width algorithm decomposes badly are unlikely to occur in real data.The greedy length and the bicriteria length algorithms perform 10-20%worse than their width counterparts.To eliminate the risk of getting a bad decomposition while using the greedy width algorithm it may make sense to run both the greedy width algorithm and our new bicirteria width algorithm and select the smaller decomposition.2)The maximum and average latency of the paths produced by the greedy width and the bicriteria width algo-rithms were comparable.Somewhat counter-intuitively the maximum and average latency of the greedy length and the bicriteria length algorithms were slightly larger.This is a side affect of the fact that the decomposition produced by these algorithms were considerably larger This shows that on the data we tested,optimizing the number of paths does not sacrifice latency substantially.3)The number of paths required increases exponentially with the fraction of the flow we decompose (again on the tested data sets).B.Related workVatinlen et al.considered exactly the same problem as we do,i.e.minimizing the number of paths.They gave an example showing that counter-intuitively,by eliminating cycles from the flow,the size of the optimal solution can increase.They defined a decomposition to be saturating if we can obtain itby taking a source-sink path,pushing as muchflow as we can along it,removing it from theflow and repeating this process. In particular the greedy width and the greedy length algorithms that we defined above produce saturating decompositions. Vatinlen et al.show that there may not be a saturating optimal solution.They also show that the size of a saturating decomposition is at most m−n+2where m is the number of arcs and n is the number of nodes,2and consists of O(n) more paths than OPT.Vatinlen et al.also defined and implemented the greedy width and the greedy length algorithms.They compared their performance on random networks and showed that for large enough networks with relatively large average degree the greedy width is slightly better(<3%),and they both were close to the upper bound of m−n+2.Hendel and Kubiak[9]in an unpublished manuscript con-sider a similar problem in which the inputflow is acyclic, there is a nonnegative cost associated with each arc,and the goal is tofind a decomposition in which the maximum cost of a path is minimum.The networking interpretation of this objective is to minimize the maximum latency of a path in the decomposition.They give several results classifying the complexity of this problem(for details see their manuscript). The problem considered by Hendel and Kubiak is easier if the lengths of all arcs are the same.In this case we want to minimize the number of hops of the path with the largest number of hops in the decomposition.One can construct a polynomial algorithm for this problem using linear programming.We write an LP for the maximum possibleflow on paths of at most k hops.This LP has a variable for each path of at most k hops so it may have exponentially many variables.The dual of this LP has a constraint per path of at most k hops which requires that the sum of the dual variables associated with the arcs on the path is at least1.There is a polynomial separation oracle for this dual so we can solve it using the ellipsoid algorithm[10],[8].Tofind the smallest k such that all the givenflow can be covered by paths with at most k hops we perform binary search on k solving an LP as above in each iteration of the search.Another possible objective is to minimize the average weighted latency of the paths in the decomposition(where the weight of each path is the amount offlow it covers).However, it is not hard to see that all the decompositions of a givenflow have the same average weighted latency.This average equals to the sum over the arcs of the latency of the arc times its flow value.Minimizing the nonweighted average latency and the maxi-mum number of paths through each vertex are also interesting objectives which we do not consider in this paper and as far as we know have not been addressed.A problem related to ours is the maximum unsplittableflow problem introduced by Kleinberg[11].In this problem we try tofind aflow which can be decomposed into a small number 2In fact they showed this for a more general family of independent decompositions,where the incidence vectors of the paths,in the space where each coordinate is an arc,are independent.of paths.In contrast,recall,that in our setting theflow is given as the input and we cannot change it.Kleinberg studied a single-source version of the problem where we are given a single source and many terminals each with demand associated with it.The problem is to decide whether the demand to each terminal can be routed on a single path so that capacity constraints are satisfied.Kleinberg gave approximation algo-rithms for some optimization versions of this problem.Many variations of the problem were introduced since Kleinberg’s work,including unsplittable multicommodityflow,variants in which we allow more than one path per commodity,and varying optimization criteria.(for recent work see[12],[2] and the references there)Mirrokni et al.[14]studied a network planning problem in which one goal is to minimize the number of paths in multi-path routing setting such as an MPLS network.They formulate the problem as a multicommodityflow problem and derive from the formulation that an optimal solution can be decomposed into at most m+k paths,where k is the number of commodities and m is the number of routes.They further try to reduce the number of routes by using an integer programming formulations.They show that their approach is effective on various simulated topologies.II.P RELIMINARIESThe input to our algorithm is an st-flow f represented as a graph G(f)with two special vertices s and t called the source and the sink,respectively.Each arc e has a nonnegativeflow-value f(e)associated with it.For every vertex v other than s and t the sum of theflow-values on the arcs incoming to v equals the sum of theflow values on the arcs outgoing from v.The totalflow on arcs outgoing from s minus the totalflow on arcs incoming to s is the value of f.This is equal to the totalflow incoming to t minus the totalflow outgoing from t. Aflow-path in f is a path p from s to t in G(f)together with a value which is smaller than theflow-value on all the arcs along p.A decomposition of an st-flow f is a collection offlow-paths with the following property:If we take the union of these paths,and associate with each arc e in this union the sum of the values of the paths containing e we get an st-flow f .Theflow f is contained in f and its value is equal to the value of f.3Note that the difference between f and f is a collection offlow cycles and if f is acyclic then f must be equal to f .Our goal is tofind a decomposition of f with the smallest possible number of paths.We point out that removing cycles from f may change the size of the smallest solution as shown in[18].Let G be a graph with capacity c(e)associated with each arc e and two special vertices s and t.An st-flow in G is an st-flow f such that G(f)is a subgraph of G and for each arc e,f(e)≤c(e).A maximum st-flow in G in an st-flow in G with maximum value.3Aflow f is contained in f if each arc in f is also in f and itsflow-value in f is smaller than itsflow-value in f.To simplify the presentation,we will useflow instead of st-flow in the rest of the paper.We will use the following basic lemma.The proof is straight forward and omitted.Lemma II.1.Let G be a graph in which all capacities are multiples of x,and let f be a maximumflow in G of value F. Then F is a multiple of x and we can decompose f into exactly F/x paths each routing xflow.Furthermore,any collection of paths each routing xflow can be completed to such a decomposition of theflow.III.T WO V ALUE C APACITIESA simpler version of theflow decomposition problem is when all the capacities in G have one of two possible values, x and y,such that x divides y,i.e.y=kx for some integer k>1.We canfind an optimal decomposition in this case with the following algorithm.Wefirst form the subgraph G y of G(which is not necessarily aflow)induced by all arcs of capacity y.Wefind a maximumflow F y from s to t in G y. We decompose F y into paths of value y,and add these paths into our decomposition.This is possible by Lemma II.1.Then we subtract F y from G and get aflow G .Notice that since x divides y all capacities in G are multiples of x.So using Lemma II.1again we decompose G into paths offlow value x and add these paths into our decomposition.We now prove that this algorithm indeedfind the optimal solution. Theorem III.1.The algorithm described above produces an optimal solution.Proof:Let F1be the part of F that OP T routes on paths of value>x.Note that each such path routes at most yflow. So F−F1of theflow OP T routes on paths of value<x.Itfollows thatOP T≥F1y+F−F1x(1)To minimize the right hand side of Equation(1)we want to have F1as large as possible.However since all paths of value>x that OP T uses must be in the subgraph induced by the arcs offlow value y,we get that F1cannot be larger than the maximumflow in that subgraph.By the definition of the algorithm this maximumflow equals to#(y)·y,where #(y)is the number of paths of value y used by the algorithm. Substituting this into equation1we get thatOP T≥F1y+F−F1x≥#(y)·yy +F−#(y)·yx=#(y)+#(x)=ALGwhich concludes the proof.IV.G REEDY A PPROACHESIt is easy to verify that when we take aflow-path and subtract it from theflow the result is still aflow.Based on this fact one can come up with many greedy algorithms to decompose aflow.Each such algorithmfinds the bestflow-path according to some criteria,add it to the decomposition, removes it from theflow and continue decomposing the remainingflow.Maybe the most natural among these greedy algorithms are the greedy length algorithm and the greedy width algorithm. Let G(f)be the graph of the currentflow f.(Note that G(f) changes as we subtractflow-paths that we accumulate in our decomposition.)In the greedy length wefind a shortest path from s to the t in G(f),according to some latency measure on the arcs such as RTT(Round Trip Time).We route the maximum possibleflow along this path,so that when we remove this path,at least one arc is completely removed from theflow.The greedy widthfinds the“thickest”path from s to t in G(f),which is a path that can carry moreflow than any other path from s and t.Again,we route the maximum possibleflow along this path,so that at least one arc is completely removed from theflow when we add the path to the decomposition. We implement greedy length and the greedy width by running a version of Dijskstra’s single source shortest path algorithm in each iteration[5].For the greedy length we run the standard version of Dijskstra’s algorithm using RTTs as the weights of the arcs.With the greedy width we need a modified version of Dijskstra for the bottleneck shortest path problem.Here the weight of each arc is theflow that it carries in G(f).The modified version of Dijskstra for the bottleneck shortest path problem uses the width of the thickestflow-path from s to v as the key of v in the heap(rather than the length of this path in the standard version).Note that asymptotically faster algorithms for the bottleneck shortest path problem are known[6]and also for the standard single source shortest path problem when we assume integer weights[17].These algorithms are more complex and may be practical only for very large graphs.Both greedy approaches remove at least one arc from G(f) for each path they pick,so they will stop after at most m iterations(where m is the number of arcs in theflow we started out with).Furthermore in the decomposition they produce we have at most m paths.A.The worst case performance of the greedy algorithms The next intriguing question that one should understand before using these greedy algorithms is how large could be the decomposition which they produce compared to the optimal one.We note that the greedy length may produce a decomposition with paths of rather small latency and we will consider this in Section VII.But our main objective in this paper is to minimize the number offlow-paths so we first consider the performance of the greedy algorithms with respect to the number of paths which they produce.Lets look at the example shown in Figure3.In thisflow there are only3possibleflow values on the arcs which are x, 2x and1.There are k+1arcs offlow2x,namely a i→a i+1, for1≤i≤k−1,s→a1,and a k→t,that together forma path from s to t.There are k/2arcs from s to every nodeFig.3.The performance of the greedy algorithm is bad on this example a2i−1,and k/2arcs going from a2i to t.Finally,there are x parallel arcs offlow1going from a2i−1to a2i.One can easily verify that this is indeed a legalflow of value kx/2+2x. The minimal number of paths thisflow can be decomposed to is k/2+x+1.This can be done by routing x units offlow on the path s→a1→...→t which is half the maximum possibleflow we can send along this path;x units offlow on each one of the k/2paths s→a2i−1→a2i→t using the remaining x units offlow of the arcs a2i−1→a2i with flow2x;andfinally,x paths s→a1→a2...→t using the parallel arcs between a2i−1and a2i,each carrying1unit of flow.Altogether,we get k/2+x+1paths.The greedy width will use kx/2+1paths.It will make a mistake and route2x units offlow on the path s→a1→a2...→t.Now,it will have to route the remaining kx/2 units offlow on paths of the form s→a2i→a2i+1→t each carrying only one unit offlow.So routing the remainingflow will take kx/2paths.Now if we set x to some value larger than say,k/2,we get that the greedy width usesΩ(k)·OP T paths to decompose theflow.The resultingflow has m=Θ(k2)arcs(and vertices if we eliminate the parallel arcs by subdividing them). Furthermore,with this rather large value of x we also get that most of theflow is routed by the greedy width on paths of value1.So even if we compare the number of paths that greedy width needs to carry only a constant fraction of theflow for anyfixed constant then we get that it usesΩ(k)·OP T paths.The following theorem summarizes this result. Theorem IV.1.There areflows G of m arcs on which the approximation ratio of greedy width isΩ(√m)even if greedy width is required to decompose only a constant fraction of the flow,for anyfixed constant.If the latency of the arcs offlow-value2x is small relative to the latency of the other arcs then the greedy length would also pick the path of width2xfirst and thereby produce a large decomposition of this example.We have a different example in which the latency of all arcs is the same but the greedy length fails in a similar fashion.So we also have the following theorem.Theorem IV.2.There areflows G of m arcs on which the approximation ratio of greedy length isΩ(√m)even if greedy length is required to decompose only a constant fraction of the flow.V.B I-CRITERIA A PPROXIMATIONLet G t be the subgraph of G containing all arcs of capacity at least t.Let t2/3be the maximum value of t such that the value of the maximumflow from s to t in G t is at least2F/3. To simplify notation we refer to G t2/3as G2/3.We round down all the capacities in G2/3to the largestpossible multiple of t2/3and call the resulting graph G2/3. We have the following lemma.Lemma V.1.The value of the maximumflow from s to t inG2/3is at least F/3.Proof:If we scale down the capacities in G2/3by a factor of2then we obtain aflow of value F/3in the resulting graph simply by scaling down theflow of value2F/3that we have in G2/3by its definition.Consider an arc e in G2/3.Let c(e)be the capacity of e inG2/3and let c (e)be the capacity of e in G2/3.We claim that c (e)≥c(e)/2.This would imply that we can route theflow of value F/3that we routed in G2/3with capacities dividedby2also in G2/3.From the definition of G2/3we know that there is some integer k≥1such that kt2/3≤c(e)≤(k+1)t2/3.Dividing the upper bound by two we obtain that c(e)/2≤(k+1)t2/3/2. Since for every k≥1,(k+1)/2≤k we have that c(e)/2≤kt2/3.But by the definition of G2/3,c (e)=kt2/3,so we get that c(e)/2≤c (e),and the claim follows.In G2/3all capacities are multiples of t2/3so by LemmaII.1the maximumflow F in G2/3is also a multiple of t2/3 and can be decomposed into F /t2/3paths each routing t2/3 units offlow.By Lemma V.1,F ≥F/3so F/3t2/3 paths of the decomposition of F route at least F/3units offlow. So we get the following theorem.Theorem V.2.Given aflow f of value F the algorithm which we described decomposes f into at most OP T paths that together route at least F/3units offlow.Proof:By the definition of G2/3we know that in the optimal decomposition of f at least F/3units offlow are routed using paths each carrying at most t2/3units offlow. Therefore OP T≥ F/3t2/3 .Since our algorithm uses at most F/3t2/3 paths the theorem follows.We can generalize this result as follows.Let t 1− be the maximum value of t such that the value of the maximum flow from s to t in G t is at least (1− )F .To simplify notation we refer to G t 1− as G 1− .We introduce a new constant 0≤δ≤1and round down all the capacities in G 1− to the largest possible multiple of δt 1− and call the resulting graph G δ1− .The following lemma generalizes Lemma V .1(which is the case where δ=1and =1/3).Lemma V .3.The value of the maximum flow from s to t inG δ1− is at least 1−F .The proof follows the same lines of V .1and is omitted.In G δ1− all capacities are multiples of δt 1− so by Lemma II.1the maximum flow F in G δ1− is also a multiple of δt 1− .By Lemma II.1we can decompose F into F /δt 1− paths each routing δt 1− flow.Furthermore,any set of paths each routing δt 1− flow can be completed by additional paths each routing δt 1− flow so that all-together we route F flow.ByLemma V .3,F ≥1−1+δF so by using (1− )F (1+δ)δt 1− paths ofthe decomposition of F we route at least (1− )F (1+δ)flow.The following theorem summarizes the properties of our algorithm.Lemma V .4.In G δ1− we can route 1−F flow in (1− )F 1− paths each routing δt 1− .Furthermore,starting with any set of paths inG δ1− each routing δt 1− flow we can find additional paths each routing δt 1− flow so that all paths together route 1−F flow.The following theorem follows from Lemma V .4.Theorem V .5.Given a flow f of value F we can decomposef into at most 1+(1+δ) δ ·OP T paths.Each path routes δt 1− units of flow and together they route at least (1− )F units offlow.Proof:By the definition of G 1− we know that in the optimal decomposition of f at least F units of flow are routed using paths each carrying at most t 1− units of flow.Therefore OP T ≥ F/t 1− .By Lemma V .4our algorithm uses at most (1− )F (1+δ)δt 1− paths therefore the ratio between the number of paths that we use and the optimal number of paths is(1− )F (1+δ)δt 1−F/t 1− = (1− ) F(1+δ) δt 1−F/t 1− ≤≤ (1− ) · F 1− F/t 1− =(1− )(1+δ)δas required.VI.H ARDNESS R ESULTSIn this section we prove the following strong hardness result.Theorem VI.1.Let F be a flow such that on each arc theflow value is either 1,2or 4,and let k be an integer.Then it is NP-complete to decide if there exists a decomposition of F into at most k paths.Proof:For a variable x let o (x )be the number of occur-rences of the literal x ,let o (x )be the number of occurrencesof x ,and let o m (x )=max {o (x ),o (x )}.Given an instance Φof 3SAT with a set Z of N variables and a set C of M clauses we construct a flow F with n =M +2+ x ∈Z (3+4o m (x ))vertices and m =4M +x ∈Z (15o m (x )+12)arcs such that there is a decomposition of F intox ∈vars 5+3o m (x )paths if and only if Φis satisfiable.For each variable x we construct the gadget shown in Figure 4.We have two vertices s (x )and t (x )which we connect with two parallel paths each containing o m (x )pairs of nodes u i (x )and u i (x ),1≤i ≤o m (x )in one path and w i (x )and wi (x ),1≤i ≤o m (x )in the other path.There is an arc from s (x )to u 1(x ),from u i (x )to u i (x )and from ui (x )to u i +1(x )for every 1≤i ≤o m (x )−1,and from u o m (x )(x )to t (x ).Similarly,there is an arc from s (x )to w 1(x ),from w i (x )to w i (x )and from w i (x )to w i +1(x )for every 1≤i ≤o m (x )−1,and from wM (x )to t (x ).The source s is connected to all vertices s (x j ),for each variable x j ,1≤j ≤N ,and all vertices t (x j ),1≤j ≤N are connected to t .The flow on all arcs we specified so far is 4,and these would be the only arcs with flow 4in F .In addition we have four arcs with flow 1from s to s (x j ),1≤j ≤N ,and from t (x j ),1≤j ≤N to t .We also have a pair of parallel arcs with flow 1from u i (x )to u i (x )and fromw i (x )to wi (x )for 1≤i ≤o m (x ).For each variable x we have an additional vertex a (x ).We connect s to a (x )with o m (x )arcs of flow 2and 2o m (x )arcs of flow 1.We also connect a (x )to u i (x )and w i (x )for 1≤i ≤o m (x )with an arc of flow 2.For each clause c we have a vertex which we also call c .Each such vertex c is connected to t by four arcs,two of flow 1and two of flow 2.Last we have to specify the connection between the variable gadget and the clause vertices.We connect each vertex u i (x )for 1≤i ≤o (x )to a clause vertex corresponding to a clause containing x .We make these connections such that each of these clause vertices is connected to a single vertex u i (x ).Similarly,we connect each vertex w i (x )for 1≤i ≤o(x )to a clause vertex corresponding to a clause containing x ,such that each of these clause vertices is connected to a single vertex w i (x ).If o (x )<o m (x )we connect each vertex u i (x )foro (x )+1≤i ≤o m (x ).If o(x )<o m (x )we connect eachvertex wi (x )for o (x )+1≤i ≤o m (x )to t .All arcs defined in this paragraph have flow 2.We now show that Φis satisfiable if and only if we can decompose F to x ∈vars (5+3o m (x ))paths.We first assume that Φis satisfiable and show a decom-position of F into x ∈vars (5+3o m (x ))paths.Consider an assignment that satisfies Φ.Each path in our decomposition saturates an edge from s .The number of these arcs is exactly x ∈vars 5+3o m (x ).For a variable x which equals 1there would be a pair of paths of value 1to each clause vertex containing the literal x ,and a path of value 2to each clause vertex containing the literal x .。

2015年CFA考生都应该了解的ROE公式For investors, one of the most important metrics of a company is return on equity (ROE), which can be calculated by taking net income and dividing it by equity."The decision to expand into the market of a competitor and seek additional return is not a decision driven by the expected profit margin, the expected return relative to the anticipated quantity of sales," said Jesse Livermore, the pseudonymous author of the Philosophical Economics blog. "Rather, it’s a decision driven instead by the expected ROE, the expected return relative to the amount of capital that will have to be invested, put at risk, in order to earn it."Unfortunately, ROE alone doesn't tell you much about a company's operating or capital structure. That's why analysts decompose the ROE into multiple components, including a measure of profit margin (see below).The Chartered Financial Analyst (CFA) exam, which will be administered on June 7, is among the advanced Wall Street exams that tests test-takers on at least two decompositions of ROE. The more complicated one is the DuPont model. Goldman Sachs' Stuart Kaiser recently included the formula for reference in an April 2 note sent out to its clients.As you can see, the DuPont model breaks ROE down to five components. From left to right:1.Operating margin: This is earnings before interest and tax. To put it another way, this is sales less cost of goods and fixed asset expenses. The higher the operating margin, the more profitable the company.2.Asset turnover: This is the amount of sales generated by a dollar's worth of assets. It's a measure of how well a company uses its stuff.3.Borrow cost: This measures financial stress by comparing a company's interest expenses to its assets. This is not a commonly used measure of leverage. Typically, an analyst will look look at interest expense in relation to EBIT and assets in relation to debt.4.Assets/Equity: This gives a measure of financial leverage. When this ratio is high, liabilities are high, which suggests a high level of leverage.5.Tax: This captures the company's effective tax rate.The DuPont framework offers much more information than what you would get from just net income and equity.You begin to understand that the negative effects of a shrinking operating profit margin can be offset by a combination of higher asset turnover, lower borrowing costs, higher leverage, or lower taxes.Everyone in the CFA program must remember and master this concept.Everyday investors who want to take their analysis to the next level should master this too.各位考生，2015年CFA备考已经开始，为了方便各位考生能更加系统地掌握考试大纲的重点知识，帮助大家充分备考，体验实战，网校开通了全免费的高顿题库（包括精题真题和全真模考系统），题库里附有详细的答案解析，学员可以通过多种题型加强练习，通过针对性地训练与模考，对学习过程进行全面总结。

磷石膏低温分解研究及进展谢龙贵马丽萍*陈宇航戴取秀崔夏许文娟（昆明理工大学环境科学与工程学院，云南昆明650093）摘要：磷石膏是磷化工中产生量最大的工业废渣，其大量堆存导致严重的环境污染，同时也造成土地和硫资源的大量浪费。

将其用于制硫酸联产水泥能很好的解决磷石膏的堆存问题，然而，磷石膏热分解的高能耗成为了该项技术推广应用的瓶颈，因此，不少学者对磷石膏的低温分解做了一定的研究，研究表明，反应气氛和外加助剂对磷石膏分解温度的降低具有显著贡献。

关键词：磷石膏；低温分解；研究进展The Research Progress of Low-temperature Decomposition ofPhosphogypsumXIE Longgui, MA Liping*,CHEN Y uhang, DAI Quxiu, CUI Xia, XU Wenjuan(Faculty of Environmental Science and Engineering ,Kunming University of Science and Technology，Kunming 650093,China)Abstract: Phosphogypsum is the largest industrial solid waste of phosphorous chemical industry. Its store up in quantity causes serious environmental pollution, moreover, great amount of land and sulfur are wasted. To use phosphogypsum to produce sulphuric acid and cement can decrease its storage volume at a large extent, however, the decomposition of phosphogypsum under high temperature becomes the limits to popularize this technology. Lots of scholars have done a great amount of work to lower the decomposition temperature, the results showed that the reaction atmosphere and some additives do a significant contribution to lower the decomposition temperature.Keywords: Phosphogypsum, Low-temperature decomposition, Research progress基金项目：国家高技术发展计划“863”项目（2007AA06Z321）。

本论文受到国家基础研究“973”计划(2013CB127700)、“十三五”国家重点研发计划（2016YFD0100602）、高等学校创新引智计划“111”计划（B07049）、国家自然科学基金(31000078)及中央高校基本科研业务费专项资金(2452017160)。

小麦条锈菌在侵染小麦过程中吸收糖的分子机制研究摘 要由小麦条锈菌（Puccinia striiformis f.sp. tritici）引起的小麦条锈病是小麦生产上最重要的真菌病害之一。

利用小麦品种抗病性防治麦类锈病是生产上最为经济而有效的重要措施。

然而，由于小麦条锈菌毒性变异频繁，新的毒性小种不断出现，往往会导致小麦品种抗病性失效。

因此对小麦条锈菌与小麦互作机理方面需要开展更广泛深入的研究，提出新的抗病策略。

以往的大量研究都主要针对小麦条锈菌侵染进程及致病相关基因的克隆等方面。

然而小麦条锈菌作为一种严格的活体营养专性寄生真菌，完全依赖于从寄主活体组织中吸收营养物质以保证自身正常生长发育过程。

但是对于小麦条锈菌从寄主吸收的主要营养物质的种类及对应转运机制仍不明确。

糖类物质可以作为碳源，为异养微生物的生长发育提供物质基础。

因此本研究主要着眼于小麦条锈菌糖类物质吸收机制的研究。

本研究首次揭示了小麦条锈菌的糖吸收过程及其分子机制，为进一步提出防控小麦条锈病的新策略提出了理论依据，同时对于小麦条锈病的可持续控制也具有重要的理论指导意义。

本研究的主要内容和结果如下：1.为了明确小麦条锈菌从小麦中所吸收的主要糖类物质，本研究首先对被小麦条锈菌所侵染的小麦叶片中的糖类进行了高效液相色谱分析。

结果表明小麦叶片中的可溶性糖，即葡萄糖、果糖与蔗糖，中只有蔗糖含量在被小麦条锈菌侵染与未被侵染的小麦叶片中有明显差异。

检测亲和与非亲和互作体系中被小麦条锈菌侵染不同阶段的小麦叶片样品中的蔗糖含量表明在亲和与非亲和体系中被侵染小麦叶片中的蔗糖含量都会相对于对照组显著升高。

Decomposition Procedures for Distributional Analysis:A Unified Framework Based on the Shapley ValueAnthony F. ShorrocksUniversity of EssexandInstitute for Fiscal StudiesFirst draft, June 1999Mailing Address:Department of EconomicsUniversity of EssexColchester CO4 3SQ, UKshora@1. IntroductionDecomposition techniques are used in many fields of economics to help disentangle and quantify the impact of various causal factors. Their use is particularly widespread in studies of poverty and inequality. In poverty analysis, most practitioners now employ decomposable poverty measures — especially the Foster et al. (1984) family of indices —which enable the overall level of poverty to be allocated among subgroups of the population, such as those defined by geographical region, household composition, labour market characteristics or education level. Recent examples include Grootaert (1995), Szekely (1995), Thorbecke and Jung (1996). Other dynamic decomposition procedures are used to examine how economic growth contributes to a reduction in poverty over time, and to assess the extent to which the impact of growth is reinforced, or attenuated, by changes in income inequality: see for example, Ravallion and Huppi (1991), Datt and Ravallion (1992) and Tsui (1996). In the context of income inequality, decomposition techniques enable researchers to distinguish the “between-group” effect due to differences in average incomes across subgroups (males and females, say), from the “within-group” effect due to inequality within the population subgroups. (See ???). Decomposition techniques have also been developed in order to measure the importance of components of income such as earnings or transfer payments.Despite their widespread use, these procedures have a number of shortcomings which have become increasingly evident as more sophisticated models and econometrics are brought to bear on distributional questions. Four broad categories of problems can be distinguished. First, the contribution assigned to a specific factor is not always interpretable in an intuitively meaningful way. As Chantreuil and Trannoy (1997) and Morduch and Sinclair (1998) point out, this is particularly true of the decomposition by income components proposed by Shorrocks (1982). In other cases, the interpretation commonly given to a component may not be strictly accurate. Foster and Shneyerov (1996), for example, question the conventional interpretation of the between-group term in the decomposition of inequality by subgroups.The second problem with conventional procedures is that they often place constraints on the kinds of poverty and inequality indices which can be used. Only certain forms of indices yield a set of contributions that sum up to the amount of poverty or inequality thatX k ,k '1,2,...,m I 'f (X 1,X 2,...,X m )f (@)requires explanation. Similar methods applied to other indices require the introduction of a vaguely defined residual or “interaction” term in order to maintain the decomposition identity. The best known example is the subgroup decomposition of the Gini coefficient,which has exercised the minds of many authors including Pyatt (1976) and Lambert and Aronson (1993).A less familiar, but potentially much more serious, problem concerns the limitations placed on the types of contributory factors which can be considered. Subgroupdecompositions can handle situations in which the population is partitioned on the basis of a single attribute, but have difficulty identifying the relevant contributions in multi-variate decompositions. Nor is there any established method of dealing with mixtures of factors,such as a simultaneous decomposition by subgroups (into, say, males and females) and income components (say, earnings and unearned income). As more sophisticated models are used to analyse distributional issues, these limitations have become increasingly evident.The studies by Cowell and Jenkins (1995), Jenkins (1995), Bourguignon et al. (1998), and Bouillon et al. (1998) illustrate the range of problems faced by those trying to apply current techniques to complex distributional questions.The final criticism of current decomposition methods is that the individual applications are viewed as different problems requiring different solutions. No attempt has been made to integrate the various techniques within a common overall framework. This is the mainreason why it is impossible at present to combine decompositions by subgroups and income components. Yet the individual applications share certain features and objectives which enable a common structure to be formulated. Let I represent an aggregate statistical indicator, such as the overall level of poverty or inequality, and let ,denote a set of contributory factors which together account for the value of I . Then we can write(1.1),where is a suitable aggregator function representing the underlying model. The objective in all types of decomposition exercises is to assign contributions C to each of the k factors X , ideally in a manner that allows the value of I to be expressed as the sum of the k factor contributions.The aim of this paper is to offer a solution to this general decomposition problem and to compare the results with the specific procedures currently applied to a number of distributional questions. In broad terms, the proposed solution considers the marginal effect on I of eliminating each of the contributory factors in sequence, and then assigns to each factor the average of its marginal contributions in all possible elimination sequences. This procedure yields an exact additive decomposition of I into m contributions.Posing the decomposition issue in the general way indicated by (1.1) highlights formal similarities with problems encountered in other areas of economics and econometrics. Of particular relevance to this paper is the classic question of cooperative game theory, which asks how a certain amount of output (or costs) should be allocated among a set of contributors (or beneficiaries). The Shapley value (Shapley, 1953) provides a popular answer to this question. The proposed solution to the general decomposition problem turns out to formally equivalent to the Shapley value, and is therefore referred to as the Shapley decomposition. Rongve (1995) and Chantreuil and Trannoy (1997) have both applied the Shapley value to the decomposition of inequality by income components, but fail to realise that a similar procedure can be used in all forms of distributional analysis, regardless of the complexity of the model, or the number and types of factors considered. Indeed, the procedure can be employed in all areas of applied economics whenever one wishes to assess the relative importance of the explanatory variables.The paper begins with a description of the general decomposition problem and the proposed solution based on the Shapley value. Section 3 shows how the procedure may be applied to three issues concerned with poverty: the effects of growth and redistribution on changes in poverty; the conventional application of decomposable poverty indices; and the impact of population shifts and changes in within-group poverty on the level of poverty over time.Section 4 looks in more detail at the features of the Shapley decomposition in the context of a hierarchical model in which groups of factors may be treated as single units. This leads to a discussion of the two-stage Shapley procedure associated with the Owen value (Owen, 1977). A number of results in this section establish the conditions under which the Shapley and Owen decompositions coincide, and indicate several ways of simplifying the calculation of the factor contributions. These results are then used tok0K'{1,2,...,m}I'f(X1,X2,...,Xm)f(@)XkX k ,kóSgenerate the Shapley solution to the multi-variate decomposition of poverty by subgroups, a problem which has not been solved before.In Sections 5 and 6, attention turns to inequality analysis, beginning with decomposition by subgroups using the Entropy and Gini measures of inequality. This is followed by a discussion of the application of the Shapley rule to decomposition by source of income.The main purpose of these applications is to see how the Shapley procedure compares with existing techniques in the context of a variety of standard decomposition problems. The overall results are encouraging. In all cases, the Shapley decomposition either replicates current practice or (arguably) provides a more satisfactory method of assigning contributions to the explanatory factors. However, the greatest attraction of the procedure proposed in this paper is that it overcomes all four of the categories of problems associated with present techniques. As a consequence, it offers a unified framework capable of handling any type of decomposition exercise. After summarising the principal findings of this paper, Section 8 briefly discusses the wide range of potential applications to issues which have not previously been considered candidates for decomposition analysis.2. A General Framework for Decomposition AnalysisConsider a statistical indicator I whose value is completely determined by a set of m contributory factors, X, indexed by , so that we may writek(2.1),where describes the underlying model. In the applications examined later, the indicator I will represent the overall level of poverty or inequality in the population, or the change in poverty over time. The factor may refer to a conventional scalar or vector variable, but other interpretations are possible and often desirable; for the moment it is best regarded as a loose descriptive label capturing influences like “uncertain returns to investments”,“differences in household composition” or “supply-side effects”.In what follows, we imagine scenarios in which some or all of the factors are eliminated, and use F(S) to signify the value that I takes when the factors , have been dropped. As each of the factors is either present or absent, it is convenient to characterise+K ,F ,F :{S |S f K }6ú+K ,F ,C k ,k 0K ,C k 'C k (K ,F ),k 0K ,+K ,F ,j k 0KC k (K ,F )'F (K ),+K ,F ,M k (K ,F )'F (K )&F (K \{k }),k 0K .F '(F 1,F 2,...,F m )S (F r ,F )'{F i |i >r }F r If F (·) is derived from an econometric model, this constraint will usually mean that one of the factors 1represents the unexplained residuals. When the constraint is not satisfied automatically, I can always be renormalised so that it measures the “surplus” due to the identified factors.the model structure in terms of the set of factors (or, more accurately, “factor indices”), K , and the function . Since that the set of factors completely accounts for I , it will also be convenient to assume throughout that F (i ) = 0: in other words, that I is zero when all the factors are removed.1A decomposition of is a set of real values indicating thecontribution of each of the factors. A decomposition rule C is a function which yields a set of factor contributions(2.2)for any possible model .In seeking to construct a decomposition rule, several desiderata come to mind. First,that it should be symmetric (or anonymous ) in the sense that the contribution assigned to any given factor should not depend on the way in which the factors are labelled or listed.Secondly, that the decomposition should be exact (and additive), so that(2.3) for all .When condition (2.3) is satisfied, it is meaningful to speak of the proportion of observed inequality or poverty attributable to factor k .It is also desirable that the contributions of the factors can be interpreted in an intuitively appealing way. In this respect, the most natural candidate is the rule which equates the contribution of each factor to its (first round) marginal impact(2.4)This decomposition rule is symmetric, but will not normally yield an exact decomposition. A second possibility is to consider the marginal impact of each of the factors when they are eliminated in sequence. Let indicate the order in which the factors are removed, and let be the set of factors that remain after factorB (|S |,|M |)|M |C Fk 'F (S (k ,F )^{k })&F (S (k ,F ))')k F (S (k ,F )),k 0K ,)k F (S )/F (S ^{k })&F (S ),S f K \{k },S (F r ,F )S (F r %1,F )^{F r %1}r '1,2,...,m &1j k 0K C F k 'j m r '1C F F r 'j mr '1[F (S (F r ,F )^{F r })&F (S (F r ,F ))]'F (S (F 1,F )^{F 1})&F (S (F m ,F ))'F (K )&F (i )'F (K ).C F k C S k (K ,F )'1m !j F 0E C F k '1m !j F 0E)k F (S (k ,F ))'j m &1s '0j S f K \{k }|S |'s 1m !j F 0E S (k ,F )'S )k F (S )'j m &1s '0j S f K \{k }|S |'s (m &1&s )!s !m !)k F (S ).B (s ,m &1)'(m &1&s )!s !/m !C S k (K ,F )'j S K \{k }B (|S |,|K \{k }|))k F (S )'õS f K \{k })k F (S ),k 0K ,õS f L is the probability of randomly selecting the subset S from M , given that all subset sizes from 0 to 2 are equally likely.has been eliminated. Then the marginal impacts are given by(2.5)where(2.6)is the marginal effect of adding factor k to the set S . Using the fact that = for , we deduce(2.7) The decomposition (2.5) is therefore exact. However, the value of the contribution assigned to any given factor depends on the order in which the factors appear in the elimination sequence F , so the factors are not treated symmetrically. This “path dependence” problem may be remedied by considering the m ! possible elimination sequences, denoted here by the set E , and by computing the expected value of when the sequences in E are chosen at random. This yields the decomposition rule C given byS (2.8)Using to indicate the relevant probability, equation (2.8)2is expressed more succinctly as(2.9) where is the expectation taken with respect to the subsets of L .From (2.7) it is clear that C is an exact decomposition rule, and also one which treats SC k (K ,F ){)k F (S )|S f K \{k }}P (µt ,L t )G 'µ2/µ1&1 Several characterisations of the Shapley value are available, and may be reinterpreted in the present3framework. For example, (2.8) is the only symmetric and exact decomposition rule which, for each k , yields contributions that depend only on the set of marginal effects relating to factor k (Young, 1985).the factors symmetrically. Furthermore, the contributions can be interpreted as the expected marginal impact of each factor when the expectation is taken over all the possibleelimination paths.Expression (2.8) will be familiar to many readers, since it corresponds to the Shapley value for the cooperative game in which “output” or “surplus” F (K ) is shared amongst the set of “inputs” or “agents” K (see, for example, Moulin (1988, Chapter 5)). The application to distributional analysis is quite different from the context in which the Shapley value was conceived, and the results therefore need to be reinterpreted. Nevertheless, it seems convenient and appropriate to refer to (2.8) as the Shapley decomposition rule .33. Applications of the Shapley Decomposition to Poverty AnalysisTo illustrate how the Shapley decomposition operates in practice, this section looks at three simple applications to poverty analysis.3.1. The Impact of Growth and Redistribution on the Change in PovertyAn important issue in development economics concerns the extent to which economic growth helps to alleviate poverty. With a fixed real poverty standard, growth is normally expected to raise the incomes of some of the poor, thereby reducing the value of anyconventional poverty index. However, thus tendency can be moderated, or even reversed,if economic growth is accompanied by redistribution in the direction of increased inequality.Datt and Ravallion (1992) suggest a method for separating out the effects of growth and redistribution on the change in poverty between two points of time. Given a fixed poverty line, the poverty level at time t (t = 1, 2) may be expressed as a function of mean income µ and the Lorenz curve L . Denoting the growth factor by , and the t tR 'L 2&L 1)P 'P (µ2,L 2)&P (µ1,L 1)'P (µ1(1%G ),L 1%R )&P (µ1,L 1)'F (G ,R ).F (G ,R )F (R )F (G )F (G ,R )&F (R )F (R ) This is a slight abuse of notation, as the growth and redistribution factors ought to be distinguished from the 4variables representing growth and redistribution. However, in this example, the growth and redistribution factors are eliminated by setting G and R equal to zero, so no serious confusion arises. Factors and variables are distinguished more carefully in later sections.redistribution factor by , the problem becomes one of identifying the4contributions of growth G and redistribution R in the decomposition of(3.1) Figure 1: The Shapley decomposition for the growth andredistribution components of the change in poverty'(('F (i ) = 0 Figure 1 illustrates the basic structure of the Shapley decomposition for this example,which is particularly simple given that there are just two factors,G and R , and hence only two possible elimination sequences. Eliminating G before R produces the path portrayed on the left, with the marginal contribution for the growth factor, and the contribution for the redistribution effect. Repeating the exercise for the right-handC S G '12[F(G,R)&F(R)%F(G)]C S R '12[F(G,R)&F(G)%F(R)]F(R)P(µ1,L2)&P(µ1,L1)DLP(µ1,L)DL P(µ,L)P(µ,L2)&P(µ,L1)F(G)P(µ2,L1)&P(µ1,L1)DµP(µ,L1)DµP(µ,L)P(µ2,L)&P(µ1,L)C SG'12[F(G,R)&F(R)%F(G)]'12[P(µ2,L2)&P(µ1,L2)%P(µ2,L1)&P(µ1,L1)]'12[DµP(µ,L1)%DµP(µ,L2)]C SR'12[DLP(µ1,L)%DLP(µ2,L)].C SGC SRpath, and then averaging the results, yields the Shapley contributions(3.2)When growth is absent, G takes the value 0 and the change in poverty becomes(3.3) = = ,where = indicates the rise in poverty due to a shift in theLorenz curve from L to L, holding mean income constant at µ. Conversely, eliminating12the redistribution factor by setting R = 0 yields(3.4) = = ,where = is the rise in poverty due to a change in meanincome from µ to µ, with the fixed Lorenz curve L. The Shapley contributions for the 12growth and redistribution effects are therefore given by(3.5)These contributions sum up, as expected, to the overall change in poverty, and have intuitively appealing interpretations. The growth component, , indicates the rise inpoverty due to a shift in mean income from µ to µ, averaged with respect to the Lorenz12curves prevailing in the base and final years, while the redistribution effect, , represents the average impact of the change in the distribution of relative incomes, with the average taken with respect to the mean income levels in the two periods.Despite the attractions of the Shapley decomposition values given in (3.5), these are not the contributions proposed by Datt and Ravallion (1992). Instead, they associate the growth and redistribution effects with the marginal change in poverty starting from the base yearC G 'M G ({G ,R },F )C R 'M R ({G ,R },F )C G 'D µP (µ,L 1)C R 'D L P (µ1,L ))P 'C G %C R %E P 'j mk '1<k P k k 0K '{1,2,...,m } These terms correspond to the first round marginal effects; in other words, and 5 in the notation of (2.4).situation. This yields the contributions and . These do 5not sum to the observed change in poverty, so Datt and Ravallion are obliged to introduce a residual term E into their decomposition equation(3.6).They acknowledge the criticisms which can be levelled against the residual component, and note that it can be made to vanish by averaging over the base and final years, as is done in (3.5). However, this solution is rejected as being arbitrary (Datt and Ravallion, 1992,footnote 3). Far from being arbitrary, the above analysis suggests that this is exactly the outcome which results from applying a systematic decomposition procedure to the growth-redistribution issue. Furthermore, the general framework outlined in Section 2 offers the chance of extending the analysis to cover not only changes in the poverty line, but also more disaggregated influences such as changes in mean incomes and income inequality within the modern and traditional sectors.3.2 Decomposable Poverty IndicesAnother standard application of decomposition techniques involves the use ofdecomposable poverty indices. When assigning contributions to subgroups of the population, such indices enable the overall degree of poverty, P , to be written(3.7) ,where < and P respectively indicate the population share and poverty level associated with k k subgroup . Indices with this propoerty — especially the family of measures proposed by Foster et al. (1984) — are nowadays used routinely to study theway in which differences according to region, household size, age, and education attainment contribute to the overall level of poverty.In many respects, this is the simplest and most clear-cut application of decomposition techniques. Suppose, for instance, that the population is partitioned into m regions. Then factor k can be interpreted as “poverty within region k ”, and the question of interest is the+K ,F ,F (S )'j k 0S<k P k )k F (S )'<k P k for all S f K \{k }<k P kC S k (K ,F )'M k (K ,F )'<k P k ,k 0K .P 'j k 0K j R 0L<k R P k R m 1%m 2+K ^L ,F ,F (S ^T )'j k 0S j R 0T <k R P k R S f K ,T f L contribution which this factor makes to poverty in the whole country. Adopting the notation of Section 2, the model structure is defined by(3.8),so(3.9).Since eliminating poverty in region k reduces aggregate poverty by the amount regardless of the order in which the regions are considered, it follows from (2.4) and (2.9)that these values yield both the first round marginal effects and the terms in the Shapley decomposition; in other words(3.10)Not surprisingly, this allocation of poverty contributions to population subgroups accords exactly with common practice.A more complex situation emerges if we wish to perform a simultaneous decomposition by more than one attribute. In fact there is no recognised procedure at present for dealing with this problem. Suppose, for instance, that the population is subdivided into m regions 1indexed by K and m age groups indexed by L . This yields a total of m m region-age cells 2 1 2which, if treated separately, can be assigned contributions as before, by replacing equation (3.7) with(3.11),where the subscripts k R refer to region k and age group R . However, we are more likely to be interested in the overall impact of poverty in region k , or in age group R , rather than the contribution of the subgroup corresponding to region k and age group R . In other words,what we really seek are the contributions associated with the regional and age factors.The Shapley procedure offers a solution to this problem by defining the model structure where(3.12).)k F (S ^T )'j R 0T <k R P k R ')k F (T ),S f K \{k },T f L {S |S f M }{M \S |S f M }B (|S |,|M |)B (|M \S |,|M |)õS f M F (S )'12õS f M [F (S )%F (M \S )])k F (S )%)k F (M \S )'j R 0S _L <k R P k R %j R 0(M \S )_L<k R P k R ')k F (M _L ),S f M f K ^L k 0K M 'L ^K \{k }C S k (K ^L ,F )'õS f M )k F (S )'12õS f M [)k F (S )%)k F (M \S )]'12õS f M )k F (L )'12)k F (L )'12j R 0L <k R P k R ,C S k (K ^L ,F )'12<k P kSee Proposition 4 in the next section.6Eliminating poverty in region k now yields(3.13).In contrast to (3.9) above, equation (3.13) shows that the factors no longer operateindependently: the marginal impact of removing poverty in region k depends on whether poverty has already been eliminated in one or more of the age groups.To obtain the Shapley contributions for the regions, first note that = and = , so(3.14).Note also that (3.13) implies(3.15)for all . So choosing any and setting yields(3.16)or equivalently, in the notation of (3.7),(3.17)Thus, in this two attribute example, each region is assigned exactly half the contribution that would be obtained in a decomposition by region alone. A similar result applies to the age group factors. More generally, in a simultaneous decomposition by n attributes, each factor is allocated one n th of the contribution obtained in the single attributedecomposition.6This result will be comforting to those who use decomposable poverty indices, for it shows that nothing is lost by looking at each attribute in isolation; the outcomes of multi-attribute decompositions can be calculated immediately from the series of single attribute< k t P k t)P'j k0K[<k2P k2&<k1P k1])Pk 'Pk2&Pk1,k0K)<k '<k2&<k1,k0KKp'{pk,k0K}Ks+Kp ^{Ks},F,F(S^T)'j k0K[<k(T)P k(S)&<k1P k1],S f K p,T f{K s}Pk(S)'9P k2if p k0SPk1if pkóS<k(T)'9<k2if T'Ks<k1if T'ipk0Kp)pkF(S^T)'<k(T)Pk(S^{pk})&<k(T)Pk(S)'<k(T))Pk,S f Kp\{pk},T f{Ks},M'{Ks}^Kp\{pk})pkF(S)%)pkF(M\S)'<k(S_{Ks}))Pk%<k((M\S)_{Ks}))Pk'(<k1%<k2))Pk,results, and the relative importance of different subgroups remains the same, regardless of the number of attributes considered.3.3 Changes in Poverty Over TimeDecomposable poverty indices can also be used to identify the subgroup contributions to poverty changes over time. If and represent the population share and poverty level of subgroup k0K at time t (t = 1, 2), equation (3.7) yields(3.18).The aim here is to account for the overall change in poverty, )P, in terms of changes in poverty within subgroups, , and the population shifts between subgroups, .The subgroup poverty values can be changed independently, so the poverty change factors may be indexed by the set . However the population shifts necessarily sum to zero. To avoid complications at this stage, the population shift factors will be treated as a single composite factor, denoted by . The model structureis then given by(3.19),where(3.20) ;For each we have(3.21)and setting , it follows that(3.22)S f MC S pk 'õS f M)pkF(S)'12õS f M[)pkF(S)%)pkF(M\S)]'12õS f M(<k1%<k2))Pk'<k1%<k22)Pk.)KsF(S)'j k0K[<k(K s)&<k(i)]P k(S)'j k0K P k(S))<k,S f K p.C S Ks 'õS f Kp)KsF(S)'12õS f Kp[)KsF(S)%)KsF(Kp\S)]'12õS f Kpjk0K[Pk(S)%Pk(Kp\S)])<k'12õS f Kpjk0K[Pk1%Pk2])<k'j k0K P k1%P k22)<k )P'j k0K<k1%<k22)P k%j k0K P k1%P k22)<kfor all . So, using (3.14), the Shapley contribution associated with the change in poverty within subgroup k is given by(3.23)Conversely(3.24)So(3.25)This is a very natural allocation of contributions given that we seek a decomposition which treats the factors in a symmetric way, and given that (3.18) may be rewritten(3.26).4. Hierarchical StructuresDespite its attractive properties, the Shapley decomposition has one major drawback for distributional analysis: the contribution assigned to any given factor is usually sensitive to the way in which the other factors are treated. In many applications, certain groups of factors naturally cluster together. This leads to a hierarchical structure comprising a set of primary factors, each of which is subdivided into a (possibly single element) group of secondary factors. For example, when income inequality is decomposed by source of income (see Section 6 below), one may first wish to regard income as the sum of labour income, investment income and transfers. Then investment income, say, might be split into interest, dividends, capital gains and rent. The Shapley decomposition does not guarantee。