Type IIA D-Branes, K-Theory, and Matrix Theory

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二维指称理论及其发展

收稿日期：2023-09-20基金项目：国家社会科学基金项目“意义与指称理论前沿问题研究”（21BZX04）作者简介：尹均怡，南开大学哲学院，主要从事逻辑哲学研究。

E-mail ：******************刘叶涛，南开大学哲学院，主要从事逻辑哲学与哲学逻辑研究。

E-mail ：**************摘要：查尔默斯以二维语义为视角提出了一种试图融合描述论和直接指称论的新型意义理论；通过重新把握认识论与形而上学的区分与关联，构建了一种较为系统的二维指称理论，以进一步强化“意义—理性—模态”的密切关联。

在系统阐释查尔默斯二维指称理论的建构动因、核心内容及其发展历程的基础上，通过深入考察国内外学界就此提出的质疑，并追踪查尔默斯针对这些质疑给出的回应，可为深度把握二维指称理论架构的成就与不足提供方向。

关键词：指称理论；金三角；二维指称理论；认知可能性；形而上学可能性中图分类号：B813文献标识码：A 文章编号：1006-2815（2024）01-0113-12DOI ： 10.19946/j.issn.1006-2815.2024.01.010二维指称理论及其发展尹均怡刘叶涛一、背景：当代意义和指称理论的对立与交锋在当代意义理论中，描述论和直接指称论的交锋旷日持久。

弗雷格（G. Frege ）明确区分了名称的含义（sense ）和指称（reference ），强调含义在给命题赋值过程中的作用不可忽视，开创了现代指称理论的先河。

“a=a ”和“a=b ”这两种同一陈述之间存在重要区别：“a=b ”传达了一种“a=a ”无法传达的信息，这种认知意义（cognitive significance ）上的差异与被命名对象关系密切：“和一个名称相联系的，不仅具有被命名的对象（或指称），还有这个名称的含义，在其名称中包含了名称提出的方式和语境”①。

摹状词理论尽管只是为了讨论限定摹状词的意义，但罗素（B. Russell ）的基本主张经提炼，也适用于名称，从而在主要观点上与弗雷格相同：名称（专名和通名）都有含义；专名的含① Gottlob Frege, “On Sense and Nominatum Reprinted ”, in Anthony Patrick Martinich (ed.), The Philosophy of Lan-guage , Oxford: Oxford University Press, 2001, p.2.SCIENCE · ECONOMY ·SOCIETY 第42卷总第178期Vol.42, Total No.1782024年第1 期No.1, 20241132024年第1期义指谓的是个体对象的独有属性，通常由限定摹状词表达；通名的含义指谓一类对象的共有且仅有属性，通常由非限定摹状词表达；含义是判定指称的依据和标准，含义决定指称。

Linear Algebra and its Applications

Linear Algebra and its Applications432(2010)2089–2099Contents lists available at ScienceDirect Linear Algebra and its Applications j o u r n a l h o m e p a g e:w w w.e l s e v i e r.c o m/l o c a t e/l aaIntegrating learning theories and application-based modules in teaching linear algebraୋWilliam Martin a,∗,Sergio Loch b,Laurel Cooley c,Scott Dexter d,Draga Vidakovic ea Department of Mathematics and School of Education,210F Family Life Center,NDSU Department#2625,P.O.Box6050,Fargo ND 58105-6050,United Statesb Department of Mathematics,Grand View University,1200Grandview Avenue,Des Moines,IA50316,United Statesc Department of Mathematics,CUNY Graduate Center and Brooklyn College,2900Bedford Avenue,Brooklyn,New York11210, United Statesd Department of Computer and Information Science,CUNY Brooklyn College,2900Bedford Avenue Brooklyn,NY11210,United Statese Department of Mathematics and Statistics,Georgia State University,University Plaza,Atlanta,GA30303,United StatesA R T I C L E I N F O AB S T R AC TArticle history:Received2October2008Accepted29August2009Available online30September2009 Submitted by L.Verde-StarAMS classiﬁcation:Primary:97H60Secondary:97C30Keywords:Linear algebraLearning theoryCurriculumPedagogyConstructivist theoriesAPOS–Action–Process–Object–Schema Theoretical frameworkEncapsulated process The research team of The Linear Algebra Project developed and implemented a curriculum and a pedagogy for parallel courses in (a)linear algebra and(b)learning theory as applied to the study of mathematics with an emphasis on linear algebra.The purpose of the ongoing research,partially funded by the National Science Foundation,is to investigate how the parallel study of learning theories and advanced mathematics inﬂuences the development of thinking of individuals in both domains.The researchers found that the particular synergy afforded by the parallel study of math and learning theory promoted,in some students,a rich understanding of both domains and that had a mutually reinforcing effect.Furthermore,there is evidence that the deeper insights will contribute to more effective instruction by those who become high school math teachers and,consequently,better learning by their students.The courses developed were appropriate for mathematics majors,pre-service secondary mathematics teachers, and practicing mathematics teachers.The learning seminar focused most heavily on constructivist theories,although it also examinedThe work reported in this paper was partially supported by funding from the National Science Foundation(DUE CCLI 0442574).∗Corresponding author.Address:NDSU School of Education,NDSU Department of Mathematics,210F Family Life Center, NDSU Department#2625,P.O.Box6050,Fargo ND58105-6050,United States.Tel.:+17012317104;fax:+17012317416.E-mail addresses:william.martin@(W.Martin),sloch@(S.Loch),LCooley@ (L.Cooley),SDexter@(S.Dexter),dvidakovic@(D.Vidakovic).0024-3795/$-see front matter©2009Elsevier Inc.All rights reserved.doi:10.1016/a.2009.08.0302090W.Martin et al./Linear Algebra and its Applications432(2010)2089–2099Thematicized schema Triad–intraInterTransGenetic decomposition Vector additionMatrixMatrix multiplication Matrix representation BasisColumn spaceRow spaceNull space Eigenspace Transformation socio-cultural and historical perspectives.A particular theory, Action–Process–Object–Schema(APOS)[10],was emphasized and examined through the lens of studying linear algebra.APOS has been used in a variety of studies focusing on student understanding of undergraduate mathematics.The linear algebra courses include the standard set of undergraduate topics.This paper reports the re-sults of the learning theory seminar and its effects on students who were simultaneously enrolled in linear algebra and students who had previously completed linear algebra and outlines how prior research has inﬂuenced the future direction of the project.©2009Elsevier Inc.All rights reserved.1.Research rationaleThe research team of the Linear Algebra Project(LAP)developed and implemented a curriculum and a pedagogy for parallel courses in linear algebra and learning theory as applied to the study of math-ematics with an emphasis on linear algebra.The purpose of the research,which was partially funded by the National Science Foundation(DUE CCLI0442574),was to investigate how the parallel study of learning theories and advanced mathematics inﬂuences the development of thinking of high school mathematics teachers,in both domains.The researchers found that the particular synergy afforded by the parallel study of math and learning theory promoted,in some teachers,a richer understanding of both domains that had a mutually reinforcing effect and affected their thinking about their identities and practices as teachers.It has been observed that linear algebra courses often are viewed by students as a collection of deﬁnitions and procedures to be learned by rote.Scanning the table of contents of many commonly used undergraduate textbooks will provide a common list of terms such as listed here(based on linear algebra texts by Strang[1]and Lang[2]).Vector space Kernel GaussianIndependence Image TriangularLinear combination Inverse Gram–SchmidtSpan Transpose EigenvectorBasis Orthogonal Singular valueSubspace Operator DecompositionProjection Diagonalization LU formMatrix Normal form NormDimension Eignvalue ConditionLinear transformation Similarity IsomorphismRank Diagonalize DeterminantThis is not something unique to linear algebra–a similar situation holds for many undergraduate mathematics courses.Certainly the authors of undergraduate texts do not share this student view of mathematics.In fact,the variety ways in which different authors organize their texts reﬂects the individual ways in which they have conceptualized introductory linear algebra courses.The wide vari-ability that can be seen in a perusal of the many linear algebra texts that are used is a reﬂection the many ways that mathematicians think about linear algebra and their beliefs about how students can come to make sense of the content.Instruction in a course is based on considerations of content,pedagogy, resources(texts and other materials),and beliefs about teaching and learning of mathematics.The interplay of these ideas shaped our research project.We deliberately mention two authors with clearly differing perspectives on an undergraduate linear algebra course:Strang’s organization of the material takes an applied or application perspective,while Lang views the material from more of a“pure mathematics”perspective.A review of the wide variety of textbooks to classify and categorize the different views of the subject would reveal a broad variety of perspectives on the teaching of the subject.We have taken a view that seeks to go beyond the mathe-matical content to integrate current theoretical perspectives on the teaching and learning of undergrad-uate mathematics.Our project used integration of mathematical content,applications,and learningW.Martin et al./Linear Algebra and its Applications432(2010)2089–20992091 theories to provide enhanced learning experiences using rich content,student meta cognition,and their own experience and intuition.The project also used co-teaching and collaboration among faculty with expertise in a variety of areas including mathematics,computer science and mathematics education.If one moves beyond the organization of the content of textbooks weﬁnd that at their heart they do cover a common core of the key ideas of linear algebra–all including fundamental concepts such as vector space and linear transformation.These observations lead to our key question“How is one to think about this task of organizing instruction to optimize learning?”In our work we focus on the conception of linear algebra that is developed by the student and its relationship with what we reveal about our own understanding of the subject.It seems that even in cases where researchers consciously study the teaching and learning of linear algebra(or other mathematics topics)the questions are“What does it mean to understand linear algebra?”and“How do I organize instruction so that students develop that conception as fully as possible?”In broadest terms, our work involves(a)simultaneous study of linear algebra and learning theories,(b)having students connect learning theories to their study of linear algebra,and(c)the use of parallel mathematics and education courses and integrated workshops.As students simultaneously study mathematics and learning theory related to the study of mathe-matics,we expect that reﬂection or meta cognition on their own learning will enable them to construct deeper and more meaningful understanding in both domains.We chose linear algebra for several reasons:It has not been the focus of as much instructional research as calculus,it involves abstraction and proof,and it is taken by many students in different programs for a variety of reasons.It seems to us to involve important mathematical content along with rich applications,with abstraction that builds on experience and intuition.In our pilot study we taught parallel courses:The regular upper division undergraduate linear algebra course and a seminar in learning theories in mathematics education.Early in the project we also organized an intensive three-day workshop for teachers and prospective teachers that included topics in linear algebra and examination of learning theory.In each case(two sets of parallel courses and the workshop)we had students reﬂect on their learning of linear algebra content and asked them to use their own learning experiences to reﬂect on the ideas about teaching and learning of mathematics.Students read articles–in the case of the workshop,this reading was in advance of the long weekend session–drawn from mathematics education sources including[3–10].APOS(Action,Process,Object,Schema)is a theoretical framework that has been used by many researchers who study the learning of undergraduate and graduate mathematics[10,11].We include a sketch of the structure of this framework and refer the reader to the literature for more detailed descriptions.More detailed and speciﬁc illustrations of its use are widely available[12].The APOS Theoretical Framework involves four levels of understanding that can be described for a wide variety of mathematical concepts such as function,vector space,linear transformation:Action,Process,Object (either an encapsulated process or a thematicized schema),Schema(Intra,inter,trans–triad stages of schema formation).Genetic decomposition is the analysis of a particular concept in which developing understanding is described as a dynamic process of mental constructions that continually develop, abstract,and enrich the structural organization of an individual’s knowledge.We believe that students’simultaneous study of linear algebra along with theoretical examination of teaching and learning–particularly on what it means to develop conceptual understanding in a domain –will promote learning and understanding in both domains.Fundamentally,this reﬂects our view that conceptual understanding in any domain involves rich mental connections that link important ideas or facts,increasing the individual’s ability to relate new situations and problems to that existing cognitive framework.This view of conceptual understanding of mathematics has been described by various prominent math education researchers such as Hiebert and Carpenter[6]and Hiebert and Lefevre[7].2.Action–Process–Object–Schema theory(APOS)APOS theory is a theoretical perspective of learning based on an interpretation of Piaget’s construc-tivism and poses descriptions of mental constructions that may occur in understanding a mathematical concept.These constructions are called Actions,Processes,Objects,and Schema.2092W.Martin et al./Linear Algebra and its Applications432(2010)2089–2099 An action is a transformation of a mathematical object according to an explicit algorithm seen as externally driven.It may be a manipulation of objects or acting upon a memorized fact.When one reﬂects upon an action,constructing an internal operation for a transformation,the action begins to be interiorized.A process is this internal transformation of an object.Each step may be described or reﬂected upon without actually performing it.Processes may be transformed through reversal or coordination with other processes.There are two ways in which an individual may construct an object.A person may reﬂect on actions applied to a particular process and become aware of the process as a totality.One realizes that transformations(whether actions or processes)can act on the process,and is able to actually construct such transformations.At this point,the individual has reconstructed a process as a cognitive object. In this case we say that the process has been encapsulated into an object.One may also construct a cognitive object by reﬂecting on a schema,becoming aware of it as a totality.Thus,he or she is able to perform actions on it and we say the individual has thematized the schema into an object.With an object conception one is able to de-encapsulate that object back into the process from which it came, or,in the case of a thematized schema,unpack it into its various components.Piaget and Garcia[13] indicate that thematization has occurred when there is a change from usage or implicit application to consequent use and conceptualization.A schema is a collection of actions,processes,objects,and other previously constructed schemata which are coordinated and synthesized to form mathematical structures utilized in problem situations. Objects may be transformed by higher-level actions,leading to new processes,objects,and schemata. Hence,reconstruction continues in evolving schemata.To illustrate different conceptions of the APOS theory,imagine the following’teaching’scenario.We give students multi-part activities in a technology supported environment.In particular,we assume students are using Maple in the computer lab.The multi-part activities,focusing on vectors and operations,in Maple begin with a given Maple code and drawing.In case of scalar multiplication of the vector,students are asked to substitute one parameter in the Maple code,execute the code and observe what has happened.They are asked to repeat this activity with a different value of the parameter.Then students are asked to predict what will happen in a more general case and to explain their reasoning.Similarly,students may explore addition and subtraction of vectors.In the next part of activity students might be asked to investigate about the commutative property of vector addition.Based on APOS theory,in theﬁrst part of the activity–in which students are asked to perform certain operation and make observations–our intention is to induce each student’s action conception of that concept.By asking students to imagine what will happen if they make a certain change–but do not physically perform that change–we are hoping to induce a somewhat higher level of students’thinking, the process level.In order to predict what will happen students would have to imagine performing the action based on the actions they performed before(reﬂective abstraction).Activities designed to explore on vector addition properties require students to encapsulate the process of addition of two vectors into an object on which some other action could be performed.For example,in order for a student to conclude that u+v=v+u,he/she must encapsulate a process of adding two vectors u+v into an object(resulting vector)which can further be compared[action]with another vector representing the addition of v+u.As with all theories of learning,APOS has a limitation that researchers may only observe externally what one produces and discusses.While schemata are viewed as dynamic,the task is to attempt to take a snap shot of understanding at a point in time using a genetic decomposition.A genetic decomposition is a description by the researchers of speciﬁc mental constructions one may make in understanding a mathematical concept.As with most theories(economics,physics)that have restrictions,it can still be very useful in describing what is observed.3.Initial researchIn our preliminary study we investigated three research questions:•Do participants make connections between linear algebra content and learning theories?•Do participants reﬂect upon their own learning in terms of studied learning theories?W.Martin et al./Linear Algebra and its Applications432(2010)2089–20992093•Do participants connect their study of linear algebra and learning theories to the mathematics content or pedagogy for their mathematics teaching?In addition to linear algebra course activities designed to engage students in explorations of concepts and discussions about learning theories and connections between the two domains,we had students construct concept maps and describe how they viewed the connections between the two subjects. We found that some participants saw signiﬁcant connections and were able to apply APOS theory appropriately to their learning of linear algebra.For example,here is a sketch outline of how one participant described the elements of the APOS framework late in the semester.The student showed a reasonable understanding of the theoretical framework and then was able to provide an example from linear algebra to illustrate the model.The student’s description of the elements of APOS:Action:“Students’approach is to apply‘external’rules toﬁnd solutions.The rules are said to be external because students do not have an internalized understanding of the concept or the procedure toﬁnd a solution.”Process:“At the process level,students are able to solve problems using an internalized understand-ing of the algorithm.They do not need to write out an equation or draw a graph of a function,for example.They can look at a problem and understand what is going on and what the solution might look like.”Object level as performing actions on a process:“At the object level,students have an integrated understanding of the processes used to solve problems relating to a particular concept.They un-derstand how a process can be transformed by different actions.They understand how different processes,with regard to a particular mathematical concept,are related.If a problem does not conform to their particular action-level understanding,they can modify the procedures necessary toﬁnd a solution.”Schema as a‘set’of knowledge that may be modiﬁed:“Schema–At the schema level,students possess a set of knowledge related to a particular concept.They are able to modify this set of knowledge as they gain more experience working with the concept and solving different kinds of problems.They see how the concept is related to other concepts and how processes within the concept relate to each other.”She used the ideas of determinant and basis to illustrate her understanding of the framework. (Another student also described how student recognition of the recursive relationship of computations of determinants of different orders corresponded to differing levels of understanding in the APOS framework.)Action conception of determinant:“A student at the action level can use an algorithm to calculate the determinant of a matrix.At this level(at least for me),the formula was complicated enough that I would always check that the determinant was correct byﬁnding the inverse and multiplying by the original matrix to check the solution.”Process conception of determinant:“The student knows different methods to use to calculate a determinant and can,in some cases,look at a matrix and determine its value without calculations.”Object conception:“At the object level,students see the determinant as a tool for understanding and describing matrices.They understand the implications of the value of the determinant of a matrix as a way to describe a matrix.They can use the determinant of a matrix(equal to or not equal to zero)to describe properties of the elements of a matrix.”Triad development of a schema(intra,inter,trans):“A singular concept–basis.There is a basis for a space.The student can describe a basis without calculation.The student canﬁnd different types of bases(column space,row space,null space,eigenspace)and use these values to describe matrices.”The descriptions of components of APOS along with examples illustrate that this student was able to make valid connections between the theoretical framework and the content of linear algebra.While the2094W.Martin et al./Linear Algebra and its Applications432(2010)2089–2099descriptions may not match those that would be given by scholars using APOS as a research framework, the student does demonstrate a recognition of and ability to provide examples of how understanding of linear algebra can be organized conceptually as more that a collection of facts.As would be expected,not all participants showed gains in either domain.We viewed the results of this study as a proof of concept,since there were some participants who clearly gained from the experience.We also recognized that there were problems associated with the implementation of our plan.To summarize ourﬁndings in relation to the research questions:•Do participants make connections between linear algebra content and learning theories?Yes,to widely varying degrees and levels of sophistication.•Do participants reﬂect upon their own learning in terms of studied learning theories?Yes,to the extent possible from their conception of the learning theories and understanding of linear algebra.•Do participants connect their study of linear algebra and learning theories to the mathematics content or pedagogy for their mathematics teaching?Participants describe how their experiences will shape their own teaching,but we did not visit their classes.Of the11students at one site who took the parallel courses,we identiﬁed three in our case studies (a detailed report of that study is presently under review)who demonstrated a signiﬁcant ability to connect learning theories with their own learning of linear algebra.At another site,three teachers pursuing math education graduate studies were able to varying degrees to make these connections –two demonstrated strong ability to relate content to APOS and described important ways that the experience had affected their own thoughts about teaching mathematics.Participants in the workshop produced richer concept maps of linear algebra topics by the end of the weekend.Still,there were participants who showed little ability to connect material from linear algebra and APOS.A common misunderstanding of the APOS framework was that increasing levels cor-responded to increasing difﬁculty or complexity.For example,a student might suggest that computing the determinant of a2×2matrix was at the action level,while computation of a determinant in the 4×4case was at the object level because of the increased complexity of the computations.(Contrast this with the previously mentioned student who observed that the object conception was necessary to recognize that higher dimension determinants are computed recursively from lower dimension determinants.)We faced more signiﬁcant problems than the extent to which students developed an understanding of the ideas that were presented.We found it very difﬁcult to get students–especially undergraduates –to agree to take an additional course while studying linear algebra.Most of the participants in our pilot projects were either mathematics teachers or prospective mathematics teachers.Other students simply do not have the time in their schedules to pursue an elective seminar not directly related to their own area of interest.This problem led us to a new project in which we plan to integrate the material on learning theory–perhaps implicitly for the students–in the linear algebra course.Our focus will be on working with faculty teaching the course to ensure that they understand the theory and are able to help ensure that course activities reﬂect these ideas about learning.4.Continuing researchOur current Linear Algebra in New Environments(LINE)project focuses on having faculty work collaboratively to develop a series of modules that use applications to help students develop conceptual understanding of key linear algebra concepts.The project has three organizing concepts:•Promote enhanced learning of linear algebra through integrated study of mathematical content, applications,and the learning process.•Increase faculty understanding and application of mathematical learning theories in teaching linear algebra.•Promote and support improved instruction through co-teaching and collaboration among faculty with expertise in a variety of areas,such as education and STEM disciplines.W.Martin et al./Linear Algebra and its Applications432(2010)2089–20992095 For example,computer and video graphics involve linear transformations.Students will complete a series of activities that use manipulation of graphical images to illustrate and help them move from action and process conceptions of linear transformations to object conceptions and the development of a linear transformation schema.Some of these ideas were inspired by material in Judith Cederberg’s geometry text[14]and some software developed by David Meel,both using matrix representations of geometric linear transformations.The modules will have these characteristics:•Embed learning theory in linear algebra course for both the instructor and the students.•Use applied modules to illustrate the organization of linear algebra concepts.•Applications draw on student intuitions to aid their mental constructions and organization of knowledge.•Consciously include meta-cognition in the course.To illustrate,we sketch the outline of a possible series of activities in a module on geometric linear transformations.The faculty team–including individuals with expertise in mathematics,education, and computer science–will develop a series of modules to engage students in activities that include reﬂection and meta cognition about their learning of linear algebra.(The Appendix contains a more detailed description of a module that includes these activities.)Task1:Use Photoshop or GIMP to manipulate images(rotate,scale,ﬂip,shear tools).Describe and reﬂect on processes.This activity uses an ACTION conception of transformation.Task2:Devise rules to map one vector to another.Describe and reﬂect on process.This activity involves both ACTION and PROCESS conceptions.Task3:Use a matrix representation to map vectors.This requires both PROCESS and OBJECT conceptions.Task4:Compare transform of sum with sum of transforms for matrices in Task3as compared to other non-linear functions.This involves ACTION,PROCESS,and OBJECT conceptions.Task5:Compare pre-image and transformed image of rectangles in the plane–identify software tool that was used(from Task1)and how it might be represented in matrix form.This requires OBJECT and SCHEMA conceptions.Education,mathematics and computer science faculty participating in this project will work prior to the semester to gain familiarity with the APOS framework and to identify and sketch potential modules for the linear algebra course.During the semester,collaborative teams of faculty continue to develop and reﬁne modules that reﬂect important concepts,interesting applications,and learning theory:Modules will present activities that help students develop important concepts rather than simply presenting important concepts for students to absorb.The researchers will study the impact of project activities on student learning:We expect that students will be able to describe their knowledge of linear algebra in a more conceptual(structured) way during and after the course.We also will study the impact of the project on faculty thinking about teaching and learning:As a result of this work,we expect that faculty will be able to describe both the important concepts of linear algebra and how those concepts are mentally developed and organized by students.Finally,we will study the impact on instructional practice:Participating faculty should continue to use instructional practices that focus both on important content and how students develop their understanding of that content.5.SummaryOur preliminary study demonstrated that prospective and practicing mathematics teachers were able to make connections between their concurrent study of linear algebra and of learning theories relating to mathematics education,speciﬁcally the APOS theoretical framework.In cases where the participants developed understanding in both domains,it was apparent that this connected learning strengthened understanding in both areas.Unfortunately,we were unable to encourage undergraduate students to consider studying both linear algebra and learning theory in separate,parallel courses. Consequently,we developed a new strategy that embeds the learning theory in the linear algebra。

“K-理论、C★-代数并流形上的拓扑”国际会议

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Ｔｃｎｌｙ，ｈｎｅｂｒｎｖｒｉｏｎ，ｌａｄｒＢｒｎｄｎｖｒｔｏｅｈｇｎ，ａｈｙｅｓａｓｎｎｔｕｆｅｈｏｏ）ＪａｎｓＥｅｔｉｓｏＢｎ）ｅｎｅｅｇ（ｉｅｉｏＣｐｎａｅ）ＫｔｒｎＨｓ（ｕａｅｉｉｔｏｇｏＵ（ｅｔｆｙＡｘｌＵｓｙｆＬｎｓｔｅ
会议邀请了一批活跃在量子代数及其在物理中的应用等领域的学者与会并作学术报告。大会共安排了３２个１小时报告。
内容主要包括李代数，量子群，顶点算子代数等及它们的表示理论和在物理中的应用等方面。
会议吸引了来自中国、国、美日本、兰、荷澳大利亚、法国和加拿大等７个国家和地区的７０余位参会代表。（南开大学陈省身数学研究所，天津３０７）００１
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量子代数与物理国际会议

“Stabilizing

INRIA Sophia Antipolis, CAFE project, 2004 route des lucioles, BP 93, 06902 Sophia Antipolis cedex, France. Alban.Quadrat@sophia.inria.fr, http://www-sop/cafe/Alban.Quadrat/index.html.
“Stabilizing” the stabilizing controllers
A. Quadrat
∗
Abstract The main purpose of this paper is to revisit the internal/simultaneous/robust stabilization problems without assuming the existence of doubly coprime factorizations for the transfer matrices. Indeed, it has been recently shown in the literature that an internally stabilizable does not generally admit doubly coprime factorizations. Firstly, we give new necessary and suﬃcient conditions for internal stabilizability by means of matrix equalities. On these characterizations, the fact that internal stabilizability does not imply the existence of coprime factorizations becomes obvious. Secondly, we recall that a necessary condition for strong stabilizability is the existence of a doubly coprime factorization for the transfer matrix. Then, using the concept of stable range sr(A) of a ring A, introduced in (algebraic, topological) K -theory, we prove that sr(A) = 1 implies that every transfer matrix deﬁned over the quotient ﬁeld of A and which admits a left- or a rightcoprime factorization is strongly stabilizable. In particular, this result holds for A = H∞ (D), H∞ (C+ ), W+ and A(D), solving a question asked by A. Feintuch in [8]. Thirdly, we point out that the simultaneous stabilization problem is not equivalent to the strong stabilization problem if the plants do not admit doubly coprime factorizations. Using the fractional ideal approach to stabilization problems, we give a necessary and suﬃcient condition for a pair of SISO plants to be simultaneously stabilizable without assuming the existence of coprime factorizations. Finally, using the parametrization of all stabilizing controllers of an internally stabilizable SISO plant (which does not necessarily admit coprime factorizations), we show how to transform the non-linear sensitivity minimization problem into an aﬃne, and thus, convex minimization problem.

2013-cancer cell-Chromatin-Bound IκBα RegulatesPolycomb Target Genes in Differentiation and Cancer

Cancer CellArticleChromatin-Bound I k B a Regulates a Subset of Polycomb Target Genes in Differentiation and CancerMarı´a Carmen Mulero,1Dolors Ferres-Marco,2Abul Islam,3,4Pol Margalef,1Matteo Pecoraro,5Agustı´Toll,6Nils Drechsel,8Cristina Charneco,8Shelly Davis,9Nicola´s Bellora,3Fernando Gallardo,6Erika Lo ´pez-Arribillaga,1Elena Asensio-Juan,1Vero´nica Rodilla,1Jessica Gonza ´lez,1Mar Iglesias,7Vincent Shih,10M.Mar Alba `,3,11Luciano Di Croce,5,11Alexander Hoffmann,10Shigeki Miyamoto,9Jordi Villa`-Freixa,8,12Nuria Lo ´pez-Bigas,3,11William M.Keyes,5Marı´a Domı´nguez,2Anna Bigas,1,13and Lluı´s Espinosa 1,13,*1Program in Cancer Research,Institut Hospital del Mar d’Investigacions Me`diques (IMIM),Barcelona 08003,Spain 2DevelopmentalNeurobiology,Instituto de Neurociencias de Alicante,CSIC-UMH,Alicante 03550,Spain3Research Program on Biomedical Informatics,Universitat Pompeu Fabra,IMIM-Hospital del Mar,Barcelona 08003,Spain 4Department of Genetic Engineering and Biotechnology,University of Dhaka,Dhaka 1000,Bangladesh 5Gene Regulation,Stem Cells and Cancer,Centre de Regulacio ´Geno `mica (CRG),Barcelona 08003,Spain 6Dermatology Department 7Pathology DepartmentHospital del Mar,Barcelona 08003,Spain8Computational Biochemistry and Biophysics Laboratory,IMIM-Hospital del Mar and Universitat Pompeu Fabra,Barcelona 08003,Spain 9McArdle Laboratory for Cancer Research,University of Wisconsin Carbone Cancer Center,University of Wisconsin-Madison,6159Wisconsin Institute for Medical Research,1111Highland Avenue,Madison,WI 53705,USA 10Signaling Systems Laboratory,UCSD,La Jolla,CA 92093-0375,USA 11Institucio ´Catalana de Recerca i Estudis Avanc ¸ats (ICREA),Barcelona 08003,Spain 12Escola Polite `cnica Superior (EPS),Universitat de Vic,Barcelona 08500,Spain 13These authors contributed equally to this work *Correspondence:lespinosa@imim.es/10.1016/r.2013.06.003SUMMARYHere,we demonstratethat sumoylated and phosphorylated of keratinocytes and interacts with histones H2A and H4at the regulatory region of HOX and IRX genes.Chromatin-bound I k B a modulates Polycomb recruitment and imparts their competence to be activated by TNF a .Mutations in the Drosophila I k B a gene cactus enhance the homeotic phenotype of Polycomb mutants,which is not counteracted by mutations in dorsal/NF-k B .Oncogenic trans-formation of keratinocytes results in cytoplasmic I k B a translocation associated with a massive activation of Hox .Accumulation of cytoplasmic I k B a was found in squamous cell carcinoma (SCC)associated with IKK activation and HOX upregulation.INTRODUCTIONNF-k B plays a crucial role in biological processes,such as native and adaptive immune responses,organ development,cell proliferation,apoptosis,or cancer (Naugler and Karin,2008;Vallabhapurapu and Karin,2009).NF-k B activation de-pends on the IKK-mediated degradation of the NF-k B inhibitors,I k B proteins,that takes place in the cytoplasm and results in the translocation of the NF-k B transcription factor to the nucleus,where it activates gene expression.Recent studies demonstrate the existence of alternative nuclear functions for regulatory ele-ments of the pathway (reviewed in Espinosa et al.,2011),but their biological implications remain poorly understood.Recently,it has been demonstrated that nuclear I k B b binds the promoterCancer Cell 24,151–166,August 12,2013ª2013Elsevier Inc.151角质形成细胞Cancer CellI k B a Is a Modulator of Polycomb Function(legend on next page) 152Cancer Cell24,151–166,August12,2013ª2013Elsevier Inc.of NF-k B target genes following lipopolysaccharide (LPS)stimu-lation to prevent I k B a -mediated inactivation,thereby sustaining cytokine expression in immune cells (Rao et al.,2010).Numerous studies have reported nuclear translocation of I k B a (Aguilera et al.,2004;Arenzana-Seisdedos et al.,1997;Huang and Miyamoto,2001;Wuerzberger-Davis et al.,2011)and various partners for nuclear I k B a ,including histone deacetylases (HDACs)and nuclear corepressors,have been identiﬁed (Agui-lera et al.,2004;Espinosa et al.,2003;Viatour et al.,2003).In ﬁbroblasts,nuclear I k B a associates with the promoter of Notch target genes correlating with their transcriptional repression,which is reverted by TNF a (Aguilera et al.,2004).Nevertheless,the mechanisms that regulate association of I k B to the chromatin and its repressive function remain unknown.I k B a -deﬁcient mice die around day 5because of skin inﬂam-mation associated with high levels of IL1b and IFN-g in the dermis,CD8+T cells,and Gr-1+neutrophils inﬁltrating the epidermis,as well as altered keratinocyte differentiation (Beg et al.,1995;Klement et al.,1996;Rebholz et al.,2007),similar to keratinocyte-speciﬁc I k B a -deﬁcient mice (I k B a k5D /k5D )(Re-bholz et al.,2007).In all cases,disruption of TNF a signaling rescued the skin phenotype (Shih et al.,2009),suggesting that lethality was associated with an excessive inﬂammatory response,likely due to increased NF-k B activity.However,mice expressing different I k B a mutants that are equally able to repress NF-k B in the skin showed divergent phenotypes.Speciﬁcally,mice expressing the nondegradable I k B a mutant,I k B a S32-36A ,developed skin tumors resembling SCC (van Hoger-linden et al.,1999),whereas mice carrying a predominantly nuclear form of I k B a show no overt skin defects (Wuerzberger-Davis et al.,2011).Skin differentiation depends on the correct establishment and maintenance of speciﬁc gene expression patterns,including genes of the HOX family,which in the basal progenitor cells are repressed by EZH2,the catalytic subunit of the Polycomb repressive complex 2(PRC2)(Ezhkova et al.,2009,2011).PRC2is composed by EZH2,the WD-repeat protein EED,RbAp48,and the zinc-ﬁnger protein SUZ12(Zhang and Reinberg,2001).Methylation of lysine 27on histone H3(H3K27me3)by EZH2imposes gene silencing in part by trig-gering recruitment of PRC1(Cao et al.,2002;Min et al.,2003)and histone deacetylases (HDACs).Here,we investigate analternative function for I k B a in the regulation of skin homeosta-sis,development,and cancer.RESULTSPhosphorylated and Sumoylated I k B a Localizes in the Nucleus of KeratinocytesTo investigate the physiological role for nuclear I k B a ,we per-formed an initial screen to determine its subcellular distribution in human tissues.We found that I k B a localizes in the cytoplasm of most tissues and cell types as expected (Figure S1A available online);yet,a distinctive nuclear staining of I k B a was found in human (Figure 1A)and mouse skin sections (Figures 1A,S1A,and S1C),more prominently in the keratin14+basal layer kerati-nocytes.I k B a distribution became more diffused in the supra-basal layer of the skin and gradually disappeared in the more differentiated cells.Speciﬁcity of nuclear I k B a staining was conﬁrmed using skin sections from newborn I k B a -knockout (KO)mice (Figure S1B)and different anti-I k B a antibodies and blocking peptides (Figure S1C).By immunoﬂuorescence (IF)and immunoblot (IB),we detected I k B a protein in both the cyto-plasmic and the nuclear/chromatin fractions of human (Figures 1B and 1C)and mouse (Figure S1D)keratinocytes.Interestingly,nuclear I k B a displayed a shift in its electrophoretic mobility (z 60kDa)detected by different anti-I k B a antibodies,including the anti-phospho-S32-36-I k B a antibody.We next precipitated I k B a from nuclear and cytoplasmic keratinocyte extracts and determined whether this low I k B a mobility was a result of ubiq-uitin or SUMO modiﬁcations.We found that nuclear I k B a was speciﬁcally recognized by anti-SUMO2/3,but not anti-SUMO1or anti-ubiquitin antibodies (Figure 1D;data not shown).Here-after,we will refer to this nuclear I k B a species as phospho-SUMO-I k B a (PS-I k B a ).By cotransfection of different SUMO plasmids in HEK293T cells,we demonstrated that SUMO2was integrated to HA-I k B a at K21,22(Figure S1E),independently of S32,36phosphorylation (Figure 1E).By subcellular fractionation,we found that most HA-PS-I k B a was distributed in the nucleus of HEK293T cells (data not shown),and both K21,22R and S32,36A I k B a mutants showed reduced association with the chromatin (Figure 1F).These results suggest that phosphorylation and sumoylation are both required for I k B a nuclear functions in vivo.Of note,PS-I k B a levels were always low in HEK293T cells whenFigure 1.Phosphorylated and Sumoylated I k B a Is Found in the Nucleus of Normal Basal Keratinocytes(A)Immunodetection of I k B a (green)in normal human skin and detail of basal layer.B,basal;S,spinous,G,granular;and C,corniﬁed layers of epidermis.Dashed line indicates the dermis interphase.DAPI was used for nuclear staining.(B)IF of I k B a in primary human keratinocytes.(C)Subcellular fractionation of human keratinocytes followed by IB with the indicated antibodies.(D)I k B a was immunoprecipitated from primary murine keratinocyte extracts followed by IB with the indicated antibodies.(E)IB analysis of His-tag precipitates from HEK293T cells transfected with the indicated plasmids.SUMO2is incorporated in I k B a when K21,22are present.(F)HEK293T cells were transfected with the indicated I k B a plasmids and processed following the ChIP protocol to obtain the whole chromatin fraction that was analyzed by IB.(G)IF of I k B a and P-IKK in skin sections.Cells with P-IKK staining do not contain nuclear I k B a .(H)IB analysis of keratinocytes transduced with myc-IKK a EE or control.(I)IF analysis of the indicated differentiation markers in skin sections of WT and I k B a KO newborn mice.(J)IB analysis of indicated proteins in control or Ca 2+-treated murine keratinocytes.Total and nuclear/chromatin fractions are shown.(K)Determination of Filaggrin ,K10,and p63mRNA levels in control and I k B a KD keratinocytes following Ca 2+treatment.Expression levels are relative to Gapdh and compared to control cells.Error bars indicate SD.I k B a protein levels were analyzed by IB.Data correspond to one representative of three experiments.N,nuclear;C,cytoplasmic.See also Figure S1.Cancer CellI k B a Is a Modulator of Polycomb FunctionCancer Cell 24,151–166,August 12,2013ª2013Elsevier Inc.153Cancer CellI k B a Is a Modulator of Polycomb FunctionFigure2.I k B a Binds Histones H2A and H4(A)PD experiment using GST-I k B a and native(lane2)or denatured-renatured(lanes3–4)human keratinocyte nuclear extracts.One representative gel stained with Coomassie blue is shown(n=3).(B)Puriﬁcation and analysis of B and C bands identiﬁed as histones H2A and H4by mass spectrometry.Table indicates the number of peptides identiﬁed and their score factor.The highest score is highlighted.(C)Coprecipitation from DSP-crosslinked nuclear extracts from human keratinocytes.(D and E)PD using different GST-H2A proteins and total lysates from HEK293T cells expressing the indicated proteins.(legend continued on next page) 154Cancer Cell24,151–166,August12,2013ª2013Elsevier Inc.compared with keratinocytes,even in overexpression conditions and cell lysates directly obtained under denaturing conditions (see inputs in Figures 1E and S1E).It is well known that IKK activity regulates the cytoplasmic levels of I k B a .By double staining of skin sections,we found that the few cells that were positive for active IKK contained I k B a ,but this I k B a was excluded from the nucleus (Figure 1G).To directly investigate whether IKK regulates subcellular distri-bution of I k B a ,we transduced primary murine keratinocytes with lentiviral IKK a EE .We found that active IKK a induced a decrease in the nuclear levels of PS-I k B a as determined by IB (Figure 1H)and IF (Figure S1G).Additional experiments comparing the effects of both IKK isoforms demonstrated that IKK a EE was more efﬁcient than IKK b EE in decreasing nuclear PS-I k B a levels (60%±5%compared with 16%±9%reduction;p <0.001)(Figure S1F).To directly address whether I k B a was required for normal skin differentiation,we performed IF analysis using different markers comparing I k B a wild-type (WT)and KO newborn skins.Consis-tent with previous reports,I k B a -deﬁcient mice do not show any obvious skin defect at birth with a normal K5-positive basal layer,although we observed a slight reduction in the thickness of the K10-positive suprabasal epidermal layer.Most importantly,I k B a mutant skins showed a severe reduction of the more differ-entiated layer of cells identiﬁed by the accumulation of ﬁlaggrin granules (Figure 1I).This is a cause for impaired barrier function (Palmer et al.,2006).Next,we aimed to distinguish between cell-autonomous and non-cell-autonomous effects of I k B a deﬁ-ciency by using an in vitro system for keratinocyte differentiation induced by high Ca 2+exposure (Hennings et al.,1980).In this model,we found that keratinocyte differentiation was associated with a decrease in both I k B a and PS-I k B a levels and activation of nuclear IKK (Figures 1J and S1H).Notably,knockdown (KD)of I k B a disturbs in vitro keratinocyte differentiation as indicated by the impaired K10and ﬁlaggrin induction in response to Ca 2+,which was accompanied by sustained expression of the progenitor marker p63(Figure 1K).Together,these results strongly suggest that I k B a plays a cell-autonomous function in skin differentiation.I k B a Directly Binds to the N-Terminal Tail of Histones H2A and H4To further investigate the mechanisms underlying nuclear I k B a functions,we searched for nuclear proteins that directly asso-ciate with I k B a .Using GST-I k B a and human keratinocyte nuclear extracts in pull-down (PD)experiments,we isolated pro-teins of estimated molecular weights of 15and 14kDa that were identiﬁed by mass spectrometry as histones H2A and H4(Fig-ures 2A and 2B).Interaction between histones H2A and H4and I k B a was further conﬁrmed by coprecipitation of endoge-nous proteins from keratinocyte nuclear extracts.Of note,the NF-k B subunit p65was absent from nuclear I k B a precipitates but coprecipitated in the cytoplasmic fraction (Figure 2C).By PD assays,we determined the speciﬁcity of I k B a binding compared to other I k B homologs (Figure 2D)and mapped the I k B a -binding domain of histone H2A to be between amino acids 2and 35(Figure 2E).Preincubation of I k B a with p65prevented its association with histones (Figure S2A),suggestive of mutually exclusive complexes.Comparative sequence analysis of the I k B a -binding region of histone H2A (AA1–36)and the homologous region of H4revealed the presence of a motif (3KXXXK/R)that was absent from other histone and nonhistone proteins (Figure 2F).To further study I k B a binding speciﬁcity,we screened a histone peptide array using nuclear HA-I k B a expressed in HEK293T cells as bait.We found that I k B a bound to peptides containing AA11–30of his-tone H4,but not the corresponding region of histone H3.Most importantly,binding of I k B a to H4was prevented by the combi-nation K12/K16Ac and K20Ac or Me2(Figure 2G).Because the equivalent peptides from histone H2A were not included in the array,we performed parallel precipitation experiments using biotin-tagged peptides (AA5–23)of histone H2A and H4(Figures 2H and 2I).We found that I k B a association was prevented by K12Ac,K16Ac,and K20me2of histone H4or the equivalent modiﬁcations of the H2A peptide (Figure 2H)and also when all K/R residues in the 3KXXXK motif were changed into A (Figure 2I).Of note that in these experiments histone-bound HA-I k B a was mostly identiﬁed as a nonsumoylated band,which opens the possibility that posttranslation modiﬁcations are not essential to mediate this interaction in vitro.However,parallel binding experiments using keratinocyte extracts,PS-I k B a ,showed a preferential binding to the histone peptides compared with the cytoplasmic 37kDa I k B a form (Figure S2B).Together,these re-sults strongly suggest that only PS-I k B a can bind the chromatin,but in HEK293T cells this molecule is then desumoylated in vivo or as a consequence of the experimental processing.To gain further insights into the molecular basis of I k B a binding to histones,we completed the structure of I k B a obtained from the Protein Data Bank (ID code 1IKN),which lacked part of the ankyrin repeat (AR)1,using RAPPER (Depristo et al.,2005)and performed docking studies with AutoDock Vina (Trott and Olson,2010)of the histone H4peptide,GKGGAKRHRKV,that contains most of the KXXXK domain.Docking calculations showed two deep pockets for K interaction in I k B a located between ARs 1-2and 2-3and an additional shallower patch between AR3and 4.Overall,the peptide bound in a clearly nega-tive region on the I k B a surface (Figure 2J),with higher afﬁnity than the modiﬁed peptide that was acetylated in the ﬁrst and(F)ClustalW alignment of the conserved KXXXK/R motifs in the N terminus of histones H2A and H4.Conserved K and R residues are in green.Red triangle indicates the last AA included in GST-H2A 2-35.(G)Histone peptide array hybridized with nuclear HEK293T extracts expressing HA-I k B a .One informative area of the blot image and the relative binding of selected peptides are shown.(H)Coprecipitation of cytoplasmic and nuclear HA-I k B a expressed in HEK293T cells with the indicated histone H2A and H4peptides.(I)HA-I k B a was precipitated using the indicated histone H2A peptide,the K/A mutant,or scrambled peptide.In (G),(H),and (I),cell lysates were denatured-renatured previous to the precipitation to disrupt preformed complexes.(J)Model for binding of the histone H4peptide (unmodiﬁed or modiﬁed)to consecutive ankyrin repeats of I k B a (3KXXXK).See also Figure S2.Cancer CellI k B a Is a Modulator of Polycomb FunctionCancer Cell 24,151–166,August 12,2013ª2013Elsevier Inc.155Cancer CellI k B a Is a Modulator of Polycomb FunctionFigure3.Analysis and Identiﬁcation of I k B a Target Genes(A)Developmentally related genes selected from I k B a targets identiﬁed in ChIP-seq analysis.Fold change over the random background is indicated.(B)Functional enrichment of target genes with p value cutoff%10À5based on gene ontology(GO)as extracted from Ensembl database using GiTools.Enriched categories are represented in heatmap with the indicated color-coded p value scale.(legend continued on next page) 156Cancer Cell24,151–166,August12,2013ª2013Elsevier Inc.second K residues (K12and K16).We experimentally validated that ARs of I k B a participate in histone binding because the I k B aD 55-106mutant,lacking part of AR1,failed to bind GST-H2A (Figure S2C).Similarly,this association was prevented by 1%deoxycholate (Figure S2D),as described for interactions involving the ARs of I k B a (Baeuerle and Baltimore,1988;Savi-nova et al.,2009).I k B a Is Speciﬁcally Recruited to the Regulatory Regions of Developmental GenesTo identify putative PS-I k B a target genes,we performed chromatin immunoprecipitation sequencing (ChIP-seq)using chromatin extracts from primary human keratinocytes and anti-I k B a antibody.We identiﬁed 2,778enriched peaks,correspond-ing to 2,433Ensembl genes that were signiﬁcantly enriched with p values %10À5.Gene ontology analysis showed that a signiﬁ-cant proportion of genes participate in biological processes associated with embryonic development and cell differentiation.I k B a targets included genes of the HOX and IRX families,ASCL4,CDX2,NEUROD4,OLIG3,and NEURL ,among others (Figures 3A and 3B).Annotation of the peak genomic positions to the closest gene demonstrated that many peaks were positioned immediately after the transcription start site (TSS),with a sharp decrease near the transcription termination site (TTS)(Figure 3C),whereas others were located far from promoter regions.Some of the latter overlapped with regions enriched in H3K4me1,a his-tone mark associated with enhancer regions (data not shown).Randomly selected I k B a targets were conﬁrmed by conventional chromatin immunoprecipitation (ChIP)using primers ﬂanking the regions identiﬁed in the ChIP-seq experiment (Figure 3D).Consistent with its overall effects on I k B a levels,sustained Ca 2+treatment caused the loss of I k B a from all tested gene pro-moters (Figure 3E).Similarly,short treatment with TNF a released chromatin-bound I k B a in keratinocytes,as we previously found in ﬁbroblasts (Aguilera et al.,2004).However,we did not detect a general effect of TNF a on PS-I k B a levels,but we consistently found a partial redistribution of PS-I k B a to the soluble nuclear fractions (Figure S3A).Next,we investigated whether TNF a and Ca 2+modulated HOX and IRX transcription in keratinocytes.All tested I k B a targets (n =12)were robustly induced by TNF a treatment (up to 12-fold)following different kinetics (Figures 3F)and to a lesser extent (up to 3-fold)by Ca 2+treatment (Fig-ure S3B)or I k B a KD (Figure S3C).Interestingly,1hr of TNF a treatment prevented Ca 2+-induced differentiation of murine keratinocytes (Figure S3D),supporting the notion that PS-I k B a integrates inﬂammatory signals with skin homeostasis (see Dis-cussion ).We also tested whether p65participated in HOX or IRX gene activation by TNF a .By ChIP analysis,we did not ﬁnd anyrecruitment of p65to I k B a target genes after TNF a treatment,in contrast to a canonical NF-k B target gene promoter (Fig-ure S3E).However,we detected low amounts of p65at HOX genes under basal conditions that might contribute to gene repression (Dong et al.,2008),although the function of chromatin-bound p65at regions distant from the TSS of both NF-k B targets and nontargets is unresolved.Binding of p65to HOX and IRX was reduced after TNF a treatment,suggesting that p65was redistributed from noncanonical to canonical NF-k B targets once activated.Silencing of HOX genes in keratinocytes involves PRC2and its core component the H3K27methyltransferase EZH2(Ezhkova et al.,2009;Mejetta et al.,2011).To explore a putative associa-tion between I k B a and PRC2function,we crossed our list of 2,433I k B a targets with available ChIP-seq data from keratino-cytes.Approximately 50%of I k B a targets corresponded to genes enriched for the H3K27me3mark (Figures 3G and S3F),although I k B a targeted only 13%of the H3K27trimethylated genes.Most importantly,genomic sequences occupied by I k B a essentially overlapped with those regions containing high H3K27me3levels (Figure 3G).We also found a statistically signif-icant overlap (p <10À16)between I k B a target genes and PRC tar-gets in ES cells (Birney et al.,2007;Ku et al.,2008)(Figure S3G).I k B a Interacts with and Regulates Association of PRC2to Target Genes in Response to TNF aIn the mass spectrometry analysis of proteins that associate with GST-I k B a ,we identiﬁed a few peptides corresponding to chro-matin modiﬁers,such as EZH2and SUZ12,and SIN3A (Figure S4A).Speciﬁcity of I k B a interactions with PRC2elements,but also I k B a association to the PRC1protein BMI1,was conﬁrmed by PD assays (Figure S4B).SUZ12was able to interact with nuclear I k B a ,whereas p65speciﬁcally associated with cyto-plasmic I k B a in the IP experiments (Figure 4A).Importantly,exogenous wild-type I k B a ,but not an I k B a mutant that failed to bind histones,facilitated the association of SUZ12to GST-H4(Figure 4B).Moreover,ChIP experiments demonstrated that TNF a treatment induced the dissociation of SUZ12from I k B a target regions,but not non-I k B a targets (Figure 4C).Sequential ChIP experiments demonstrated that I k B a and SUZ12simultaneously bound to I k B a target genes (Figure 4D).To test the functional relevance of I k B a in PRC-mediated repression,we used WT murine embryonic ﬁbroblast (MEFs),which expressed detectable levels of PS-I k B a (Figure 4E)and I k B a KO MEFs.By ChIP-on-chip experiments using three different I k B a antibodies,we conﬁrmed that several Hox genes were also targets of I k B a in MEFs (Table S1).By ChIP,we found that SUZ12and EZH2bound I k B a targets efﬁciently in WT MEFs(C)Graphs show the relative distance to the nearest ChIPed region,3kb upstream and downstream of the RefSeq gene’s TSS and TTS.(D)Validation of the identiﬁed DNA regions (À222to À200for HOXA10,À9,380to À9,360for HOXB2,+6,166to +6,186for HOXB5,+4,451to +4,471for HOXB3,and À18,820to À18,800for IRX3)by conventional ChIP.Ampliﬁcation of 2kb distant regions was used as negative controls.(E)ChIP analysis of I k B a after 20min of TNF a or 48hr Ca 2+treatments.In (D)and (E),graphs represent mean enrichment relative to nonspeciﬁc immunoglobulin G (IgG)(n =2).(F)Expression levels of I k B a target genes following TNF a treatment analyzed by qRT-PCR.Gene represented is in black,whereas genes following the same kinetics are indicated in red.(G)ChIP-seq proﬁles of endogenous I k B a occupancy in three enriched loci (HOXA ,HOXB ,and IRX5)and one negative locus (JARID1B/KDM5B )compared to H3K27me3(from the UW ENCODE Project)in keratinocytes.(D–F)Bars represent mean,and error bars indicate SD.See also Figure S3.Cancer CellI k B a Is a Modulator of Polycomb FunctionCancer Cell 24,151–166,August 12,2013ª2013Elsevier Inc.157Figure 4.I k B a Interacts with and Regulates Association of PRC2in Response to TNF a(A)IB analysis of I k B a precipitates from nuclear and cytoplasmic primary murine keratinocyte extracts.Five percent of the input and 25%of the IP was loaded in all cases except for detection of I k B a input that represents 0.5%.(B)PD using GST-H4and cell lysates from HEK293T expressing different combinations of HA-I k B a and SUZ12.(C)Relative recruitment assessed by ChIP of SUZ12to different genes 40min after TNF a in primary murine keratinocytes.(legend continued on next page)Cancer CellI k B a Is a Modulator of Polycomb Function158Cancer Cell 24,151–166,August 12,2013ª2013Elsevier Inc.but only weakly in I k B a KO MEFs (Figure 4F,time 0).In WT cells,TNF a treatment induced a signiﬁcant but temporary release of SUZ12and EZH2from these loci,which peaked after 30–60min of treatment (Figure 4F).The binding of PRC2proteins at Hox genes inversely correlated with the expression of these Hox genes (Figure 4G).In contrast,I k B a KO cells failed to activate Hox transcription in response to TNF a (Figure 4G),which is consistent with a defective release of PRC2proteins (Figure 4F).Unexpectedly,we did not detect changes in H3K27me3levels in these Hox genes upon TNF a treatment at any of the time points studied (20min,2hr,and 7hr)(data not shown),likely reﬂecting the high stability of this histone modiﬁca-tion (De Santa et al.,2007).Supporting the possibility that activa-tion of I k B a targets is independent of the enzymatic activity of EZH2,a 24hr treatment with the EZH2inhibitor DZNep does not affect Hoxb8or Irx3messenger RNA (mRNA)levels in kera-tinocytes (data not shown).Together,these results suggest that transcriptional repression-activation of these genes does not strictly depend on EZH2enzymatic activity but rather PRC2release might modulate the dissociation of PRC1or HDACs (van der Vlag and Otte,1999)that associate with more dynamic chromatin modiﬁcations.In agreement with this possibility,his-tone H3is rapidly acetylated following TNF a treatment at different Hox gene promoters (Figure S4C).To further study the involvement of NF-k B in the regulation of I k B a targets by TNF a ,we attempted to use different mutant MEFs,including the p65,Ikk a ,Ikk b ,and the triple p65;p50;c-Rel KO.We found that TNF a induced Hox and Irx expression in both p65and Ikk b KO cells (Figure S4D)suggesting that it was NF-k B independent.However,speciﬁc mutants contained vari-able levels of I k B a and PS-I k B a (Figures S4E and S4F),which make a more accurate quantitative analysis unproductive.Inter-estingly,phosphorylation of nuclear I k B a was not reduced in the Ikk a or Ikk b KO cells (Figures S4F),indicating that other kinases are involved in generating PS-I k B a .Only triple KO cells,which essentially lacked I k B a (Figure S4G)and the Ikk a -deﬁcient MEFs,showed a strong defect on Hox and Irx transcription (Figures S4D and S4H).To better understand the contribution of NF-k B to Hox regulation,we next performed luciferase re-porter assays measuring the ability of different I k B a mutant pro-teins to repress a Hoxb8-promoter construct compared with a reporter containing three consensus sites for NF-k B (3x k B).We found that WT I k B a and the nuclear I k B a NES mutant (Huang et al.,2000)signiﬁcantly repressed both promoters.However,mutations that affect chromatin-association (Figure 1F)pre-vented Hoxb8repression but still inhibited the expression of the 3x k B reporter (Figure 4H).Consistently,Hoxb8mRNA levels were signiﬁcantly reduced in the skin of mice expressing I k B a NES(Wuerzberger-Davis et al.,2011)(Figure 4I).Moreover,these animals showed an expansion of the K14-positive basal layer of keratinocytes containing nuclear I k B a (Figure 4J),associated with increased proliferation measured by ki67staining and impaired differentiation as indicated by the reduced thickness of the suprabasal K10-expressing layer (Figure 4K).Genetic Interaction between I k B a /cactus and polycomb in DrosophilaPolycomb group (PcG)I k B and NF-k B proteins are conserved from ﬂies to humans.In addition,Drosophila contains one Hox cluster,compared with four clusters in vertebrates,which facili-tates studying genetic interactions.We ﬁrst conﬁrmed that the single Drosophila I k B homolog,cactus (cact)(Geisler et al.,1992),maintained the capacity to associate with histones (Figure 5A).By IF,we detected colocalization of cactus and Polycomb (Pc),a PRC1protein that is essential for the repressive PRC2function,in speciﬁc bands of polytene chromosomes (Figure 5B).Most of the cactus staining overlapped with Pc,but only a few Pc-positive bands contained cactus.Based on our mammalian data,our ﬁrst attempt was to generate single PRC2mutants and combine them with cactus -deﬁcient mutants.All mutants were tested in heterozygosis because homozygous mutations in cactus or PcG genes are lethal.We found that heterozygous mutations in PRC2genes (e.g.,the null mutations of E (z ))as well as the composed E (z )and cact exhibit no overt homeotic phenotypes.In contrast,hetero-zygous Pc mutants exhibit a variety of characteristic homeotic transformations,including the partial transformation of the sec-ond and third (mesothoracic and metathoracic)legs toward ﬁrst (also known as prothoracic)legs that in males are characterized by the presence of sex combs.Modiﬁcations of this phenotype (also called ‘‘extra sex comb’’)have been extensively used as a functional assay to validate new PcG proteins in vivo.Thus,we tested the effect of reducing cact on Pc -induced homeotic transformation using 12recessive mutations of cact and two independently generated Pc alleles.All cact mutant alleles,but more prominently cact 1,enhanced the ‘‘extra sex comb’’pheno-type of Pc mutations (Pc 3and Pc XT109)(Figure 5C;Table S2).Because cells lacking cact exhibit massive nuclear localization of the transcription factor dorsal (dl,Drosophila NF-k B/Rel/p65ortholog)during postembryonic stages (Lemaitre et al.,1995),we tested whether enhancement of homeotic defect of Pc by cact mutations is due to increased dorsal activity.Because sta-bility of cact is under NF-k B/Dorsal control (Kubota and Gay,1995),we anticipated that for phenotypes due to increased dorsal,dl mutations would counteract cact mutations,whereas for phenotypes independent of dorsal,reducing dl should yield(D)Sequential ChIP using the indicated combinations of antibodies.An analysis of two different Hox regulatory regions is shown.(E)IB analysis of WT ﬁbroblasts showing the presence of cytoplasmic and nuclear I k B a .(F)Relative chromatin binding of PRC2and I k B a in WT and I k B a KO MEFs treated with TNF a .ChIP values were normalized by IgG precipitation.(G)Relative levels of the indicated genes in WT and I k B a KO MEFs.(H)Luciferase assays to determine the effect of different I k B a constructs on the activity of HoxB8compared to the 3x k B reporter.Lower panels show expression levels of different constructs.(I)Expression levels of HoxB8in the skin of WT and I k B a NES/NES mice by qRT-PCR (n =2).(J and K)Analysis of skin sections from 7-to 8-week-old WT and I k B a NES/NES mice by IF.K14labels the basal layer keratinocytes (J).Immunostaining of ki67,the suprabasal marker K10,and ﬁlaggrin (K).Throughout the ﬁgure,bars represent mean,and error bars indicate SD.See also Figure S4and Table S1.Cancer CellI k B a Is a Modulator of Polycomb FunctionCancer Cell 24,151–166,August 12,2013ª2013Elsevier Inc.159。

Universities in Evolutionary Systems(系统变革中的大学)

Universities in Evolutionary Systems of InnovationMarianne van der Steen and Jurgen EndersThis paper criticizes the current narrow view on the role of universities in knowledge-based economies.We propose to extend the current policy framework of universities in national innovation systems(NIS)to a more dynamic one,based on evolutionary economic principles. The main reason is that this dynamic viewﬁts better with the practice of innovation processes. We contribute on ontological and methodological levels to the literature and policy discussions on the effectiveness of university-industry knowledge transfer and the third mission of uni-versities.We conclude with a discussion of the policy implications for the main stakeholders.1.IntroductionU niversities have always played a major role in the economic and cultural devel-opment of countries.However,their role and expected contribution has changed sub-stantially over the years.Whereas,since1945, universities in Europe were expected to con-tribute to‘basic’research,which could be freely used by society,in recent decades they are expected to contribute more substantially and directly to the competitiveness ofﬁrms and societies(Jaffe,2008).Examples are the Bayh–Dole Act(1982)in the United States and in Europe the Lisbon Agenda(2000–2010) which marked an era of a changing and more substantial role for universities.However,it seems that this‘new’role of universities is a sort of universal given one(ex post),instead of an ex ante changing one in a dynamic institutional environment.Many uni-versities are expected nowadays to stimulate a limited number of knowledge transfer activi-ties such as university spin-offs and university patenting and licensing to demonstrate that they are actively engaged in knowledge trans-fer.It is questioned in the literature if this one-size-ﬁts-all approach improves the usefulness and the applicability of university knowledge in industry and society as a whole(e.g.,Litan et al.,2007).Moreover,the various national or regional economic systems have idiosyncratic charac-teristics that in principle pose different(chang-ing)demands towards universities.Instead of assuming that there is only one‘optimal’gov-ernance mode for universities,there may bemultiple ways of organizing the role of univer-sities in innovation processes.In addition,we assume that this can change over time.Recently,more attention in the literature hasfocused on diversity across technologies(e.g.,King,2004;Malerba,2005;Dosi et al.,2006;V an der Steen et al.,2008)and diversity offormal and informal knowledge interactionsbetween universities and industry(e.g.,Cohenet al.,1998).So far,there has been less atten-tion paid to the dynamics of the changing roleof universities in economic systems:how dothe roles of universities vary over time andwhy?Therefore,this article focuses on the onto-logical premises of the functioning of univer-sities in innovation systems from a dynamic,evolutionary perspective.In order to do so,we analyse the role of universities from theperspective of an evolutionary system ofinnovation to understand the embeddednessof universities in a dynamic(national)systemof science and innovation.The article is structured as follows.InSection2we describe the changing role ofuniversities from the static perspective of anational innovation system(NIS),whereasSection3analyses the dynamic perspective ofuniversities based on evolutionary principles.Based on this evolutionary perspective,Section4introduces the characteristics of a LearningUniversity in a dynamic innovation system,summarizing an alternative perception to thestatic view of universities in dynamic economicsystems in Section5.Finally,the concludingVolume17Number42008doi:10.1111/j.1467-8691.2008.00496.x©2008The AuthorsJournal compilation©2008Blackwell Publishingsection discusses policy recommendations for more effective policy instruments from our dynamic perspective.2.Static View of Universities in NIS 2.1The Emergence of the Role of Universities in NISFirst we start with a discussion of the literature and policy reports on national innovation system(NIS).The literature on national inno-vation systems(NIS)is a relatively new and rapidly growingﬁeld of research and widely used by policy-makers worldwide(Fagerberg, 2003;Balzat&Hanusch,2004;Sharif,2006). The NIS approach was initiated in the late 1980s by Freeman(1987),Dosi et al.(1988)and Lundvall(1992)and followed by Nelson (1993),Edquist(1997),and many others.Balzat and Hanusch(2004,p.196)describe a NIS as‘a historically grown subsystem of the national economy in which various organizations and institutions interact with and inﬂuence one another in the carrying out of innovative activity’.It is about a systemic approach to innovation,in which the interaction between technology,institutions and organizations is central.With the introduction of the notion of a national innovation system,universities were formally on the agenda of many innovation policymakers worldwide.Clearly,the NIS demonstrated that universities and their interactions with industry matter for innova-tion processes in economic systems.Indeed, since a decade most governments acknowl-edge that interactions between university and industry add to better utilization of scienti-ﬁc knowledge and herewith increase the innovation performance of nations.One of the central notions of the innovation system approach is that universities play an impor-tant role in the development of commercial useful knowledge(Edquist,1997;Sharif, 2006).This contrasts with the linear model innovation that dominated the thinking of science and industry policy makers during the last century.The linear innovation model perceives innovation as an industry activity that‘only’utilizes fundamental scientiﬁc knowledge of universities as an input factor for their innovative activities.The emergence of the non-linear approach led to a renewed vision on the role–and expectations–of universities in society. Some authors have referred to a new social contract between science and society(e.g., Neave,2000).The Triple Helix(e.g.,Etzkowitz &Leydesdorff,1997)and the innovation system approach(e.g.,Lundvall,1988)and more recently,the model of Open Innovation (Chesbrough,2003)demonstrated that innova-tion in a knowledge-based economy is an inter-active process involving many different innovation actors that interact in a system of overlapping organizationalﬁelds(science, technology,government)with many interfaces.2.2Static Policy View of Universities in NIS Since the late1990s,the new role of universi-ties in NIS thinking emerged in a growing number of policy studies(e.g.,OECD,1999, 2002;European Commission,2000).The con-tributions of the NIS literature had a large impact on policy makers’perception of the role of universities in the national innovation performance(e.g.,European Commission, 2006).The NIS approach gradually replaced linear thinking about innovation by a more holistic system perspective on innovations, focusing on the interdependencies among the various agents,organizations and institutions. NIS thinking led to a structurally different view of how governments can stimulate the innovation performance of a country.The OECD report of the national innovation system (OECD,1999)clearly incorporated these new economic principles of innovation system theory.This report emphasized this new role and interfaces of universities in knowledge-based economies.This created a new policy rationale and new awareness for technology transfer policy in many countries.The NIS report(1999)was followed by more attention for the diversity of technology transfer mecha-nisms employed in university-industry rela-tions(OECD,2002)and the(need for new) emerging governance structures for the‘third mission’of universities in society,i.e.,patent-ing,licensing and spin-offs,of public research organizations(OECD,2003).The various policy studies have in common that they try to describe and compare the most important institutions,organizations, activities and interactions of public and private actors that take part in or inﬂuence the innovation performance of a country.Figure1 provides an illustration.Theﬁgure demon-strates the major building blocks of a NIS in a practical policy setting.It includesﬁrms,uni-versities and other public research organiza-tions(PROs)involved in(higher)education and training,science and technology.These organizations embody the science and tech-nology capabilities and knowledge fund of a country.The interaction is represented by the arrows which refer to interactive learn-ing and diffusion of knowledge(Lundvall,Volume17Number42008©2008The AuthorsJournal compilation©2008Blackwell Publishing1992).1The building block ‘Demand’refers to the level and quality of demand that can be a pull factor for ﬁrms to innovate.Finally,insti-tutions are represented in the building blocks ‘Framework conditions’and ‘Infrastructure’,including various laws,policies and regula-tions related to science,technology and entre-preneurship.It includes a very broad array of policy issues from intellectual property rights laws to ﬁscal instruments that stimulate labour mobility between universities and ﬁrms.The ﬁgure demonstrates that,in order to improve the innovation performance of a country,the NIS as a whole should be conducive for innovative activities in acountry.Since the late 1990s,the conceptual framework as represented in Figure 1serves as a dominant design for many comparative studies of national innovation systems (Polt et al.,2001;OECD,2002).The typical policy benchmark exercise is to compare a number of innovation indicators related to the role of university-industry interactions.Effective performance of universities in the NIS is judged on a number of standardized indica-tors such as the number of spin-offs,patents and licensing.Policy has especially focused on ‘getting the incentives right’to create a generic,good innovative enhancing context for ﬁrms.Moreover,policy has also inﬂu-enced the use of speciﬁc ‘formal’transfer mechanisms,such as university patents and university spin-offs,to facilitate this collabo-ration.In this way best practice policies are identiﬁed and policy recommendations are derived:the so-called one-size-ﬁts-all-approach.The focus is on determining the ingredients of an efﬁcient benchmark NIS,downplaying institutional diversity and1These organizations that interact with each other sometimes co-operate and sometimes compete with each other.For instance,ﬁrms sometimes co-operate in certain pre-competitive research projects but can be competitors as well.This is often the case as well withuniversities.Figure 1.The Benchmark NIS Model Source :Bemer et al.(2001).Volume 17Number 42008©2008The AuthorsJournal compilation ©2008Blackwell Publishingvariety in the roles of universities in enhanc-ing innovation performance.The theoretical contributions to the NIS lit-erature have outlined the importance of insti-tutions and institutional change.However,a further theoretical development of the ele-ments of NIS is necessary in order to be useful for policy makers;they need better systemic NIS benchmarks,taking systematically into account the variety of‘national idiosyncrasies’. Edquist(1997)argues that most NIS contribu-tions are more focused onﬁrms and technol-ogy,sometimes reducing the analysis of the (national)institutions to a left-over category (Geels,2005).Following Hodgson(2000), Nelson(2002),Malerba(2005)and Groenewe-gen and V an der Steen(2006),more attention should be paid to the institutional idiosyncra-sies of the various systems and their evolution over time.This creates variety and evolving demands towards universities over time where the functioning of universities and their interactions with the other part of the NIS do evolve as well.We suggest to conceptualize the dynamics of innovation systems from an evolutionary perspective in order to develop a more subtle and dynamic vision on the role of universities in innovation systems.We emphasize our focus on‘evolutionary systems’instead of national innovation systems because for many universities,in particular some science-based disciplinaryﬁelds such as biotechnology and nanotechnology,the national institutional environment is less relevant than the institu-tional and technical characteristics of the technological regimes,which is in fact a‘sub-system’of the national innovation system.3.Evolutionary Systems of Innovation as an Alternative Concept3.1Evolutionary Theory on Economic Change and InnovationCharles Darwin’s The Origin of Species(1859)is the foundation of modern thinking about change and evolution(Luria et al.,1981,pp. 584–7;Gould,1987).Darwin’s theory of natural selection has had the most important consequences for our perception of change. His view of evolution refers to a continuous and gradual adaptation of species to changes in the environment.The idea of‘survival of theﬁttest’means that the most adaptive organisms in a population will survive.This occurs through a process of‘natural selection’in which the most adaptive‘species’(organ-isms)will survive.This is a gradual process taking place in a relatively stable environment, working slowly over long periods of time necessary for the distinctive characteristics of species to show their superiority in the‘sur-vival contest’.Based on Darwin,evolutionary biology identiﬁes three levels of aggregation.These three levels are the unit of variation,unit of selection and unit of evolution.The unit of varia-tion concerns the entity which contains the genetic information and which mutates fol-lowing speciﬁc rules,namely the genes.Genes contain the hereditary information which is preserved in the DNA.This does not alter sig-niﬁcantly throughout the reproductive life-time of an organism.Genes are passed on from an organism to its successors.The gene pool,i.e.,the total stock of genetic structures of a species,only changes in the reproduction process as individuals die and are born.Par-ticular genes contribute to distinctive charac-teristics and behaviour of species which are more or less conducive to survival.The gene pool constitutes the mechanism to transmit the characteristics of surviving organisms from one generation to the next.The unit of selection is the expression of those genes in the entities which live and die as individual specimens,namely(individual) organisms.These organisms,in their turn,are subjected to a process of natural selection in the environment.‘Fit’organisms endowed with a relatively‘successful’gene pool,are more likely to pass them on to their progeny.As genes contain information to form and program the organisms,it can be expected that in a stable environment genes aiding survival will tend to become more prominent in succeeding genera-tions.‘Natural selection’,thus,is a gradual process selecting the‘ﬁttest’organisms. Finally,there is the unit of evolution,or that which changes over time as the gene pool changes,namely populations.Natural selec-tion produces changes at the level of the population by‘trimming’the set of genetic structures in a population.We would like to point out two central principles of Darwinian evolution.First,its profound indeterminacy since the process of development,for instance the development of DNA,is dominated by time at which highly improbable events happen (Boulding,1991,p.12).Secondly,the process of natural selection eliminates poorly adapted variants in a compulsory manner,since indi-viduals who are‘unﬁt’are supposed to have no way of escaping the consequences of selection.22We acknowledge that within evolutionary think-ing,the theory of Jean Baptiste Lamarck,which acknowledges in essence that acquired characteris-tics can be transmitted(instead of hereditaryVolume17Number42008©2008The AuthorsJournal compilation©2008Blackwell PublishingThese three levels of aggregation express the differences between ‘what is changing’(genes),‘what is being selected’(organisms),and ‘what changes over time’(populations)in an evolutionary process (Luria et al.,1981,p.625).According to Nelson (see for instance Nelson,1995):‘Technical change is clearly an evolutionary process;the innovation generator keeps on producing entities superior to those earlier in existence,and adjustment forces work slowly’.Technological change and innovation processes are thus ‘evolutionary’because of its characteristics of non-optimality and of an open-ended and path-dependent process.Nelson and Winter (1982)introduced the idea of technical change as an evolutionary process in capitalist economies.Routines in ﬁrms function as the relatively durable ‘genes’.Economic competition leads to the selection of certain ‘successful’routines and these can be transferred to other ﬁrms by imitation,through buy-outs,training,labour mobility,and so on.Innovation processes involving interactions between universities and industry are central in the NIS approach.Therefore,it seems logical that evolutionary theory would be useful to grasp the role of universities in innovation pro-cesses within the NIS framework.3.2Evolutionary Underpinnings of Innovation SystemsBased on the central evolutionary notions as discussed above,we discuss in this section how the existing NIS approaches have already incor-porated notions in their NIS frameworks.Moreover,we investigate to what extent these notions can be better incorporated in an evolu-tionary innovation system to improve our understanding of universities in dynamic inno-vation processes.We focus on non-optimality,novelty,the anti-reductionist methodology,gradualism and the evolutionary metaphor.Non-optimality (and Bounded Rationality)Based on institutional diversity,the notion of optimality is absent in most NIS approaches.We cannot deﬁne an optimal system of innovation because evolutionary learning pro-cesses are important in such systems and thus are subject to continuous change.The system never achieves an equilibrium since the evolu-tionary processes are open-ended and path dependent.In Nelson’s work (e.g.,1993,1995)he has emphasized the presence of contingent out-comes of innovation processes and thus of NIS:‘At any time,there are feasible entities not present in the prevailing system that have a chance of being introduced’.This continuing existence of feasible alternative developments means that the system never reaches a state of equilibrium or ﬁnality.The process always remains dynamic and never reaches an optimum.Nelson argues further that diversity exists because technical change is an open-ended multi-path process where no best solu-tion to a technical problem can be identiﬁed ex post .As a consequence technical change can be seen as a very wasteful process in capitalist economies with many duplications and dead-ends.Institutional variety is closely linked to non-optimality.In other words,we cannot deﬁne the optimal innovation system because the evolutionary learning processes that take place in a particular system make it subject to continuous change.Therefore,comparisons between an existing system and an ideal system are not possible.Hence,in the absence of any notion of optimality,a method of comparing existing systems is necessary.According to Edquist (1997),comparisons between systems were more explicit and systematic than they had been using the NIS approaches.Novelty:Innovations CentralNovelty is already a central notion in the current NIS approaches.Learning is inter-preted in a broad way.Technological innova-tions are deﬁned as combining existing knowledge in new ways or producing new knowledge (generation),and transforming this into economically signiﬁcant products and processes (absorption).Learning is the most important process behind technological inno-vations.Learning can be formal in the form of education and searching through research and development.However,in many cases,innovations are the consequence of several kinds of learning processes involving many different kinds of economic agents.According to Lundvall (1992,p.9):‘those activities involve learning-by-doing,increasing the efﬁciency of production operations,learning-characteristics as in the theory of Darwin),is acknowledged to ﬁt better with socio-economic processes of technical change and innovation (e.g.,Nelson &Winter,1982;Hodgson,2000).Therefore,our theory is based on Lamarckian evolutionary theory.However,for the purpose of this article,we will not discuss the differences between these theo-ries at greater length and limit our analysis to the fundamental evolutionary building blocks that are present in both theories.Volume 17Number 42008©2008The AuthorsJournal compilation ©2008Blackwell Publishingby-using,increasing the efﬁciency of the use of complex systems,and learning-by-interacting, involving users and producers in an interac-tion resulting in product innovations’.In this sense,learning is part of daily routines and activities in an economy.In his Learning Economy concept,Lundvall makes learning more explicit,emphasizing further that ‘knowledge is assumed as the most funda-mental resource and learning the most impor-tant process’(1992,p.10).Anti-reductionist Approach:Systems and Subsystems of InnovationSo far,NIS approaches are not yet clear and systematic in their analysis of the dynamics and change in innovation systems.Lundvall’s (1992)distinction between subsystem and system level based on the work of Boulding implicitly incorporates both the actor(who can undertake innovative activities)as well as the structure(institutional selection environment) in innovation processes of a nation.Moreover, most NIS approaches acknowledge that within the national system,there are different institu-tional subsystems(e.g.,sectors,regions)that all inﬂuence each other again in processes of change.However,an explicit analysis of the structured environment is still missing (Edquist,1997).In accordance with the basic principles of evolutionary theory as discussed in Section 3.1,institutional evolutionary theory has developed a very explicit systemic methodol-ogy to investigate the continuous interaction of actors and institutional structures in the evolution of economic systems.The so-called ‘methodological interactionism’can be per-ceived as a methodology that combines a structural perspective and an actor approach to understand processes of economic evolu-tion.Whereas the structural perspective emphasizes the existence of independent institutional layers and processes which deter-mine individual actions,the actor approach emphasizes the free will of individuals.The latter has been referred to as methodological individualism,as we have seen in neo-classical approaches.Methodological indi-vidualism will explain phenomena in terms of the rational individual(showingﬁxed prefer-ences and having one rational response to any fully speciﬁed decision problem(Hodgson, 2000)).The interactionist approach recognizes a level of analysis above the individual orﬁrm level.NIS approaches recognize that national differences exist in terms of national institu-tions,socio-economic factors,industries and networks,and so on.So,an explicit methodological interactionist approach,explicitly recognizing various insti-tutional layers in the system and subsystem in interaction with the learning agents,can improve our understanding of the evolution of innovation.Gradualism:Learning Processes andPath-DependencyPath-dependency in biology can be translated in an economic context in the form of(some-times very large)time lags between a technical invention,its transformation into an economic innovation,and the widespread diffusion. Clearly,in many of the empirical case studies of NIS,the historical dimension has been stressed.For instance,in the study of Denmark and Sweden,it has been shown that the natural resource base(for Denmark fertile land,and for Sweden minerals)and economic history,from the period of the Industrial Revolution onwards,has strongly inﬂuenced present specialization patterns(Edquist& Lundvall,1993,pp.269–82).Hence,history matters in processes of inno-vation as the innovation processes are inﬂu-enced by many institutions and economic agents.In addition,they are often path-dependent as small events are reinforced and become crucially important through processes of positive feedback,in line with evolutionary processes as discussed in Section3.1.Evolutionary MetaphorFinally,most NIS approaches do not explicitly use the biological metaphor.Nevertheless, many of the approaches are based on innova-tion theories in which they do use an explicit evolutionary metaphor(e.g.,the work of Nelson).To summarize,the current(policy)NIS approaches have already implicitly incorpo-rated some evolutionary notions such as non-optimality,novelty and gradualism.However, what is missing is a more explicit analysis of the different institutional levels of the economic system and innovation subsystems (their inertia and evolution)and how they change over time in interaction with the various learning activities of economic agents. These economic agents reside at established ﬁrms,start-upﬁrms,universities,govern-ments,undertaking learning and innovation activities or strategic actions.The explicit use of the biological metaphor and an explicit use of the methodological interactionst approach may increase our understanding of the evolu-tion of innovation systems.Volume17Number42008©2008The AuthorsJournal compilation©2008Blackwell Publishing4.Towards a Dynamic View of Universities4.1The Logic of an Endogenous‘Learning’UniversityIf we translate the methodological interaction-ist approach to the changing role of universities in an evolutionary innovation system,it follows that universities not only respond to changes of the institutional environment(government policies,business demands or changes in scientiﬁc paradigms)but universities also inﬂuence the institutions of the selection envi-ronment by their strategic,scientiﬁc and entre-preneurial actions.Moreover,these actions inﬂuence–and are inﬂuenced by–the actions of other economic agents as well.So,instead of a one-way rational response by universities to changes(as in reductionist approach),they are intertwined in those processes of change.So, universities actually function as an endogenous source of change in the evolution of the inno-vation system.This is(on an ontological level) a fundamental different view on the role of universities in innovation systems from the existing policy NIS frameworks.In earlier empirical research,we observed that universities already effectively function endogenously in evolutionary innovation system frameworks;universities as actors (already)develop new knowledge,innovate and have their own internal capacity to change,adapt and inﬂuence the institutional development of the economic system(e.g., V an der Steen et al.,2009).Moreover,univer-sities consist of a network of various actors, i.e.,the scientists,administrators at technology transfer ofﬁces(TTO)as well as the university boards,interacting in various ways with indus-try and governments and embedded in various ways in the regional,national or inter-national environment.So,universities behave in an at least partly endogenous manner because they depend in complex and often unpredictable ways on the decision making of a substantial number of non-collusive agents.Agents at universities react in continuous interaction with the learn-ing activities ofﬁrms and governments and other universities.Furthermore,the endogenous processes of technical and institutional learning of univer-sities are entangled in the co-evolution of institutional and technical change of the evo-lutionary innovation system at large.We propose to treat the learning of universities as an inseparable endogenous variable in the inno-vation processes of the economic system.In order to structure the endogenization in the system of innovation analysis,the concept of the Learning University is introduced.In thenext subsection we discuss the main character-istics of the Learning University and Section5discusses the learning university in a dynamic,evolutionary innovation system.An evolution-ary metaphor may be helpful to make theuniversity factor more transparent in theco-evolution of technical and institutionalchange,as we try to understand how variouseconomic agents interact in learning processes.4.2Characteristics of the LearningUniversityThe evolution of the involvement of universi-ties in innovation processes is a learningprocess,because(we assume that)universitypublic agents have their‘own agenda’.V ariousincentives in the environment of universitiessuch as government regulations and technol-ogy transfer policies as well as the innovativebehaviour of economic agents,compel policymakers at universities to constantly respondby adapting and improving their strategiesand policies,whereas the university scientistsare partly steered by these strategies and partlyinﬂuenced by their own scientiﬁc peers andpartly by their historically grown interactionswith industry.During this process,universityboards try to be forward-looking and tobehave strategically in the knowledge thattheir actions‘inﬂuence the world’(alsoreferred to earlier as‘intentional variety’;see,for instance,Dosi et al.,1988).‘Intentional variety’presupposes that tech-nical and institutional development of univer-sities is a learning process.University agentsundertake purposeful action for change,theylearn from experience and anticipate futurestates of the selective environment.Further-more,university agents take initiatives to im-prove and develop learning paths.An exampleof these learning agents is provided in Box1.We consider technological and institutionaldevelopment of universities as a process thatinvolves many knowledge-seeking activitieswhere public and private agents’perceptionsand actions are translated into practice.3Theinstitutional changes are the result of inter-actions among economic agents deﬁned byLundvall(1992)as interactive learning.Theseinteractions result in an evolutionary pattern3Using a theory developed in one scientiﬁc disci-pline as a metaphor in a different discipline mayresult,in a worst-case scenario,in misleading analo-gies.In the best case,however,it can be a source ofcreativity.As Hodgson(2000)pointed out,the evo-lutionary metaphor is useful for understandingprocesses of technical and institutional change,thatcan help to identify new events,characteristics andphenomena.Volume17Number42008©2008The AuthorsJournal compilation©2008Blackwell Publishing。

斯普林格数学研究生教材丛书

《斯普林格数学研究生教材丛书》（Graduate Texts in Mathematics）GTM001《Introduction to Axiomatic Set Theory》Gaisi Takeuti, Wilson M.Zaring GTM002《Measure and Category》John C.Oxtoby（测度和范畴）（2ed.）GTM003《Topological Vector Spaces》H.H.Schaefer, M.P.Wolff（2ed.）GTM004《A Course in Homological Algebra》P.J.Hilton, U.Stammbach（2ed.）（同调代数教程）GTM005《Categories for the Working Mathematician》Saunders Mac Lane（2ed.）GTM006《Projective Planes》Daniel R.Hughes, Fred C.Piper（投射平面）GTM007《A Course in Arithmetic》Jean-Pierre Serre（数论教程）GTM008《Axiomatic set theory》Gaisi Takeuti, Wilson M.Zaring（2ed.）GTM009《Introduction to Lie Algebras and Representation Theory》James E.Humphreys（李代数和表示论导论）GTM010《A Course in Simple-Homotopy Theory》M.M CohenGTM011《Functions of One Complex VariableⅠ》John B.ConwayGTM012《Advanced Mathematical Analysis》Richard BealsGTM013《Rings and Categories of Modules》Frank W.Anderson, Kent R.Fuller（环和模的范畴）（2ed.）GTM014《Stable Mappings and Their Singularities》Martin Golubitsky, Victor Guillemin （稳定映射及其奇点）GTM015《Lectures in Functional Analysis and Operator Theory》Sterling K.Berberian GTM016《The Structure of Fields》David J.Winter（域结构）GTM017《Random Processes》Murray RosenblattGTM018《Measure Theory》Paul R.Halmos（测度论）GTM019《A Hilbert Space Problem Book》Paul R.Halmos（希尔伯特问题集）GTM020《Fibre Bundles》Dale Husemoller（纤维丛）GTM021《Linear Algebraic Groups》James E.Humphreys（线性代数群）GTM022《An Algebraic Introduction to Mathematical Logic》Donald W.Barnes, John M.MackGTM023《Linear Algebra》Werner H.Greub（线性代数）GTM024《Geometric Functional Analysis and Its Applications》Paul R.HolmesGTM025《Real and Abstract Analysis》Edwin Hewitt, Karl StrombergGTM026《Algebraic Theories》Ernest G.ManesGTM027《General Topology》John L.Kelley（一般拓扑学）GTM028《Commutative Algebra》VolumeⅠOscar Zariski, Pierre Samuel（交换代数）GTM029《Commutative Algebra》VolumeⅡOscar Zariski, Pierre Samuel（交换代数）GTM030《Lectures in Abstract AlgebraⅠ.Basic Concepts》Nathan Jacobson（抽象代数讲义Ⅰ基本概念分册）GTM031《Lectures in Abstract AlgebraⅡ.Linear Algabra》Nathan.Jacobson（抽象代数讲义Ⅱ线性代数分册）GTM032《Lectures in Abstract AlgebraⅢ.Theory of Fields and Galois Theory》Nathan.Jacobson（抽象代数讲义Ⅲ域和伽罗瓦理论）GTM033《Differential Topology》Morris W.Hirsch（微分拓扑）GTM034《Principles of Random Walk》Frank Spitzer（2ed.）（随机游动原理）GTM035《Several Complex Variables and Banach Algebras》Herbert Alexander, John Wermer（多复变和Banach代数）GTM036《Linear Topological Spaces》John L.Kelley, Isaac Namioka（线性拓扑空间）GTM037《Mathematical Logic》J.Donald Monk（数理逻辑）GTM038《Several Complex Variables》H.Grauert, K.FritzsheGTM039《An Invitation to C*-Algebras》William Arveson（C*-代数引论）GTM040《Denumerable Markov Chains》John G.Kemeny, urie Snell, Anthony W.KnappGTM041《Modular Functions and Dirichlet Series in Number Theory》Tom M.Apostol （数论中的模函数和Dirichlet序列）GTM042《Linear Representations of Finite Groups》Jean-Pierre Serre（有限群的线性表示）GTM043《Rings of Continuous Functions》Leonard Gillman, Meyer JerisonGTM044《Elementary Algebraic Geometry》Keith KendigGTM045《Probability TheoryⅠ》M.Loève（概率论Ⅰ）（4ed.）GTM046《Probability TheoryⅡ》M.Loève（概率论Ⅱ）（4ed.）GTM047《Geometric Topology in Dimensions 2 and 3》Edwin E.MoiseGTM048《General Relativity for Mathematicians》Rainer.K.Sachs, H.Wu伍鸿熙（为数学家写的广义相对论）GTM049《Linear Geometry》K.W.Gruenberg, A.J.Weir（2ed.）GTM050《Fermat's Last Theorem》Harold M.EdwardsGTM051《A Course in Differential Geometry》Wilhelm Klingenberg（微分几何教程）GTM052《Algebraic Geometry》Robin Hartshorne（代数几何）GTM053《A Course in Mathematical Logic for Mathematicians》Yu.I.Manin（2ed.）GTM054《Combinatorics with Emphasis on the Theory of Graphs》Jack E.Graver, Mark E.WatkinsGTM055《Introduction to Operator TheoryⅠ》Arlen Brown, Carl PearcyGTM056《Algebraic Topology：An Introduction》W.S.MasseyGTM057《Introduction to Knot Theory》Richard.H.Crowell, Ralph.H.FoxGTM058《p-adic Numbers, p-adic Analysis, and Zeta-Functions》Neal Koblitz（p-adic 数、p-adic分析和Z函数）GTM059《Cyclotomic Fields》Serge LangGTM060《Mathematical Methods of Classical Mechanics》V.I.Arnold（经典力学的数学方法）（2ed.）GTM061《Elements of Homotopy Theory》George W.Whitehead（同论论基础）GTM062《Fundamentals of the Theory of Groups》M.I.Kargapolov, Ju.I.Merzljakov GTM063《Modern Graph Theory》Béla BollobásGTM064《Fourier Series：A Modern Introduction》VolumeⅠ（2ed.）R.E.Edwards（傅里叶级数）GTM065《Differential Analysis on Complex Manifolds》Raymond O.Wells, Jr.（3ed.）GTM066《Introduction to Affine Group Schemes》William C.Waterhouse（仿射群概型引论）GTM067《Local Fields》Jean-Pierre Serre（局部域）GTM069《Cyclotomic FieldsⅠandⅡ》Serge LangGTM070《Singular Homology Theory》William S.MasseyGTM071《Riemann Surfaces》Herschel M.Farkas, Irwin Kra（黎曼曲面）GTM072《Classical Topology and Combinatorial Group Theory》John Stillwell（经典拓扑和组合群论）GTM073《Algebra》Thomas W.Hungerford（代数）GTM074《Multiplicative Number Theory》Harold Davenport（乘法数论）（3ed.）GTM075《Basic Theory of Algebraic Groups and Lie Algebras》G.P.HochschildGTM076《Algebraic Geometry：An Introduction to Birational Geometry of Algebraic Varieties》Shigeru IitakaGTM077《Lectures on the Theory of Algebraic Numbers》Erich HeckeGTM078《A Course in Universal Algebra》Stanley Burris, H.P.Sankappanavar（泛代数教程）GTM079《An Introduction to Ergodic Theory》Peter Walters（遍历性理论引论）GTM080《A Course in_the Theory of Groups》Derek J.S.RobinsonGTM081《Lectures on Riemann Surfaces》Otto ForsterGTM082《Differential Forms in Algebraic Topology》Raoul Bott, Loring W.Tu（代数拓扑中的微分形式）GTM083《Introduction to Cyclotomic Fields》Lawrence C.Washington（割圆域引论）GTM084《A Classical Introduction to Modern Number Theory》Kenneth Ireland, Michael Rosen（现代数论经典引论）GTM085《Fourier Series A Modern Introduction》Volume 1（2ed.）R.E.Edwards GTM086《Introduction to Coding Theory》J.H.van Lint（3ed .）GTM087《Cohomology of Groups》Kenneth S.Brown（上同调群）GTM088《Associative Algebras》Richard S.PierceGTM089《Introduction to Algebraic and Abelian Functions》Serge Lang（代数和交换函数引论）GTM090《An Introduction to Convex Polytopes》Ame BrondstedGTM091《The Geometry of Discrete Groups》Alan F.BeardonGTM092《Sequences and Series in BanachSpaces》Joseph DiestelGTM093《Modern Geometry-Methods and Applications》（PartⅠ.The of geometry Surfaces Transformation Groups and Fields）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov （现代几何学方法和应用）GTM094《Foundations of Differentiable Manifolds and Lie Groups》Frank W.Warner（可微流形和李群基础）GTM095《Probability》A.N.Shiryaev（2ed.）GTM096《A Course in Functional Analysis》John B.Conway（泛函分析教程）GTM097《Introduction to Elliptic Curves and Modular Forms》Neal Koblitz（椭圆曲线和模形式引论）GTM098《Representations of Compact Lie Groups》Theodor Breöcker, Tammo tom DieckGTM099《Finite Reflection Groups》L.C.Grove, C.T.Benson（2ed.）GTM100《Harmonic Analysis on Semigroups》Christensen Berg, Jens Peter Reus Christensen, Paul ResselGTM101《Galois Theory》Harold M.Edwards（伽罗瓦理论）GTM102《Lie Groups, Lie Algebras, and Their Representation》V.S.Varadarajan（李群、李代数及其表示）GTM103《Complex Analysis》Serge LangGTM104《Modern Geometry-Methods and Applications》（PartⅡ.Geometry and Topology of Manifolds）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM105《SL₂ (R)》Serge Lang（SL₂ (R)群）GTM106《The Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术理论）GTM107《Applications of Lie Groups to Differential Equations》Peter J.Olver（李群在微分方程中的应用）GTM108《Holomorphic Functions and Integral Representations in Several Complex Variables》R.Michael RangeGTM109《Univalent Functions and Teichmueller Spaces》Lehto OlliGTM110《Algebraic Number Theory》Serge Lang（代数数论）GTM111《Elliptic Curves》Dale Husemoeller（椭圆曲线）GTM112《Elliptic Functions》Serge Lang（椭圆函数）GTM113《Brownian Motion and Stochastic Calculus》Ioannis Karatzas, Steven E.Shreve （布朗运动和随机计算）GTM114《A Course in Number Theory and Cryptography》Neal Koblitz（数论和密码学教程）GTM115《Differential Geometry：Manifolds, Curves, and Surfaces》M.Berger, B.Gostiaux GTM116《Measure and Integral》Volume1 John L.Kelley, T.P.SrinivasanGTM117《Algebraic Groups and Class Fields》Jean-Pierre Serre（代数群和类域）GTM118《Analysis Now》Gert K.Pedersen（现代分析）GTM119《An introduction to Algebraic Topology》Jossph J.Rotman（代数拓扑导论）GTM120《Weakly Differentiable Functions》William P.Ziemer（弱可微函数）GTM121《Cyclotomic Fields》Serge LangGTM122《Theory of Complex Functions》Reinhold RemmertGTM123《Numbers》H.-D.Ebbinghaus, H.Hermes, F.Hirzebruch, M.Koecher, K.Mainzer, J.Neukirch, A.Prestel, R.Remmert（2ed.）GTM124《Modern Geometry-Methods and Applications》（PartⅢ.Introduction to Homology Theory）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM125《Complex Variables：An introduction》Garlos A.Berenstein, Roger Gay GTM126《Linear Algebraic Groups》Armand Borel（线性代数群）GTM127《A Basic Course in Algebraic Topology》William S.Massey（代数拓扑基础教程）GTM128《Partial Differential Equations》Jeffrey RauchGTM129《Representation Theory：A First Course》William Fulton, Joe HarrisGTM130《Tensor Geometry》C.T.J.Dodson, T.Poston（张量几何）GTM131《A First Course in Noncommutative Rings》m（非交换环初级教程）GTM132《Iteration of Rational Functions：Complex Analytic Dynamical Systems》AlanF.Beardon（有理函数的迭代：复解析动力系统）GTM133《Algebraic Geometry：A First Course》Joe Harris（代数几何）GTM134《Coding and Information Theory》Steven RomanGTM135《Advanced Linear Algebra》Steven RomanGTM136《Algebra：An Approach via Module Theory》William A.Adkins, Steven H.WeintraubGTM137《Harmonic Function Theory》Sheldon Axler, Paul Bourdon, Wade Ramey（调和函数理论）GTM138《A Course in Computational Algebraic Number Theory》Henri Cohen（计算代数数论教程）GTM139《Topology and Geometry》Glen E.BredonGTM140《Optima and Equilibria：An Introduction to Nonlinear Analysis》Jean-Pierre AubinGTM141《A Computational Approach to Commutative Algebra》Gröbner Bases, Thomas Becker, Volker Weispfenning, Heinz KredelGTM142《Real and Functional Analysis》Serge Lang（3ed.）GTM143《Measure Theory》J.L.DoobGTM144《Noncommutative Algebra》Benson Farb, R.Keith DennisGTM145《Homology Theory：An Introduction to Algebraic Topology》James W.Vick（同调论：代数拓扑简介）GTM146《Computability：A Mathematical Sketchbook》Douglas S.BridgesGTM147《Algebraic K-Theory and Its Applications》Jonathan Rosenberg（代数K理论及其应用）GTM148《An Introduction to the Theory of Groups》Joseph J.Rotman（群论入门）GTM149《Foundations of Hyperbolic Manifolds》John G.Ratcliffe（双曲流形基础）GTM150《Commutative Algebra with a view toward Algebraic Geometry》David EisenbudGTM151《Advanced Topics in the Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术高级选题）GTM152《Lectures on Polytopes》Günter M.ZieglerGTM153《Algebraic Topology：A First Course》William Fulton（代数拓扑）GTM154《An introduction to Analysis》Arlen Brown, Carl PearcyGTM155《Quantum Groups》Christian Kassel（量子群）GTM156《Classical Descriptive Set Theory》Alexander S.KechrisGTM157《Integration and Probability》Paul MalliavinGTM158《Field theory》Steven Roman（2ed.）GTM159《Functions of One Complex Variable VolⅡ》John B.ConwayGTM160《Differential and Riemannian Manifolds》Serge Lang（微分流形和黎曼流形）GTM161《Polynomials and Polynomial Inequalities》Peter Borwein, Tamás Erdélyi（多项式和多项式不等式）GTM162《Groups and Representations》J.L.Alperin, Rowen B.Bell（群及其表示）GTM163《Permutation Groups》John D.Dixon, Brian Mortime rGTM164《Additive Number Theory：The Classical Bases》Melvyn B.NathansonGTM165《Additive Number Theory：Inverse Problems and the Geometry of Sumsets》Melvyn B.NathansonGTM166《Differential Geometry：Cartan's Generalization of Klein's Erlangen Program》R.W.SharpeGTM167《Field and Galois Theory》Patrick MorandiGTM168《Combinatorial Convexity and Algebraic Geometry》Günter Ewald（组合凸面体和代数几何）GTM169《Matrix Analysis》Rajendra BhatiaGTM170《Sheaf Theory》Glen E.Bredon（2ed.）GTM171《Riemannian Geometry》Peter Petersen（黎曼几何）GTM172《Classical Topics in Complex Function Theory》Reinhold RemmertGTM173《Graph Theory》Reinhard Diestel（图论）（3ed.）GTM174《Foundations of Real and Abstract Analysis》Douglas S.Bridges（实分析和抽象分析基础）GTM175《An Introduction to Knot Theory》W.B.Raymond LickorishGTM176《Riemannian Manifolds：An Introduction to Curvature》John M.LeeGTM177《Analytic Number Theory》Donald J.Newman（解析数论）GTM178《Nonsmooth Analysis and Control Theory》F.H.clarke, Yu.S.Ledyaev, R.J.Stern, P.R.Wolenski（非光滑分析和控制论）GTM179《Banach Algebra Techniques in Operator Theory》Ronald G.Douglas（2ed.）GTM180《A Course on Borel Sets》S.M.Srivastava（Borel 集教程）GTM181《Numerical Analysis》Rainer KressGTM182《Ordinary Differential Equations》Wolfgang WalterGTM183《An introduction to Banach Spaces》Robert E.MegginsonGTM184《Modern Graph Theory》Béla Bollobás（现代图论）GTM185《Using Algebraic Geomety》David A.Cox, John Little, Donal O’Shea（应用代数几何）GTM186《Fourier Analysis on Number Fields》Dinakar Ramakrishnan, Robert J.Valenza GTM187《Moduli of Curves》Joe Harris, Ian Morrison（曲线模）GTM188《Lectures on the Hyperreals：An Introduction to Nonstandard Analysis》Robert GoldblattGTM189《Lectures on Modules and Rings》m（模和环讲义）GTM190《Problems in Algebraic Number Theory》M.Ram Murty, Jody Esmonde（代数数论中的问题）GTM191《Fundamentals of Differential Geometry》Serge Lang（微分几何基础）GTM192《Elements of Functional Analysis》Francis Hirsch, Gilles LacombeGTM193《Advanced Topics in Computational Number Theory》Henri CohenGTM194《One-Parameter Semigroups for Linear Evolution Equations》Klaus-Jochen Engel, Rainer Nagel（线性发展方程的单参数半群）GTM195《Elementary Methods in Number Theory》Melvyn B.Nathanson（数论中的基本方法）GTM196《Basic Homological Algebra》M.Scott OsborneGTM197《The Geometry of Schemes》David Eisenbud, Joe HarrisGTM198《A Course in p-adic Analysis》Alain M.RobertGTM199《Theory of Bergman Spaces》Hakan Hedenmalm, Boris Korenblum, Kehe Zhu（Bergman空间理论）GTM200《An Introduction to Riemann-Finsler Geometry》D.Bao, S.-S.Chern, Z.Shen GTM201《Diophantine Geometry An Introduction》Marc Hindry, Joseph H.Silverman GTM202《Introduction to Topological Manifolds》John M.LeeGTM203《The Symmetric Group》Bruce E.SaganGTM204《Galois Theory》Jean-Pierre EscofierGTM205《Rational Homotopy Theory》Yves Félix, Stephen Halperin, Jean-Claude Thomas（有理同伦论）GTM206《Problems in Analytic Number Theory》M.Ram MurtyGTM207《Algebraic Graph Theory》Chris Godsil, Gordon Royle（代数图论）GTM208《Analysis for Applied Mathematics》Ward CheneyGTM209《A Short Course on Spectral Theory》William Arveson（谱理论简明教程）GTM210《Number Theory in Function Fields》Michael RosenGTM211《Algebra》Serge Lang（代数）GTM212《Lectures on Discrete Geometry》Jiri Matousek（离散几何讲义）GTM213《From Holomorphic Functions to Complex Manifolds》Klaus Fritzsche, Hans Grauert（从正则函数到复流形）GTM214《Partial Differential Equations》Jüergen Jost（偏微分方程）GTM215《Algebraic Functions and Projective Curves》David M.Goldschmidt（代数函数和投影曲线）GTM216《Matrices：Theory and Applications》Denis Serre（矩阵：理论及应用）GTM217《Model Theory An Introduction》David Marker（模型论引论）GTM218《Introduction to Smooth Manifolds》John M.Lee（光滑流形引论）GTM219《The Arithmetic of Hyperbolic 3-Manifolds》Colin Maclachlan, Alan W.Reid GTM220《Smooth Manifolds and Observables》Jet Nestruev（光滑流形和直观）GTM221《Convex Polytopes》Branko GrüenbaumGTM222《Lie Groups, Lie Algebras, and Representations》Brian C.Hall（李群、李代数和表示）GTM223《Fourier Analysis and its Applications》Anders Vretblad（傅立叶分析及其应用）GTM224《Metric Structures in Differential Geometry》Gerard Walschap（微分几何中的度量结构）GTM225《Lie Groups》Daniel Bump（李群）GTM226《Spaces of Holomorphic Functions in the Unit Ball》Kehe Zhu（单位球内的全纯函数空间）GTM227《Combinatorial Commutative Algebra》Ezra Miller, Bernd Sturmfels（组合交换代数）GTM228《A First Course in Modular Forms》Fred Diamond, Jerry Shurman（模形式初级教程）GTM229《The Geometry of Syzygies》David Eisenbud（合冲几何）GTM230《An Introduction to Markov Processes》Daniel W.Stroock（马尔可夫过程引论）GTM231《Combinatorics of Coxeter Groups》Anders Bjröner, Francesco Brenti（Coxeter 群的组合学）GTM232《An Introduction to Number Theory》Graham Everest, Thomas Ward（数论入门）GTM233《Topics in Banach Space Theory》Fenando Albiac, Nigel J.Kalton（Banach空间理论选题）GTM234《Analysis and Probability：Wavelets, Signals, Fractals》Palle E.T.Jorgensen（分析与概率）GTM235《Compact Lie Groups》Mark R.Sepanski（紧致李群）GTM236《Bounded Analytic Functions》John B.Garnett（有界解析函数）GTM237《An Introduction to Operators on the Hardy-Hilbert Space》Rubén A.Martínez-Avendano, Peter Rosenthal（哈代-希尔伯特空间算子引论）GTM238《A Course in Enumeration》Martin Aigner（枚举教程）GTM239《Number Theory：VolumeⅠTools and Diophantine Equations》Henri Cohen GTM240《Number Theory：VolumeⅡAnalytic and Modern Tools》Henri Cohen GTM241《The Arithmetic of Dynamical Systems》Joseph H.SilvermanGTM242《Abstract Algebra》Pierre Antoine Grillet（抽象代数）GTM243《Topological Methods in Group Theory》Ross GeogheganGTM244《Graph Theory》J.A.Bondy, U.S.R.MurtyGTM245《Complex Analysis：In the Spirit of Lipman Bers》Jane P.Gilman, Irwin Kra, Rubi E.RodriguezGTM246《A Course in Commutative Banach Algebras》Eberhard KaniuthGTM247《Braid Groups》Christian Kassel, Vladimir TuraevGTM248《Buildings Theory and Applications》Peter Abramenko, Kenneth S.Brown GTM249《Classical Fourier Analysis》Loukas Grafakos（经典傅里叶分析）GTM250《Modern Fourier Analysis》Loukas Grafakos（现代傅里叶分析）GTM251《The Finite Simple Groups》Robert A.WilsonGTM252《Distributions and Operators》Gerd GrubbGTM253《Elementary Functional Analysis》Barbara D.MacCluerGTM254《Algebraic Function Fields and Codes》Henning StichtenothGTM255《Symmetry Representations and Invariants》Roe Goodman, Nolan R.Wallach GTM256《A Course in Commutative Algebra》Kemper GregorGTM257《Deformation Theory》Robin HartshorneGTM258《Foundation of Optimization》Osman GülerGTM259《Ergodic Theory：with a view towards Number Theory》Manfred Einsiedler, Thomas WardGTM260《Monomial Ideals》Jurgen Herzog, Takayuki HibiGTM261《Probability and Stochastics》Erhan CinlarGTM262《Essentials of Integration Theory for Analysis》Daniel W.StroockGTM263《Analysis on Fock Spaces》Kehe ZhuGTM264《Functional Analysis, Calculus of Variations and Optimal Control》Francis ClarkeGTM265《Unbounded Self-adjoint Operatorson Hilbert Space》Konrad Schmüdgen GTM266《Calculus Without Derivatives》Jean-Paul PenotGTM267《Quantum Theory for Mathematicians》Brian C.HallGTM268《Geometric Analysis of the Bergman Kernel and Metric》Steven G.Krantz GTM269《Locally Convex Spaces》M.Scott Osborne。

卡拉西奥多里实变函数论参考文献

卡拉西奥多里实变函数论参考文献在深入探讨卡拉西奥多里和实变函数论之前，我们应该先了解这两个主题的基本概念。

卡拉西奥多里（Carathéodory）是20世纪著名的希腊数学家，他在实变函数论领域做出了重要贡献。

实变函数论是数学分析中的一个重要领域，研究的是实数域上的函数的性质和性质。

在本文中，我们将以从简到繁、由浅入深的方式来探讨卡拉西奥多里和实变函数论这两个主题。

我们将介绍卡拉西奥多里的生平和学术成就，然后深入探讨实变函数论的基本概念和重要定理。

我们将分析卡拉西奥多里对实变函数论的影响，探讨他对该领域的重要贡献。

我们将总结和回顾本文的内容，共享个人对这两个主题的理解和观点。

一、卡拉西奥多里简介卡拉西奥多里（1873-1950）是一位具有希腊和德国血统的数学家，他在数学分析、复变函数论和实变函数论等领域做出了杰出的贡献。

他曾在德国、俄罗斯和希腊等国家任教，是一位颇具国际影响力的学者。

他对复变函数论和实变函数论的研究成果为这两个领域的发展做出了重要贡献。

二、实变函数论基本概念实变函数论是数学分析中研究实数域上的函数的性质和性质的一个重要分支。

它涉及到实数域上的函数序列、级数、连续性、导数、积分等内容。

实变函数论的基本概念包括实数、实函数、集合论、度量空间、拓扑空间、测度论等知识。

在实变函数论中，有一些重要的定理，如连续映射的性质、一致收敛的性质、傅里叶级数的收敛性等。

三、卡拉西奥多里对实变函数论的贡献卡拉西奥多里在实变函数论领域进行了深入的研究，他提出了许多重要的理论和定理。

其中，卡拉西奥多里收敛定理是他最为著名的成果之一。

这个定理在实变函数论和复变函数论中都有重要的应用。

卡拉西奥多里还对测度论、拓扑空间、可测函数等问题做出了深刻的研究，为实变函数论的发展做出了重要贡献。

四、总结和回顾通过对卡拉西奥多里和实变函数论的深入探讨，我们对这两个主题有了更加全面、深刻和灵活的理解。

卡拉西奥多里作为一位杰出的数学家，在实变函数论领域做出了重要贡献，他的成果对后人的研究产生了深远的影响。

Modeling, analysis and simulation

The research in MAS concentrates on the qualitative, numerical and computational aspects of mathema-tical models arising in a wide range of applications within the Dutch society.Particular attention is gi-ven to models describing continuum processes.As examples we mention freeﬂuid and porous mediaﬂow,chemical reactions in the atmosphere and cir-cuit analysis.The applications are combined into two extensive programmes or themes,each contai-ning a number of characteristic projects.They are accounted for in detail in the theme descriptions of MAS1and MAS2.New developments include the projects MAS2.1(Computationalﬂuid dynamics,a collaboration with MARIN)and MAS2.7(Mathe-matics of Finance).Preliminary contacts have been made with TNO-TPD and across the border with GMD in Germany.It is expected that new projects will emerge from these contacts.In the midst of much applied work,MAS conti-nues to contribute signiﬁcantly to the scientiﬁc de-velopment in the areas of qualitative,numerical and computational methods for partial differential equati-ons.The number of papers appeared in international journals is certainly satisfactory,and withﬁve PhD theses MAS contributes substantially to the scientiﬁc output of the CWI.Separately from the themes,the Dynamical Sys-tems Laboratory(DSL)operated as an independent unit.During the past years DSL has played a leading role in the software development for bifurcation of equilibria of systems of ordinary differential equa-tions.An extensive description of their activities is given in the DSL contribution.This year was the last year of DSL,which ofﬁcially ended when Dr.Yu.A. Kuznetsov left the CWI(November1st,1997). MAS is very pleased to have Professor Piet van der Houwen as a CWI Fellow in its ranks.His high level scientiﬁc input,interaction with young re-searchers and involvement with several project is very much appreciated.Dynamical Systems Laboratory–DSL-Yu.A.Kuznetsov-V.V.Levitin -J.A.SandersEnvironmental Modelling and Porous Media Re-search–MAS1-J.G.Verwer-C.J.van Duijn-J.Hulshof-P.J.van der Houwen-J.G.Blom-C.Cuesta-M.I.J.van Dijke-G.Galiano-W.Hundsdorfer-J.Koknserstdrager-M.A.A.van Leeuwen-W.M.Lioen-M.van Loon-J.Molenaar-R.J.Schotting-B.P.Sommeijer-E.J.Spee-J.de Vries-P.M.de ZeeuwIndustrial Processes–MAS2-E.H.van Brummelen-S.Cavallar-M.K.C¸amlıbel-M.Genseberger-T.Hantke-P.W.Hemker-J.Hoogland-P.J.van der Houwen-K.Karamazen-M.Kirkilionis-A.de Koeijer-B.Korennserstdrager-W.M.Lioen-P.L.Montgomery-M.Nool-J.Noordmans-O.Penninga-A.van der Ploeg-H.J.J.te Riele-A.J.van der Schaft-J.M.Schumacher-B.P.Sommeijer-W.J.H.Stortelder-J.B.de Swart-N.M.Temme-W.A.van der Veen-D.T.Winter-P.WesselingSecretary:N.MitrovicJ.A.SandersYu.A.KuznetsovV.V.LevitinDuring1997,two versions of CONTENT,1.3and1.4,were released on ftp.cwi.nl in directorypub/CONTENT.Release1.4–December,1997New features:-new class of dynamical systems is supported: differential-algebraic equations(DAEs)Mx’=f(x,p) with possibly singular matrix M;-numerical integration of stiff ODEs and DAEs using RADAU5code;-three-parameter continuation of all codim2bifur-cations of ODEs(cusp,Bogdanov-Takens,genera-lized Hopf,zero-Hopf,and double Hopf);-two-parameter continuation of Hopf bifurcation using the bordered squared Jacobian matrix;-symbolical calculation of derivatives of the4th or-der using the Maple system.Release1.3–August,1997New features:-one-parameter continuation of limit cycles in ODEs;-detection and branch switching at codim1bifurca-tions of limit cycles;-one-parameter continuation ofﬁxed points of ite-rated maps;-detection and normal form analysis of codim1bi-furcations of iterated maps;-3D graphic windows;-Staircase window for scalar iterated maps.At the moment,CONTENT completely supportsone-parameter analysis of equilibria and cycles in ODEs and iterated maps.To complete the support of two-parameter analysis of ODEs,one has to imple-ment the continuation of codim1bifurcations of cy-cles(i.e.,fold,period-doubling,and Neimark-Sacker bifurcations),as well as the homoclinic bifurcati-ons of saddle and saddle-node equilibria,and branch switching between them.CONTENT provides the environment to implement all of these continuations. Also,the computation of the remaining normal form coefﬁcients at codim2bifurcations of equilibria has to be implemented in CONTENT(see below).Yu.A.Kuznetsov derived explicit normal form coefﬁcients for the reduced to the central manifold equations for all codim2equilibrium bifurcation of ODEs.A CWI Report is published.Yu.A.Kuznetsov(together with aerts and B. Sijnave,Gent University)developed new algorithms to continue codim1and2bifurcations of equilibria in2and3parameters,respectively.The algorithms were implemented in CONTENT.Yu.A.Kuznetsov(together with A.Champneys (Bristol)and B.Sandstede(Berlin))implemented the homoclinic continuation into AUTO97,the latest version of the continuation/bifurcation software by E.Doedel(Concordia University,Montreal).Now it is the standard part of AUTO.The text of the second edition of the book by Yu.A. Kuznetsov Elements of Applied Bifurcation The-ory has been sent to the Production Departmentof Springer-Verlag in September1997and will be published in1998.E.Doedel(Concordia University,Montreal,Ca-nada)June15–28.The visit was devoted to discussions of new me-thods to continue codim1bifurcations of limit cy-cles.In particular,a new method to continue the period-doubling bifurcation was proposed that combines orthogonal collocation technique with matrix bordering.O.De Feo(Swiss Federal Institute of Technology, Lausanne,Switzerland)September1–30.During the visit,RADAU5method for the numeri-cal integration of ODEs and DAEs was translated from FORTRAN to C and implemented into CON-TENT.A paper on homoclinic bifurcations in a3D food chain model was practicallyﬁnished.aerts and B.Sijnave(University of Gent, Belgium)June19–22.Bordering methods to continue codim1and2bi-furcation of equilibria were discussed.A proto-type method to continue the Bogdanov-Takens bi-furcation was implemented during the visit.The continuation of all other codim2bifurcations of equilibria in three parameters was implemented la-ter in1997.A.Shilnikov(Institute of Applied Mathematics and Cybernetics,Nizhnii Novgorod,Russia)May30.A lecture was given at CWI on‘A blue-sky cata-strophe model’,demonstrating a new way of a de-struction of a limit cycle.Yu.A.Kuznetsov gave invited lectures on CON-TENT at the International Workshop‘Numerical analysis of Dynamical Systems’,IMA,Minneapo-lis,USA,September15–19,and at the Workshop ‘Hybrid-methods for Bifurcation and Dynamics of Partial Differential Equations’,University of Mar-burg,June8–11.MAS-R9730.Y U.A.K UZNETSOV.Explicit nor-mal form coefﬁcients for all codim2bifurcations of equilibria in ODEs.E.J.D OEDEL,A.R.C HAMPNEYS,T.F.F AIR-GRIEVE,Y U.A.K UZNETSOV,B.S ANDSTEDE,X.-J.W ANG(1997).AUTO97:Continuation and Bifurcation Software for Ordinary Differential Equa-tions(with HomCont).User’s Guide,Concordia University,Montreal,Canada.W.G OVAERTS,Y U.A.K UZNETSOV,B.S IJNAVE (1997).Implementation of Hopf and double Hopf Continuation Using Bordering Methods,Department of Applied Mathematics and Computer Science,Uni-versity of Ghent,Belgium.W.G OVAERTS,Y U.A.K UZNETSOV,B.S IJNAVE (1997).Computation and Continuation of Codimen-sion2Bifurcations in CONTENT,Department of Ap-plied Mathematics and Computer Science,University of Ghent,Belgium.Y U.A.K UZNETSOV(1997).Centre manifold; Codimension-two bifurcations;Equivalence of dyna-mical systems;Homoclinic bifurcations;Hopf bifur-cation;Saddle-node bifurcation.M.H AZEWINKEL (ed.).Encyclopaedia of Mathematics.Supplement Volume I,Kluwer Academic Publishers,The Nether-lands,179–181,190–101,240,293–294,296–297, 444–445.Dr.J.G.Verwer,researcher,theme leader Prof.dr.ir.C.J.van Duijn,researcher,cluster leaderDr.J.Hulshof,advisorProf.dr.P.J.van der Houwen,researcher,CWI fellowDrs.J.G.Blom,researcherMrs.C.Cuesta,Ph.D.studentDrs.M.I.J.van Dijke,Ph.D.studentDr.G.Galiano,postdocDr.W.Hundsdorfer,researcherDrs.J.Kok,researchernser,Ph.D.studentstdrager,Ph.D.studentDr.M.A.A.van Leeuwen,postdocDrs.W.M.Lioen,programmerDr.M.van Loon,postdocDr.J.Molenaar,postdocIr.R.J.Schotting,researcherDr.B.P.Sommeijer,researcherDrs.E.J.Spee,Ph.D.studentDr.J.de Vries,researcherDrs.P.M.de Zeeuw,programmer,till February1The general purpose of this research theme is to develop,analyze and implement mathematical and numerical models for application to complex pro-blems arising in environmental modelling and porous media research.MAS1is particular concerned with ordinary and partial differential equations,descri-bingﬂuidﬂow,transport of pollutants and chemical and bio-chemical processes.These differential equa-tions lie at the heart of simulation models used in atmospheric air quality modelling,in surface wa-ter and groundwater water quality modelling,and in porous media research directed for example at en-hanced oil recovery.The research subthemes cover a wide range of scientiﬁc activities,ranging from fun-damental mathematical and numerical analysis of differential equations and development of new com-putational techniques for use on vector/parallel and massively parallel computers and heterogeneous net-works(HPCN),to implementation of fully integrated models and application to real life problems.Exten-sive co-operations and contacts are maintained with researchers from the academic world and from the environmental and porous media applicationﬁelds. Externalﬁnancing comes from a variety of sources, such as industry,special programs from the Nether-lands Organization for Scientiﬁc Research,research programs from the European Union and the national HPCN program funded through the Ministry of Eco-nomic Affairs.In1997research was organized in four subthemes:The research concerns the numerical modelling of the long range transport and chemical exchange of atmospheric air pollutants.Within the Netherlandsco-operation has existed with KEMA,NLR,RIVM, TUD,TNO and UU/IMAU.At the international le-vel,two joint papers with CGRER(Center for Global and Regional Environmental Research,Universityof Iowa),have been published in Atmospheric Envi-ronment(See Sandu et al.).The CWI group is also active within the European network GLOREAM and among others involved in the organization of an In-ternational Conference on Air Pollution in Paris in 1998.January23,1998,Edwin Spee will defend his Ph.D.Thesis Numerical Methods in Global Trans-port Models at the University of Amsterdam.Two new Ph.D.students have recently joined the group, viz.Debby Lanser and Boris Lastdrager.In1997 MAS1worked on the following projects:RIFTOZ–The technique of data-assimilation has been examined for improving results of model simulations by usage of actual measurements.A special implementation of an extended Kalmanﬁlter has been shown promising,see Report MAS-R9702 for details.The project forms part of an EU project in which CWI has been active through a subcon-tract with TUD.At CWI the project has now been terminated with the departure of Dr.M.van Loonto TNO.The Kalmanﬁlter will be further tested by TNO for use in their dispersion model LOTOS. LOTOS–Here the objective is to develop a regio-nal,three-dimensional,long term ozone simulation model.This LOTOS model should replace at due time an existing regional forecasting model in use at TNO.The model is developed in co-operation with TNO researchers.At TNO the focus lies on physi-cal,meteorological and chemical aspects.The CWI research focuses on the design of the mathematical model for a so-called hybrid(terrain following and pressure based)coordinate system and,in particular, of tailored numerical algorithms and implementa-tions on super and parallel computers.The project is part of the TASC project‘HPCN for Environ-mental Applications’which is funded by the Dutch HPCN program.At the end of1997the project was halfway.Aﬁrst running operational prototype im-plemented at CWI has recently been transferred to TNO.Research details are found in the reports MAS-N9701,R9717.NCF–This one-year project is linked with the LOTOS project and concerns aspects of massive parallelism,in particular for T3E implementations. Special attention has been given to the question to which extent massive(meteo)I/O can degrade the parallel performance of models used in atmospheric simulations.Results will be reported early1998. Support is provided by the NCF/Cray University Grant program.The project lasts until April next year.Early1997the Report MAS-R9702wasﬁnish-ed.This publication concerns research in a similar NCF project terminated in February,1997.CIRK–This Ph.D.project has been terminatedat the end of1997with the departure of Drs.Edwin Spee.He will defend his Thesis at the Universityof Amsterdam on January23,1998.The project is similar to the LOTOS project,but here the particu-lar objective was to develop numerical algorithms for use in3D models for the whole of the global troposphere/stratosphere.In this last year we have worked on various aspects of a Rosenbrock method (see Report MAS-R9717),including stiff chemis-try integration and a factorization approach within the Rosenbrock framework.The factorization idea was investigated to provide an alternative for time or operator splitting.A second main activity has been the validation of various advection schemes in a real life radon experiment,using analyzed windﬁelds from the ECMWF(see Report MAS-R9710).Sup-port for this project was obtained from the RIVM and very fruitful scientiﬁc co-operation has existed with IMAU/UU.This co-operation will continue in a following project,planned for the next three years. The new project is centered around the existing mo-del TM3.With support from SWON two postdocs will be hired for algorithmic and parallel software research.GOA–This activity concerns a new Ph.D.project on the‘Analysis and Validation of Operator Split-ting in Air Quality Modeling’.This project has been granted by GOA,the Netherlands Geoscien-ces Foundation.It started September1,1997with the employment of Ir.Debby Lanser.Aﬁrst article on the analysis of Strang-splitting for PDEs of the advection-diffusion-reaction type is already in prepa-ration.SWON–A second new project Ph.D.project started December1,1997with the employment of Drs.Boris Lastdrager.This project has been granted by SWON and concerns‘Sparse Grid Methods for Time-Dependent PDEs’.Atmospheric transport-chemistry problems provide a highly useful applica-tion for sparse-grid research.The project is a joint activity between MAS1and MAS2(Dr.ir.B.Ko-ren).The research concentrates on the design of parallel numerical methods for the simulation of water pol-lution(calamitous releases),the marine eco-system,dispersion of river water,sediment transport,etc. Our activities in1997included:HPCN–In1996we started the development of a special purpose3D transport model based onﬁnite difference space-discretization and unconditionally stable,implicit time-discretization.In1997we ana-lyzed an iterative approach for solving the implicit relations.This iteration process is based on approxi-mate factorization such that only one-dimensionally implicit,linear systems occur in the algorithm.Inco-operation with C.Eichler-Liebenow from the University of Halle,the convergence region of the iteration method and its effect on the overall stabi-lity of the integration method has been analyzed, see Report MAS-R9718.Furthermore,we started the development of tools for domain decomposition with domains of varying grid resolutions.Part of the research was carried out within the research con-sortium TASC,with support from the Dutch HPCN programme.SWEM–The velocityﬁeld needed by transport models either is read from inputﬁles or is computed simultaneously with the computation of the pollu-tant concentrations by means of a hydrodynamical model.In view of the complicated data structures in-volved,we decided to focus on the second approach, because the hydrodynamical model can be designed such that it uses the same data structures as the trans-port model.Such an approach is justiﬁed,because the underlying partial differential equations are to a large extent identical.By choosing the same type of spatial and temporal discretizations,the same decom-position in domains with the same resolutions,and the same stepsizes in both algorithms,we achieve that the data structures are exactly the same.Since the transport solver is designed and tuned with paral-lel computer systems in mind,the velocityﬁeld sol-ver will also be tuned to parallel computer systems. Moreover,each velocity-ﬁeld-solver step can be per-formed in parallel with the corresponding transport-solver step.In1997aﬁrst analysis of the underlying numerical model has been performed.This subtheme coordinates a number of research ac-tivities in analysis of nonlinear partial differential equations and in mathematical modelling ofﬂow and transport through porous media.The character of the research ranges from very applied to theoretical.An example of an applied activity is the NAM-project, where software was developed to study the mixingof gases in underground reservoirs.An example of a theoretical activity is the collaboration withH.W.Alt(Universit¨a t Bonn),which involves a de-tailed study of a free boundary problem with a cusp. This project participates in the interaction platform ‘Nonlinear Transport Phenomena in Porous Media’, which brings together researchers from TUD,RUL, LUW,RIVM and CWI,and which is supported by the NWO Priority Programme‘Nonlinear Systems’. There are also numerous international contacts.The scientiﬁc output in1997includes two Ph.D.theses: Problems in Degenerate Diffusion by Mark Peletier and Multi-Phase Flow Modeling of Soil Contamina-tion and Soil Remediation by Rink van Dijke.PDE RESEARCH–Nonlinear PDEs arising in models for porous mediaﬂow form the backbone of this project.Particular attention was given to sys-tems consisting of a convection-diffusion equation coupled with an ordinary differential equation.The general case,in which the ODE is in the time vari-able,is treated in the thesis of M.A.Peletier.A par-ticular case,where the ODE is in a space coordinate, appears in a model for salt uptake by mangroves,see Report MAS-R9728.This leads to non-local con-vection,which is shown to imply non-uniqueness.A second activity involves the collaboration Alt-Van Duijn.In a series of papers they study the behaviour of the interface between fresh and salt groundwater in the presence of wells.The interface appears asa free boundary in an elliptic problem.Depending on the pumping rate of the wells,a singularity de-velops in the free boundary in the form of a cusp.A detailed local analysis of the free boundary near such a cusp is presented in Report MAS-R9703.FTPM–This project deals with density drivenﬂow in porous media.In1997research concentrated on brine transport problems that are related to high-level radioactive waste disposal in salt domes.High salt concentrations give rise to nonlinear transport phenomena such as enhancedﬂow due to volume (compressibility)effects and the reduction of hydro-dynamical dispersion due to gravity forces.Mainly (semi)analytical techniques(similarity and V on Mi-ses transformations)were used to study the volume effects,see Report MAS-R9724.Report AM-R9616 (Brine transport in porous media:Self-similar solu-tions)has been accepted for publication in Advances in Water ing experimental data of Dr.H.Moser(Technische Universit¨a t Berlin)we also ve-riﬁed a nonlinear dispersion theory proposed by Dr. S.M.Hassanizadeh(Delft University of Technology), which includes the effect of dispersion reduction due to local high salt concentrations.The nonlinear the-ory is in excellent agreement with the experimentalresults,see Report MAS-R9734.We further consi-dered the interface between fresh and salt groundwa-ter in heterogeneous media.This subject is relatedto salt water intrusion problems in coastal aquifers. The interface approximation can be justiﬁed when the width of the mixing zone between theﬂuids is small compared to the vertical extension of the aqui-fer.We studied the resulting set of interface equa-tions numerically,using a moving mesh Finite Ele-ment Method.Moreover,several simpliﬁed Dupuit problems were studied and the results were compa-red with FEM solutions,see Report MAS-R9735. NAM–This project deals with the mathematical modelling of gas injection.The dispersion is studied for gas injection into a reservoir.The aim is to un-derstand and quantify the relevant physical processes that lead to mixing of injected gas with residual gas in old reservoirs.The project is sponsored by the NAM(Nederlandse Aardolie Maatschappij).In co-operation with the Faculty of Mining and Petroleum Engineering of the Delft University of Technology a numerical model is being developed at CWI to study the mixing of the gases in detail.NOBIS–Within this project we study soil reme-diation anic contaminants may be removed from the soil either by pumping methods or by injecting air,which enhances biodegradation and volatilization.The correspondingﬂow of groundwa-ter,organic contaminant and air is described using multi-phaseﬂow models.Air injection into ground-water(air sparging)in a horizontally layered medium has been studied in Report MAS-R9729.Accurate numerical simulations of the full transient two-phase ﬂow equations were carried out and an almost ex-plicit solution for the steady state airﬂow just below a less permeable soil layer was derived.The latter solution showed almost perfect agreement with the numerical results when heterogeneity of the layers was increased.To model pumping of a lens of light organic liquid from an aquifer,multi-phase seepage face conditions were applied at the well boundary (Report MAS-R9725).For two different geome-tries of the lens similarity solutions provided good approximations of the removal rate and the location of the remaining contaminant as a function of time. The above results and other work on behaviour ofa lens of organic contaminant and on air sparging have been gathered in Rink van Dijke’s Ph.D.thesis:‘Multi-phaseﬂow modeling of soil contamination and soil remediation’,which was defended at Wage-ningen Agricultural University on December5,1997. NWO-NLS–This is the Ph.D.project‘Mathemati-cal Analysis of Dynamic Capillary Pressure Relati-ons in Porous Media Flow’.It started in November 1997,with the employment of C.M.Cuesta.It is supported by the NWO Priority Programme‘Nonli-near Systems’.The aim is to study PDEs with higher order mixed derivatives.Such equations arise in mo-dels for unsaturated groundwaterﬂow,taking into account dynamic capillary pressure.In1997two different subjects have been studied. Report MAS-R9721contains the results of an inves-tigation to the stability of approximate factorization for-methods.Approximate factorization seems for certain multi-space dimensional PDEs a viable alter-native to time-splitting as a splitting error is avoided. The investigation,however,has revealed limitations of the approximate factorization technique with re-gard to numerical stability.The second subject con-cerns RKC(Runge-Kutta-Chebyshev),an explicit time integrator speciﬁcally suitable for multi-space dimensional parabolic PDEs.In RKC the stability limitation inherent in explicit methods is greatly re-duced by the use of a three-step Chebyshev recur-sion.The current study has speciﬁcally dealt with the development of a production-grade code for non-experencied users.The work has been carried outin co-operation with Prof.L.Shampine,University of Dallas.Details are given in Report MAS-R9715. This report has been accepted for publication in the Journal of Computational and Applied Mathematics. Mini-symposium on Numerical Analysis,Wage-ningen,April3–anizer:P.M.de Zeeuw. Speakers:W.Hundsdorfer(Stability of the Doug-las Splitting Method),E.J.Spee(Advectieschema’s op een Bol voor Atmosferische Ttransport Model-len).Meeting of the Steering Committee of the ESF-Programme‘Free Boundary Problems,Theory and Applications’,CWI,anizer:C.J.van Duijn.TASC Symposium7,CWI,anizers: J.G.Verwer and J.Kok.Speakers:P.J.H.Builtjes (MEP-TNO)(Atmospheric Transport-chemistry Modelling and HPCN),J.G.Blom(LOTOS,a3D Atmospheric Air Pollution Model),J.Kok(Por-ting Atmospheric Transport-Chemistry Software to the NEC SX–4),K.Dekker(TUD)(Modiﬁcation of Flow Fields to Recover the Property of Divergence Freedom),G.S.Stelling(WL)(NonhydrostaticPressure in Free Surface Flows)and B.P.Som-meijer(Recent Progress in an Implicit Shallow Water Transport Solver).Colloquium‘Flow and Transport in Porous Me-dia’,CWI,September10.Speakers:G.Dagan (Tel-Aviv)and S.E.A.T.M.van der Zee(LUW). Organizers:R.J.Schotting and C.J.van Duijn. TASC Symposium8(‘HPCN-Platformdag’),CWI, anizers:J.G.Verwer andJ.Kok.Speakers:J.G.Verwer(Het TASC Pro-ject HPCN voor Milieutoepassingen),M.van Loon(MEP-TNO)(Langetermijnsimulatie van Ozon),J.G.Blom(Rekenen aan Ozon),B.P.Som-meijer(Simulatie van Transport in Ondiep Water), E.A.H.V ollebregt(TUD)(Parallelle Software voor Stromings-en Transportmodellen)and G.S.Stel-ling(WL)(Simulatie van Afvalwaterlozingen). Mini-symposium on Partial Differential Equati-ons at SciCADE97–International Conference on Scientiﬁc Computation and Differential Equations, Grado,September15–anizer:J.G.Ver-wer.Speakers:K.Dekker(TUD)(Parallel GM-RES and Domain Decomposition),W.Hundsdor-fer(Trapezoidal and Midpoint Splittings for Initial Boundary-value Problems),B.P.Sommeijer(RKC, an Explicit Solver for Parabolic PDEs)and J.M. Hyman(Los Alamos)(Minimizing Numerical Er-rors Introduced by Operator Splitting Methods) Colloquium‘Flow and Transport in Porous Me-dia’,CWI,September24.Speakers:A.de Wit (Brussels)and R.J.Schotting(CWI).Organizers: R.J.Schotting and C.J.van Duijn.Workshop‘Interfaces and Parabolic Regularisa-tion’,Lorentz Center(RUL),November5–7.In-ternational workshop with25speakers anizers:J.Hulshof and C.J.van Duijn.MAS Colloquium,CWI,anizer: C.J.van Duijn.Speakers:C.N.Dawson(UT at Austin)(Dynamic Adaptive Methods for Chemi-cally Reactive Transport in Porous Media),P.Wes-seling(TUD)(Numerical Solution of Hyperbolic Systems with Nonconvex Equation of State)and W.A.Mulder(Shell Rijswijk)(Finite Differences and Finite Elements for Seismic Simulation). TASC Symposium9,CWI,ani-zers:J.G.Verwer and J.Kok.Speakers:A.Peter-sen(IMAU)(More Efﬁcient Advection Schemes for the Global Atmospheric Tracer Model),H.Elbern (EURAD)(A Parallel Implementation of a4D-variational Chemistry Data Similation Scheme), E.J.Spee(Rosenbrock Methods for Atmospheric Dispersion Problems)and M.Krol(IMAU)(The TM3Model:Numerical Aspects of Atmospheric Chemistry Aplications).2nd Annual Meeting MMARIE Concerted Action, Barcelona,January15–17:B.P.Sommeijer(Do-main Decomposition for an Implicit Shallow-water Transport Solver).Meeting of the DFG Panel for the Sonderforsbe-reich1578,M¨u nchen,January16–17:Participa-tion by C.J.van Duijn.Meeting of the Scientiﬁc Council of the Weier-strass Institut f¨u r Angewandte Analysis und Sto-chastik,Berlin,January24:C.J.Van Duijn partici-pates and is elected vice-chairman of this council. Guest Lectures at the University of Amsterdam, within the framework of the course‘Parallel Scientiﬁc Computing and Simulation’,February21 and26:B.P.Sommeijer(Parallel ODE solvers). Harburger Sommerschulen,TU Hamburg-Harburg, February24–28:J.G.Verwer invited speaker (three lectures on the Method of Lines). Universidad Complutense de Madrid,Madrid, March19–23:C.J.van Duijn visits J.I.Diaz.32e Nederlands Mathematisch Congres,Wage-ningen,April3–4:W.Hundsdorfer(Stability of the Douglas Splitting Method),E.J.Spee(Ad-vectieschema’s op een Bol voor Atmosferische Transport-modellen).Istituto per le Applicazioni del Calcolo‘Mauro Pi-cone’,Rome,April7–11:C.J.van Duijn visits M. Bertsch.1st ERCIM Environmental Modelling Group Workshop on Air Pollution Modelling,GMD FIRST,Berlin,April7–8:J.G.Blom(An Evalua-tion of the Cray T3D Programming Paradigms in Atmospheric Chemistry/transport Problems),J.G. Verwer(A Numerical Study for Atmospheric Che-mistry/transport Problems).Both invited. Measurements and Modelling in Environmen-tal Pollution,Madrid,April22–24:M.van Loon (Data Assimilation for Atmospheric Chemistry Models).22nd General Assembly of the European Geo-physical Society,Vienna,April21–25:B.P.Som-meijer(A Fully Implicit3D Transport-chemistry Solver Combined with Domain Decomposition). NWO Symposium Massaal Parallel Rekenen, Veldhoven,May22:J.G.Verwer invited speaker (High Performance Computing and Environmental Pollutions).。

凯莱矩阵论的研究报告

凯莱矩阵论的研究报告
矩阵论是数学中的一个重要分支，它主要研究矩阵之间的运算规律和性质。

凯莱矩阵论是矩阵论的一个重要分支，它以法国数学家亚瑟·凯莱的名字命名。

凯莱矩阵论的研究对象是矩阵的特征值和特征向量。

凯莱定理是凯莱矩阵论的核心结果之一，它表明一个矩阵的特征值等于其特征多项式的根。

凯莱矩阵论还研究了矩阵的相似变换、对角化和正交化等性质。

研究凯莱矩阵论主要有以下几个方面：
1. 特征值与特征向量的计算：通过凯莱定理，可以通过求解特征多项式的根来计算矩阵的特征值。

特征向量可以通过解特征方程来得到。

研究如何高效地计算特征值和特征向量是凯莱矩阵论的一个重要课题。

2. 矩阵对角化：对于一个可对角化的矩阵，可以通过相似变换将其转化为对角矩阵，从而简化矩阵的运算和分析。

凯莱矩阵论研究如何确定一个矩阵是否可对角化，以及如何求解对角化的变换矩阵。

3. 矩阵正交化：正交矩阵在很多应用领域中具有重要的作用，如信号处理、图像处理等。

凯莱矩阵论研究如何将一个一般的矩阵正交化，从而得到一个正交矩阵。

4. 应用领域：凯莱矩阵论在很多领域中有广泛的应用，如量子力学、振动力学、系统控制等。

研究凯莱矩阵论在这些领域中的应用是该研究的重要方向之一。

总之，凯莱矩阵论是矩阵论的一个重要分支，它研究矩阵的特
征值和特征向量，以及相关的运算规律和性质。

通过研究凯莱矩阵论，可以深入理解和应用矩阵理论的基本概念和方法。

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a rXiv:h ep-th/9812135v414M a y1999c 1998International Press Adv.Theor.Math.Phys.2(1998)1373–1404Type IIA D-Branes,K-Theory and Matrix Theory Petr Hoˇr ava a a California Institute of Technology,Pasadena,CA 91125,USA horava@ Abstract We show that all supersymmetric Type IIA D-branes can be con-structed as bound states of a certain number of unstable non-supersym-metric Type IIA D9-branes.This string-theoretical construction demon-strates that D-brane charges in Type IIA theory on spacetime manifold X are classiﬁed by the higher K-theory group K −1(X ),as suggested recently by Witten.In particular,the system of N D0-branes can be obtained,for any N ,in terms of sixteen Type IIA D9-branes.This sug-gests that the dynamics of Matrix theory is contained in the physics of magnetic vortices on the worldvolume of sixteen unstable D9-branes,described at low energies by a U (16)gauge theory.1374TYPE IIA D-BRANES,K-THEORY,AND MATRIX THEORY 1IntroductionWhen we consider individual D-branes in Type IIA or Type IIB string theory on R10,we usually require that the branes preserve half of the original su-persymmetry,and that they carry one unit of the corresponding RR charge. These requirements limit the D-brane spectrum to p-branes with all even values of p in Type IIA theory,and odd values of p in Type IIB theory.Once we relax these requirements,however,we can consider D p-branes with all values of p.In Type IIA theory,we can consider p-branes with p odd,and in particular,a spacetime-ﬁlling9-brane.All these states are non-supersymmetric unstable excitations in the corresponding supersymmetric string theory.Indeed,there is always a tachyon in the spectrum of the open string connecting one such Type IIA(2p−1)-brane to itself.Thus,such D-brane conﬁgurations(and their counterparts on spacetimes of non-trivial topology)are highly unstable,and one expects that they should rapidly decay to the supersymmetric vacuum,by a process that involves tachyon condensation on the worldvolume.This is of course in perfect agreement with theﬁeld content of the corresponding low-energy supergravity in spacetime –there are no RRﬁelds that could couple to any conserved charges carried by such non-supersymmetric branes.This does not seem to leave much room for surprises,but in fact,the full story is much more interesting.Conﬁgurations of unstable D-branes can sometimes carry lower-dimensional D-brane charges,and therefore,when the tachyon rolls down to the minimum of its potential and the state decays,it can leave behind a supersymmetric state that diﬀers from the vacuum by a lower-dimensional D-brane charge–in other words,the state decays into a supersymmetric D-brane conﬁguration.Typically,one can then represent the supersymmetric D-brane state as a bound state of the original system of unstable D-branes.This setup generalizes a special case studied in[1,2],where one starts with an unstable conﬁguration of an equal number of stable p-branes and stable anti-p-branes(orp-brane system is unstable,and the instability manifests itself by the presence of a tachyon in the spectrum of the p-9-brane pairs[1,2].In some cases,the unstable non-supersymmetric state decays into a stablePETR HOˇRAVA1375 state that is not supersymmetric,but is protected from further decay by charge conservation.One typical example of such states is the SO(32)spinor of Type I theory[1],which is non-supersymmetric but stable,since it is the lowest spinor state in the theory.Such non-supersymmetric D-branes can be found using a direct boundary state construction[8],or alternatively as bound states of p-brane9-brane pairs wrapping the whole spacetime.It is certainly desirable to have,on the Type IIA side,a similar construction that would enable us to study stable D-branes as bound states in unstable conﬁgurations of branes of maximal di-mension.However,there are no stable D9-branes in Type IIA string theory! While we can indeed represent any stable D-brane of Type IIA as a bound state of an8-brane8-brane system.Therefore,it breaks some of the spacetime symmetries that we want to keep manifest in the theory,and limits the kinematics of branes that can be studied this way.One of the main points of this paper is to present a string-theoretical construction that keeps all spacetime symmetries manifest.This construction enables us to consider any stable D-brane of Type IIA theory as a bound state of a system of the unstable9-branes discussed above.In fact,this construction turns out to be intimately related to the statement that D-brane charges in Type IIA string theory are classiﬁed by the higher K-theory group K−1(X)of the spacetime manifold X,suggested recently by Witten in[2].This paper is organized as follows.In section2we brieﬂy review the relation of Type IIB D-brane charges1376TYPE IIA D-BRANES,K-THEORY,AND MATRIX THEORYand bound-state constructions to K-theory,and preview the Type IIA case.In section3.1we introduce the unstable9-brane of Type IIA string the-ory.General9-brane conﬁgurations wrapping spacetime manifold X are studied in section3.2.We argue that inequivalent conﬁgurations of9-branes –modulo9-branes that can be created from or annihilated to the vacuum –are classiﬁed by the higher K-theory group K−1(X).In section3.3we show that any given stable D-brane conﬁguration of Type IIA string the-ory can be represented as a bound state of a certain number of unstable Type IIA9-branes.In the worldvolume of the9-branes,the bound state appears as a stable vortex in the tachyonﬁeld,accompanied by a non-trivial gaugeﬁeld carrying a generalized magnetic charge.In the particular case of bound states in codimension three,this precisely corresponds to the’t Hooft-Polyakov magnetic monopole.We generalize our discussion to the case of Type I′theory in section3.4,and argue that Type I′D-brane charges are similarly classiﬁed by the Real K-theory group KR−1(X).In section4we focus on possible implications of our construction to Ma-trix theory[19].We use our9-brane bound-state construction to study a general system of N D0-branes in Type IIA theory.First we show that a D0-brane can be constructed as a bound state of sixteen unstable D9-branes.The low-energy worldvolume theory on the9-branes is a certain non-supersymmetric U(16)gauge theory,with a tachyon in the adjoint repre-sentation of U(16).In this worldvolume theory,the D0-brane is represented as a topologically stable vortex-monopole conﬁguration of the tachyon and the gaugeﬁeld.Multiple D0-brane conﬁgurations are in general also de-scribed by sixteen9-branes,and appear as multi-vortex conﬁgurations in the U(16)gauge theory on the spacetime-ﬁlling worldvolume.This construction thus leads to the intriguing possibility that the dynamics of Matrix theory–as described by a particular limit of the system of N D0-branes of Type IIA string theory–can be contained in the dynamics of vortices on the worldvol-ume of aﬁxed system of sixteen D9-branes,with the individual D0-branes represented by vortices in the worldvolumeﬁeld theory.This construction exhibits some striking similarities with the holographicﬁeld theory of[20].While this paper was beingﬁnished,another paper appeared[21]whose section6partially overlaps with some parts of our section3.3.PETR HOˇRAVA1377 2K-Theory and Type IIA D-BranesIn this section weﬁrst review some highlights of[2](mostly in the contextof Type IIB theory),which will give us the opportunity to present somebackground on K-theory[22]-[26]that will be useful later in the paper.Insection2.2we set the stage for our further discussion of D-brane charges andbound states in Type IIA theory.2.1Type IIB on X and K(X)Consider supersymmetric p-branes andp-branes will carry a Chan-Paton bundle E′ofdimension N′and a U(N′)gaugeﬁeld.The open string connecting a p-braneto aN′)of the gauge group U(N)×U(N′).The D-brane charge of the conﬁguration will be preserved in processeswhere p-branep-branes with a bundle F′that is topologically equiva-lent to F.Thus,invariant D-brane charges correspond to equivalence classesof pairs of bundles(E,E′),where two pairs(E1,E′1)and(E2,E′2)are equiv-alent if(E1⊕F,E′1⊕F)is isomorphic to(E2⊕G,E′2⊕G)for some F and G.(If F corresponds to brane antibrane pairs being created from the vac-uum,G corresponds to pairs annihilated to the vacuum.)The set K(X)ofsuch equivalence classes of pairs of bundles on X forms a group,called theK-theory group of X.The image of(E,0)in K(X)is sometimes denoted by[E].Each element in K(X)can be written as[E]−[E′]for some bundles Eand E′.Consider conﬁgurations of9-branes andp-branes.For a non-compact manifold Y,one deﬁnes K(Y)= K( Y),where Y is a compactiﬁcation of Y by adding a point at inﬁnity.Thus,on a general spacetime manifold X,D-brane charges1378TYPE IIA D-BRANES,K-THEORY,AND MATRIX THEORY of tadpole cancelling Type IIB9-brane9-branes wrapping X,with the class in K(X)given by the diﬀerence of the Chan-Paton bundles on9-branes andN) representation of the gauge group.Together,these bosonicﬁelds form an object A T9-branes,which locally near Y looks like a topologically stable vortex of the tachyonﬁeld.1This can be seen as follows.Stable values of T0correspond to the vacuum manifoldV0(N)=(U(N)×U(N))/U(N),(2.2) which is topologically equivalent to U(N).Thus,the tachyon will support stable defects in codimension2k,classiﬁed by the non-zero homotopy groups of the vacuum manifold,π2k−1(V0(N))=Z(for stable values of N).In order for these defects to carryﬁnite energy,the vortex of winding number n in the tachyonﬁeld must be accompanied by a non-trivial gaugeﬁeld conﬁguration carrying n units of the corresponding topological charge.In the simplest case of one p-brane1For the purposes of this paper,it will be suﬃcient to consider only branes stretching along submanifolds R m of theﬂat spacetime R10.The general case of Type IIB D-branes wrapping general submanifolds Y in general spacetime X is discussed in[2].PETR HOˇRAVA1379 in magnitude to the positive energy density due to the non-zero tension of the p-brane9-brane conﬁguration wrapping X.This embedding is realized by a classic K-theory construction [25],which selects–for a Y of codimension2k in X–a preferred value of the number of9-brane9-branes)a K-theory class [S+]−[S−].In the construction of the bound state,we identify S+with the Chan-Paton bundle carried by9-branes,and S−with the bundle supported by2Our discussion has been local in X;when global topology is taken into account,one sometimes has to“stabilize”(in the K-theory sense)the conﬁguration of9-branes and9-branes,thus leading to a conﬁguration of 9-brane1380TYPE IIA D-BRANES,K-THEORY,AND MATRIX THEORY where we have used a particular convenient representation of the twoΓ-matricesσ1,2in two transverse dimensions x1,2.This bound-state construction deﬁnes a map K(Y)→K(X)for anysubmanifold Y of X that admits a Spin c structure[2].Some other details ofthis construction,together with more details about its relation to the Thomisomorphism,the Gysin map,and the Atiyah-Bott-Shapiro construction inK-theory,can be found in[2].For some general K-theory background,see[22]-[26].2.2Type IIA on X and K−1(X)It has been suggested in[2]that D-brane charges in Type IIA theory shouldbe similarly classiﬁed by a certain higher K-theory group K−1(X).Thisconjecture is supported by the following argument.Consider the reducedK-theory groups of spheres, K(S n).These groups classify possible(9−n)-branes in Type IIB theory on R10[2].Using Bott periodicity,one can showthat K(S2n)=Z and K(S2n+1)=0.The higher K-theory group K−1(X) will be deﬁned precisely below,but now we only invoke the fact thatK−1(S n)= K(S n+1).(2.5) Hence,K−1(S2n+1)=Z and K−1(S2n)=0.This is in accord with the fact that Type IIB theory contains supersymmetric p-branes for p odd,while Type IIA theory has p-branes with p even.Thus,the higher K-theory group K−1(X)of spacetime X is a naturalcandidate for the K-theory group that classiﬁes D-brane charges in Type IIAtheory.For a manifold X of dimension d,the higher K-theory group K−1(X) is usually deﬁned using the ordinary K-theory group of a d+1dimensional extension X′of X.If X is a spacetime manifold of string theory,X′will be eleven-dimensional,and we may suspect a connection to M-theory.The deﬁnition of K−1(X)that is most suggestive of M-theory sets X′=X×S1, and deﬁnes K−1(X)using the K-theory group K(X×S1).More precisely, K−1(X)is deﬁned as the subgroup in K(X×S1)that maps to the trivial class in K(X),by the map induced from the embedding of X as X×point in X×S1.3This deﬁnition of K−1(X)that uses X′=X×S1is somewhat awkward,and we can deﬁne K−1(X)more directly by choosing a slightlyPETR HOˇRAVA1381 diﬀerent X′as follows.Considerﬁrst the product of X with a unit interval, X×I,and deﬁne the so-called“suspension”S′(X)of X by identifying all points in each boundary component of X×I.Thus,for example,the suspension of the m-sphere S m is the(m+1)-dimensional sphere,S′(S m)= S m+1.One can deﬁne K−1(X)by starting with X′=S′(X),and settingK−1(X)= K(S′(X)).(2.6) In the particular case of X=S m,we obtain2.5.In Type IIB theory,the fact that K(X)classiﬁes D-brane charges leads to the construction of all possible D-branes as bound states of spacetime-ﬁlling9-brane1382TYPE IIA D-BRANES,K-THEORY,AND MATRIX THEORY 3Type IIA D-Branes as Bound States of Unstable 9-Branes3.1Unstable9-Branes in Type IIA TheoryWe have pointed out in section1that once we relax the condition thatD-branes be supersymmetric and carry a RR charge,we can construct D p-branes of any p≤9,at the cost of sometimes obtaining unstable conﬁgura-tions.In particular,in Type IIA theory we can construct a spacetime-ﬁlling9-brane.Its structure can be easily understood from the form of its bound-ary state|B ,which–as a particular coherent state in the Hilbert spaceof the Type IIA closed string–represents the boundary conditions on theclosed string annihilated into the9-brane.The D9-brane boundary state(we will only consider the9-brane of Type IIA theory,generalizations to(2k−1)-branes of lower dimensions are obvious)is thus given by|B =|B,+ NS NS−|B,− NS NS.(3.1) Here|B,± NS NS represents the two possible implementations of the Neu-mann boundary conditions on all spacetime coordinates,[31]-[33].There is no RR component in the boundary state,as none is invariantunder the Type IIA GSO projection in the closed string channel.Indeed,there are two RR states|B,± RR that implement Neumann boundary con-ditions on all coordinates.These states transform into each other under theworldsheet fermion number operators(−1)F L,R,as follows(see e.g.[8]): (−1)F L|B,± RR=|B,∓ RR,(−1)F R|B,± RR=|B,∓ RR.(3.2) However,the GSO projection in Type IIA theory chooses opposite chirali-ties in the left-moving and the right-moving sector,and no combination of|B,± RR is invariant under the Type IIA GSO operator(1−(−1)F L)(1+ (−1)F R).4The absence of a RR boundary state means that no RR tadpoleis associated with our9-brane,and therefore,no spacetime anomalies re-lated to RR tadpoles can arise.Unlike in Type IIB theory,where tadpolecancellation requires an equal number of9-branes and4I am grateful to Oren Bergman for discussions on this subject.PETR HOˇRAVA1383 Therefore,the open string connecting one such9-brane to itself will contain –in the NS sector–both the U(1)gaugeﬁeld that a supersymmetric brane would carry,and the tachyonﬁeld T that would,in the case of supersymmet-ric branes,be projected out by the GSO projection.In the Ramond sector of the open string,both spacetime chiralities of the ground state spinor are retained,again due to the absence of any GSO projection.A more precise way of implementing this boundary-state construction of unstable D-branes in a way compatible with the general Type IIA GSO projections on higher-genus worldsheets has been proposed in a similar case of unstable Type IIB D0-branes by Witten in[2].In this procedure,one introduces an extra fermionηat each boundary component that corresponds to the string worldsheet ending on the9-brane.(Similar boundary fermions were introduced some time ago in a diﬀerent context in[28]).This extra fermion is described by the Lagrangian η(dη/dt)dt,with t a periodic co-ordinate along the worldsheet boundary component.Quantization of this√fermion gives an extra factor of√2|B +|B,R .(3.3)√The extra factor of1384TYPE IIA D-BRANES,K-THEORY,AND MATRIX THEORY site spacetime chiralities.5This should be contrasted with the worldvolume ﬁeld content of the Type IIB system of N pairs of9-branes andN).Notice the intriguing fact that thisﬁeld content3.4on N9-branes of Type IIA theory coincides with the ten-dimensional decomposition of a system in eleven dimensions,consisting of a U(N)gaugeﬁeld A M and a 32-component spinorΨin the adjoint of U(N).In particular,the adjoint tachyon plays the role of an eleventh component of the U(N)gaugeﬁeld, and the ten-dimensional decomposition gives3.4asA M=(Aµ,T),Ψ=(χ,χ′).(3.5) Of course,this hidden eleven-dimensional symmetry of the lowest open-string states is broken already at the level of freeﬁelds by the tachyon mass.3.29-Brane Conﬁgurations and K−1(X)In analogy with our understanding of Type IIB D-branes in K-theory,we want to achieve two separate things:(1)classiﬁcation of branes in Type IIA theory on general X,(2)construction of branes in terms of bound states of higher-dimensional branes.First,we will consider possible conﬁgurations of N9-branes in Type IIA string theory,up to possible creation and annihilation of9-branes from and to the vacuum.Recalling our discussion in Section1of a system of such unstable9-branes in Type IIA theory,we expect that the system will rapidly decay to the supersymmetric vacuum,whenever it does not carry lower-dimensional D-brane charges.We will call such9-brane conﬁgurations“elementary.”Any such“elementary”conﬁguration of N′branes wrapping X will give rise to a U(N′)bundle F,together with a U(N′)gaugeﬁeld on F and a tachyon T in the adjoint representation of U(N′).The bound-state construction that we discuss below indicates that the presence or absence of lower D-brane charges can be measured by the tachyon condensate T.Thus,we will assume(cf.[1,2])that a bundle E with tachyonﬁeld T can be deformed–by processes that involve only creation and annihilationPETR HOˇRAVA1385 of“elementary”9-branes–into a bundle isomorphic to E⊕F,with F the Chan-Paton bundle of an elementary9-brane conﬁguration.This deﬁnition of equivalence classes of9-branes with tachyon conden-sate,up to creation or annihilation of“elementary”9-brane conﬁgurations from and to the vacuum,corresponds to the following construction in K-theory.It turns out[22]that in K-theory,one can deﬁne the higher K-theory group K−1(X)without using an eleven-dimensional extension of X.Instead, one starts with pairs(E,α),where E is a U(N)bundle for some N,andαis an automorphism on E.(In fact,we do not lose generality if we consider only trivial bundles E on X.)A pair(F,β)is called“elementary”if the automorphismβcan be continuously deformed to the identity automorphism on F,within automorphisms of F.One deﬁnes an equivalence relation on pairs(E,α),as follows.Two pairs(E1,α1)and(E2,α2)are equivalent if there are two elementary pairs(F1,β1)and(F2,β2)such that(E1⊕F1,α1⊕β1)∼=(E2⊕F2,α2⊕β2).(3.6) The set of all such equivalence classes of pairs(E,α)on X is a group:the inverse element to the class of(E,α)is the class of(E,α−1).This group of classes of pairs(E,α)on X is precisely K−1(X)(as deﬁned e.g.in[22], Section II.3).This“string theory”deﬁnition of K−1(X)–which uses bun-dles with automorphisms on the ten-dimensional spacetime X–is equiva-lent to the deﬁnition of K−1(X)reminiscent of M-theory(and reviewed in section2.2)which uses pairs of bundles on the eleven-dimensional extension X×S1.This rather non-trivial fact can be found e.g.in[22],Theorem II.4.8.In string theory,the role of the N-dimensional bundle E is played by the Chan-Paton bundle carried by a system of N unstable Type IIA9-branes.The automorphismαis a little harder to see directly in the9-brane.However,we will see below that in the bound-state construction of supersymmetric D-branes as bound states in a system of9-branes,the role ofαis played byU=−eπiT,(3.7) where T is the adjoint U(N)tachyon on the9-brane worldvolume.Elemen-tary pairs(F,α)correspond to elementary brane conﬁgurations that do not carry any lower-dimensional D-brane charge,and therefore can be created from and annihilated to the vacuum.Thus,possible9-brane conﬁgurations up to creation and annihilation of“elementary”9-branes are classiﬁed by K−1(X).This,together with our explicit bound-state construction below, demonstrates that K−1(X)indeed classiﬁes D-brane charges in Type IIA theory.1386TYPE IIA D-BRANES,K-THEORY,AND MATRIX THEORY In contrast to Type IIB theory,where one is supposed to consider tadpole cancelling conﬁguration of an equal number of9-branes and9-brane pairs correspond to the reduced K-theory group, K(X),related to the full K(X)by K(X)=Z⊕ K(X).One can deﬁne a“reduced”higher K-theory group K−1(X)[22,26],but it turns out that(for the class of spacetime manifolds that one encounters in string theory) K−1(X)is always equal to K−1(X).3.3Type IIA D-Branes as Bound States of9-BranesSo far,we have suggested a classiﬁcation of all conﬁgurations of9-branes up to creation or annihilation of“elementary”9-branes that do not carry any lower D-brane charge.Here we present a construction that allows one to embed any lower-dimensional branes into a system of9-branes in Type IIA theory:thus,just as in Type IIB theory[2],whatever can be done with stable lower-dimensional branes can be done with unstable9-branes of Type IIA theory.Even though we will mostly focus on bound states of unstable9-branes, one could also start with any lower-dimensional unstable(2k−1)-branes,and construct stable2p-branes for p≤k−1as their bound states.In turn,each such lower-dimensional unstable(2k−1)-brane can be viewed as an unstable bound state of a2k-brane8-brane pairs.There seems to be no gain in representing this D p-brane for example as a bound state of unstable7-branes.However,while not every conﬁguration of stable D-branes of Type IIAﬁts into the worldvolume of a given8-branePETR HOˇRAVA1387values,T=±T0.We will assume–in close analogy with a similar assumption made in[1] in the related case of pairs of stable p-branes and8-brane,depending on the sign in3.8(or, in other words,the sign of the diﬀerence between the asymptotic vacuum values of the tachyon T(−∞)−T(+∞)on the two sides of the domain wall). Notice that only one8-brane or one1388TYPE IIA D-BRANES,K-THEORY,AND MATRIX THEORY the U(N)gauge symmetry is broken to U(N−k)×U(k).Just as in the case of a single9-brane,the tachyonﬁeld can form kinks of codimension one. One particularly interesting case corresponds to the kink in all eigenvalues of T,localized at a common domain wall Y of codimension one in spacetime, which near Y can be written(again,up to a convergence factor),as.(3.11)T= x9·1N−k00−x9·1kWe conjecture that this conﬁguration should be interpreted as N−k8-branes and k8-branes can be constructed by letting each eigenvalue vanish along a separate manifold of codimension one in spacetime.Thus,any number of8-branes and8-branes from9-branes with the construction discussed in[2,1],and construct all lower-dimensional D-2p-branes as bound states of a suﬃcient number of8-brane6The interpretation of this multiple kink conﬁguration as a set of8-branesandPETR HOˇRAVA1389 which represents the worldvolume of the8-brane8-brane pairs may require–when realized via the two-step construction involving8-branes –that extra9-branes be introduced,due to the fact that each8-brane or8-branes,and therefore avoids the degeneracy of the codimension-one bound-state construction,leading to a more powerful description of lower-dimensional D-branes as bound states.Along the way, we will discover many intriguing connections to K-theory.The General Bound State ConstructionFrom now on,we will consider9-brane systems whose tachyon condensate T0has an equal number of positive and negative eigenvalues.Thus,the number of9-branes is2N for some N,and the gauge group U(2N)is broken to U(N)×U(N).The vacuum manifold isV1(2N)=U(2N)/(U(N)×U(N))(3.13) We are interested in stable,vortex-like conﬁgurations in the tachyonﬁeld. Away from the core of such a stable vortex,the tachyonﬁeld(almost)as-sumes its vacuum values.This deﬁnes a map of the sphere S m surrounding the core of a vortex of codimension m+1into the vacuum manifold V1(2N). Possible candidates for stable tachyon vortices in this codimension are thus classiﬁed by elements of the homotopy groupπm(V1(2N)).It turns out that homotopy groups of the vacuum manifold3.13are non-trivial in even di-mensions,π2k(V1(2N))=Z,and trivial in all odd dimensions.(Here we are assuming that N is large enough,so that it belongs to the“stable”range.) This should be contrasted with the case of Type IIB9-brane9-brane pairs supports bound states of codimension2k,our Type IIA9-brane system will exhibit bound states in codimensions2k+1.1390TYPE IIA D-BRANES,K-THEORY,AND MATRIX THEORY This structure of homotopy groups is not coincidental,and in fact reﬂects a deep connection of our construction to K-theory.Our vacuum manifold V1(2N)can be thought of as a Grassmannian manifold whose points are N-dimensional complex subspaces in C2N.This Grassmannian plays an important role in K-theory,as it represents a standardﬁnite-dimensional approximation to the“universal classifying space”BU(see e.g.[23]).The importance of BU in K-theory stems from the fact that the K-theory group K(X)is canonically isomorphic,for any(reasonable)X,to the set of homo-topy classes of maps from X to this universal classifying space,K(X)=[X,BU].(3.14) Similarly,the higher K-theory group K−1(X)is related to the set of homo-topy classes of maps from X to the inﬁnite unitary group U,K−1(X)=[X,U].(3.15) Thus,the vacuum manifold V1(N)of the tachyon on Type IIA9-branes is aﬁnite-dimensional approximation to the classifying space BU,and the vacuum manifold V0(N)=U(N)of the tachyon in the Type IIB9-brane8This is precisely one half of the statement of Bott periodicity[22,26,30].The other half of Bott periodicity similarly relates odd homotopy groups of V1and even homotopy groups of ing these relations,together with3.14and3.15,one can for example derive all K-theory groups of spheres,used in section2.2to classify supersymmetric Type II D-branes in R10.PETR HOˇRAVA1391 U(2k)Chan-Paton bundle,which we identify with the spinor bundle S of the group SO(2k+1)of rotations in the transverse dimensions.The tachyon condensate is then given by the vortex conﬁgurationT(x)=Γm x m.(3.17)As in the Type IIB case[2],Γm are theΓ-matrices of the group of rotations in transverse dimensions x m,m=1,...,2k+1. 3.17describes a stable vortex in codimension2k+1,which we interpret as the supersymmetric (8−2k)-brane of Type IIA theory.Even though the expression for the tachyon vortex3.17on Type IIA 9-branes looks formally identical to the tachyon vortex2.3on the system of9-brane1392TYPE IIA D-BRANES,K-THEORY,AND MATRIX THEORY T=Γ·x,we can construct an element ofπ2k+1(U(2k))as follows.ConsiderU=−eπiT.(3.18) Since the tachyon is in the adjoint of U(2k),U deﬁnes a map from the unit ball|x|≤1to U(2k).The group U(2k)as well as its Lie algebra can be represented by2k×2k matrices which are unitary and hermitian,respectively.We will use a particularly useful description of the coset3.13 ,as the set of all2k×2k matrices that are simultaneously hermitian and unitary[30].In particular,all such matrices square to one,as elements in U(2k).(Incidentally,this proves that far from the core of the vortex, for an appropriate convergence factor f(|x|)omitted in3.17,the tachyon condensate3.17indeed takes values in the vacuum manifold V1(2k).) We can now apply this understanding to the tachyon ing T2=|x|2,one can show that U of3.18maps the origin x=0to−1in U(2k), and each point with x2=1to the identity in U(2k).Thus,3.18indeed deﬁnes a map from S2k+1to U(2k),and hence an element ofπ2k+1(U(2k)). This element inπ2k+1(U(2k))maps under3.16to the element inπ2k(V1(2k)) that corresponds to the tachyon vortex3.17.(For details on this K-theory construction,see[30].)In terms of K-theory,this proves that our tachyon condensate actually represents the generator of the relative K-theory group K−1(B2k+1,S2k),and our bound-state construction is precisely the analog of the ABS construc-tion[25],now mapping classes in K(Y)to classes in K−1(X)for Y of odd codimension in the spacetime manifold X wrapped by the ustable9-branes of Type IIA theory.9This one-step construction of Type IIA D-branes as bound states of codi-mension2k+1in a system of unstable9-branes suggests the following hi-erarchy of bound state constructions.Consider a supersymmetric D p-brane in Type IIA or Type IIB theory.This D-brane can be constructed as a bound state(tachyon kink)in the worldvolume of an unstable D-(p+1)-brane.Alternatively,it can be constructed[1,2]as a bound state of a (p+2)-brane9The relative K-theory group K−1(B2k+1,S2k)is deﬁned as the group of equivalence classes of bundles with automorphisms(E,α)on B2k+1,withα=1when restricted to the boundary S2k(see e.g.[22],section II.3.25.).In our string theory construction,E is the Chan-Paton bundle,and as we have just seen,U of3.18has just the right properties to be identiﬁed withα.。

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Type IIA D-Branes, K-Theory, and Matrix Theory

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