Normed BCI-algebras

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。

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TIME SER ANAL J OURNAL OF TIME SERIES ANALYSIS0143-9782统计学与概率论LIFETIME DATA ANALYSIS1380-7870统计学与概率论LIFETIME DATA ANMATH POPUL STUD M athematical Population Studies0889-8480统计学与概率论METHODOL COMPUT1387-5841统计学与概率论METHODOLOGY AND COMPUTING IN APPLIED PROMETRIKA METRIKA0026-1335统计学与概率论PAK J STAT Pakistan Journal of Statistics1012-9367统计学与概率论0269-9648统计学与概率论PROBABILITY IN THE ENGINEERING AND INFORPROBAB ENG INFORRevista Colombiana de Estadistica0120-1751统计学与概率论REV COLOMB ESTADREVSTAT-STAT JREVSTAT-Statistical Journal1645-6726统计学与概率论SCAND ACTUAR JScandinavian Actuarial Journal0346-1238统计学与概率论Statistical Methods and Applications1618-2510统计学与概率论STAT METHOD APPLSTAT MODEL STATISTICAL MODELLING1471-082X统计学与概率论STAT NEERL STATISTICA NEERLANDICA0039-0402统计学与概率论STAT PAP STATISTICAL PAPERS0932-5026统计学与概率论STATISTICS & PROBABILITY LETTERS0167-7152统计学与概率论STAT PROBABIL LESTAT SINICA STATISTICA SINICA1017-0405统计学与概率论STATISTICS STATISTICS0233-1888统计学与概率论STOCH ANAL APPL S TOCHASTIC ANALYSIS AND APPLICATIONS0736-2994统计学与概率论STOCH DYNAM Stochastics and Dynamics 0219-4937统计学与概率论STOCH MODELS STOCHASTIC MODELS1532-6349统计学与概率论1744-2508统计学与概率论STOCHASTICS Stochastics-An International Journal ofSURV METHODOL Survey Methodology 0714-0045统计学与概率论0040-585X统计学与概率论THEORY OF PROBABILITY AND ITS APPLICATIOTHEOR PROBAB APPUTILITAS MATHEMAUTILITAS MATHEMATICA0315-3681统计学与概率论Journal of Noncommutative Geometry1661-6952物理：数学物理J NONCOMMUT GEOMJ NONLINEAR SCI J OURNAL OF NONLINEAR SCIENCE0938-8974物理：数学物理MULTISCALE MODELING & SIMULATION1540-3459物理：数学物理MULTISCALE MODELINVERSE PROBL INVERSE PROBLEMS0266-5611物理：数学物理Inverse Problems and Imaging1930-8337物理：数学物理INVERSE PROBL IMAdvances in Applied Clifford Algebras0188-7009物理：数学物理ADV APPL CLIFFOR1935-0090物理：数学物理Applied Mathematics & Information SciencAPPL MATH INFORM1559-3940物理：数学物理Communications in Applied Mathematics anCOMM APP MATH COCOMPUTATIONAL MATHEMATICS AND MATHEMATIC0965-5425物理：数学物理COMP MATH MATH PINFINITE DIMENSIONAL ANALYSIS QUANTUM PR0219-0257物理：数学物理INFIN DIMENS ANA0219-8916物理：数学物理Journal of Hyperbolic Differential EquatJ HYPERBOL DIFFE1812-9471物理：数学物理Journal of Mathematical Physics AnalysisJ MATH PHYS ANAL1385-0172物理：数学物理MATHEMATICAL PHYSICS ANALYSIS AND GEOMETMATH PHYS ANAL GNONLINEAR OSCIL N onlinear Oscillations1536-0059物理：数学物理1223-7027物理：综合University Politehnica of Bucharest ScieU POLITEH BUCH SABSTR APPL ANAL A bstract and Applied Analysis 1085-3375应用数学0010-3640应用数学COMMUN PUR APPLCOMMUNICATIONS ON PURE AND APPLIED MATHE0021-7824应用数学JOURNAL DE MATHEMATIQUES PURES ET APPLIQJ MATH PURE APPL0218-2025应用数学MATHEMATICAL MODELS & METHODS IN APPLIEDMATH MOD METH APNONLINEAR ANALYSIS-REAL WORLD APPLICATIO1468-1218应用数学NONLINEAR ANAL-RSIAM REV SIAM REVIEW0036-1445应用数学ADV COMPUT MATH A DVANCES IN COMPUTATIONAL MATHEMATICS1019-7168应用数学Aequationes Mathematicae0001-9054应用数学AEQUATIONES MATHANAL APPL Analysis and Applications0219-5305应用数学ANAL PDE Analysis & PDE1948-206X应用数学ANNALI DI MATEMATICA PURA ED APPLICATA 0373-3114应用数学ANN MAT PUR APPLAPPL MATH COMPUTAPPLIED MATHEMATICS AND COMPUTATION0096-3003应用数学Boundary Value Problems 1687-2762应用数学BOUND VALUE PROBCALCULUS OF VARIATIONS AND PARTIAL DIFFE0944-2669应用数学CALC VAR PARTIALCarpathian Journal of Mathematics1584-2851应用数学CARPATHIAN J MAT0219-1997应用数学COMMUN CONTEMP MCOMMUNICATIONS IN CONTEMPORARY MATHEMATI0360-5302应用数学COMMUN PART DIFFCOMMUNICATIONS IN PARTIAL DIFFERENTIAL E0925-7721应用数学COMPUTATIONAL GEOMETRY-THEORY AND APPLICCOMP GEOM-THEOR1078-0947应用数学DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMDISCRETE CONT DY0764-583X应用数学ESAIM-MATHEMATICAL MODELLING AND NUMERICESAIM-MATH MODELFixed Point Theory 1583-5022应用数学FIXED POINT THEOFixed Point Theory and Applications 1687-1820应用数学FIXED POINT THEO1615-3375应用数学FOUNDATIONS OF COMPUTATIONAL MATHEMATICSFOUND COMPUT MATFUZZY SET SYSTFUZZY SETS AND SYSTEMS0165-0114应用数学IMA JOURNAL OF NUMERICAL ANALYSIS0272-4979应用数学IMA J NUMER ANALInternational Journal of Numerical Analy1705-5105应用数学INT J NUMER ANAL1040-7294应用数学Journal of Dynamics and Differential EquJ DYN DIFFER EQU1435-9855应用数学J EUR MATH SOCJOURNAL OF THE EUROPEAN MATHEMATICAL SOCJ FOURIER ANAL A1069-5869应用数学JOURNAL OF FOURIER ANALYSIS AND APPLICATJ GLOBAL OPTIMJOURNAL OF GLOBAL OPTIMIZATION0925-5001应用数学JOURNAL OF MATHEMATICAL ANALYSIS AND APP0022-247X应用数学J MATH ANAL APPLJ MOD DYNAM Journal of Modern Dynamics1930-5311应用数学Journal of Noncommutative Geometry1661-6952应用数学J NONCOMMUT GEOMJ NONLINEAR SCI J OURNAL OF NONLINEAR SCIENCE0938-8974应用数学J SCI COMPUT JOURNAL OF SCIENTIFIC COMPUTING0885-7474应用数学KINET RELAT MOD K inetic and Related Models1937-5093应用数学。

现代电子技术Modern Electronics Technique2023年12月1日第46卷第23期Dec. 2023Vol. 46 No. 230 引 言脑机接口（Brain⁃Computer Interface, BCI ）是一种通过分析神经元电信号，促进人脑与外部电子设备直接通信的技术[1]。

BCI 系统最初是为帮助患有身体或认知障碍的患者而开发的，现已在神经医学、智能家居、自动驾驶和娱乐等领域得到广泛应用[2]。

BCI 系统包括收集大脑信号、对其进行解码和控制外部设备（例如计算机、智能轮椅或假肢）三部分组件[3]。

记录人脑活动意图的技术分为侵入式和非侵入式两种。

侵入性技术需要植入微电极阵列，存在一定的风险[4]；非侵入性技术如脑电图（Electroencephalography, EEG ）是主要采用的研究方基于通道选择的多尺度Inception 网络的脑电信号分类研究刘 培， 宋耀莲（昆明理工大学 信息工程与自动化学院， 云南 昆明 650500）摘 要： 基于运动想象脑电信号的脑机接口系统有可能在大脑和外部设备之间创建通信通道。

然而，特征提取的局限性、通道选择的复杂性和被试者之间的可变性使得脑电信号分类模型难以有效泛化。

在这项研究中，文中提出一种端到端的深度学习模型，该模型使用并行多尺度Inception 卷积神经网络在6个通道选择区域中进行多分类运动想象任务。

为了解决被试者间可变性，实验进行了跨被试和跨被试微调两种评估场景。

在BCI 竞赛IV 2a 数据集上的实验和测试结果表明：ROI F 达到了98.49%的最高分类精度，比最低准确率高17.26%；且跨被试微调场景分类性能优于被试内和跨被试场景，分类准确率分别提高了1.82%和1.69%。

此外，并行多尺度Inception 卷积神经网络模型的平均分类准确率比单尺度Inception CNN 模型高5.17%。

总之，文中提出一种基于通道选择的端到端的脑电信号分类框架，可以促进高性能和稳健的脑机接口系统的开发。

bcl-2 PCR Primers Set Product No. B9179 Store at –20 ºCProduct DescriptionIn recent years, several genes have been linked to apoptosis. The bcl-2 family of genes regulates PCD either positively or negatively. Bcl-2 and members of its family have been found to block apoptotic cell death. Bcl-2 protein heterodimerizes with Bax (Bcl-2 Associated X protein), which is a potent mediator of programmed cell death. The Bcl-2/Bax ratio appears to determine whether some cells live or die.1-4The bcl-2 PCR Pr imers Set contains both sense and antisense primers for the amplification of the bcl-2αgene. It is designed for PCR† detection of human, rat and mouse cDNA levels (representing mRNA expression) of the bcl-2α apoptotic gene. No ampli-fication of the genomic DNA has been observed.The size of the amplified product resulting from the use of the bcl-2 PCR Primers Set is 127 bp.Component• bcl-2 PCR Primers Set, Product No. B9179 1 vial Equipment and Reagents Required but Not Provided (Sigma product numbers have been given where appropriate)• Thermal cycler• Taq DNA polymerase, Product No. D4545 or equivalent• Deoxynucleotide mix, 10 mM, Product No. D7295 or equivalent• Agarose• Ethidium bromide, 500 µg/ml, Product No. E1385 • PCR 100 bp low ladder, Product No. P1473• Gel loading solutions, Product No. G2526 or G7654• PCR grade water, Product No. W1754• Mineral oil, Product No. M8662• PCR microtubes, Product No. Z37,487-3 or Z37,496-2StorageStore the vial at −20 °C.Preparation InstructionsThe bcl-2 PCR Primers Set contains 1 nmoles of each primer (sense and antisense). Centrifuge the tube briefly in order to collect the tube contents. For the following procedure, resuspend the primers set in100 µl deionized water to a final concentration of10 pmole/µl. Mix until the solution is homogenous. Once suspended, store the solution at –20 °C. To avoid repeated freeze-thaw cycles, aliquot the primer solution for long-term storage.ProcedureNote: Use aseptic techniques and use aerosol barrier tips while performing PCR experiments.1. Thaw the bcl-2 PCR Primers Set on ice, beingsure that the solution is homogenous.2. Add the following reagents to a PCRmicrocentrifuge tube in the following order:Amount for50 µl singlePCRreactionFinalconcentrationin the PCRreaction Water To 50 µl ----10X PCR Buffer 5 µl 1X2 mM dNTP solution 5 µl 0.2 mM ofeach dNTP25 mM MgCl2*solution3 µl 1.5 mMbcl-2 PCR PrimersSet, 10 pmole/µl2 µl 0.4 µM cDNA** 2 µl ~30 ngTaq DNAPolymerase,5 units/µl1 µl 0.1 units/µl Total volume50 µl -* When using the bcl-2 PCR Primers Set for thefirst time, you may set two additional reactiontubes with a higher and a lower MgCl2concentrations (see Note at the end of thissection).** Optimize this parameter with your own cDNA.3. Mix gently by finger tapping and centrifuge briefly to collect the mixture in the bottom of the tube. Overlay the reaction mixture with 2 drops (~30 µl) of mineral oil to cover the surface of the reaction mixture if not using a thermal cycler with a heated lid. Place the tube in the thermal cycler when the thermal cycler reaches 95 oC, and run the following PCR program. 95 oC for 2 min 94 oC for 45 sec 53 oC for 45 sec x 30 cycles 72 oC for 1.5 min 72 oC for 7 min The amplified DNA can be evaluated by agarose gel electrophoresis.Note: Using different thermal cyclers:For a better detection of the amplified product you may increase the number of amplification cycles. In case you do not see differences in the amount of the amplified DNA fragments, decrease the number of cycles to verify your results.In rare cases, some of the parameters should beoptimized for the specific thermal cycler or cDNA samples. The most frequently adjusted factors are MgCl 2 concentration and annealing temperature. You may prepare three different reactions using MgCl 2 at a concentration of 0.5-3 mM (e.g., 0.5-0.8 mM, 1.5 mM and 3 mM). Optimize the MgCl 2 and/or the annealing temperature on your instrument using the positive control cDNA provided before using your own cDNA.Troubleshooting GuideProblem Cause Solution A PCR component may be missing or degraded. Try to isolate the problematic reagent by replacing it with a fresh one. A checklist is also recommended when assembling reactions.No PCR products cDNA or MgCl 2concentration is not optimal .Optimize the cDNA and MgCl 2 concentrations.Highbackground, smearing or nonspecific bandsIncrease the annealing temperature or decrease the MgCl 2concentration. Another solution for avoiding high background is to decrease the amount of cDNA template used for amplification.Contamination with other DNAUse sterile techniques while performing PCR experiments. cDNA quality is not sufficientUse a different cDNA preparation.Amplifiedproducts are not the correct sizeNon-optimal PCR conditions Optimize PCR conditions especially cDNA and MgCl 2 concentrations and annealing temperature. Poor resolution of products in agarose gelUse 2% agarose gel and increase run time.References1. Oltavi, Z.N., et al ., Cell, 74, 609 (1993)2. Korsmeyer, S.J., Cancer Research (Suppl), 59,1693s (1999)3. Agarwal, N. and Metha, K., Biochem. Biophys.Res. Commun., 230, 251 (1997)4. Aggarwal, S. and Gupta, S., J. Immunol., 160,1627 (1998)†The PCR process is covered by patents owned by Hoffman-LaRoche, Inc. Purchase of this product does not convey a license under these patents.ya 6/00Sigma brand products are sold through Sigma -Aldrich, Inc.Sigma-Aldrich, Inc. warrants that its products conform to the information contained in this and other Sigma -Aldrich publications. Purchaser must determine the suitability of the product(s) for their particular use. Additional terms and conditions may apply.Please see reverse side of the invoice or packing slip.。

(0,2) 插值||(0,2) interpolation0#||zero-sharp; 读作零井或零开。

0+||zero-dagger; 读作零正。

1-因子||1-factor3-流形||3-manifold; 又称“三维流形”。

AIC准则||AIC criterion, Akaike information criterionAp 权||Ap-weightA稳定性||A-stability, absolute stabilityA最优设计||A-optimal designBCH 码||BCH code, Bose-Chaudhuri-Hocquenghem codeBIC准则||BIC criterion, Bayesian modification of the AICBMOA函数||analytic function of bounded mean oscillation; 全称“有界平均振动解析函数”。

BMO鞅||BMO martingaleBSD猜想||Birch and Swinnerton-Dyer conjecture; 全称“伯奇与斯温纳顿－戴尔猜想”。

B样条||B-splineC*代数||C*-algebra; 读作“C星代数”。

C0 类函数||function of class C0; 又称“连续函数类”。

CA T准则||CAT criterion, criterion for autoregressiveCM域||CM fieldCN 群||CN-groupCW 复形的同调||homology of CW complexCW复形||CW complexCW复形的同伦群||homotopy group of CW complexesCW剖分||CW decompositionCn 类函数||function of class Cn; 又称“n次连续可微函数类”。

Cp统计量||Cp-statisticC。

专利名称：癌症分期与治疗的系统和方法专利类型：发明专利
发明人：格伦·韦斯
申请号：CN200980113822.8
申请日：20090219
公开号：CN102027131A
公开日：
20110420
专利内容由知识产权出版社提供
摘要：本发明公开了评估癌细胞对酪氨酸激酶抑制剂敏感性的方法。

所述方法包括评估miR-497的表达，并将降低表达与对酪氨酸激酶抑制剂的敏感性相关联。

本发明还公开了评估细胞对酪氨酸激酶抑制剂敏感性的方法，该方法包括评估FGF1、HOXC10、和/或LHFP的表达。

本发明还公开了基于由所公开的方法获得的结果使用酪氨酸激酶抑制剂例如舒尼替尼治疗患者的方法和有助于所述方法的试剂盒。

申请人：翻译基因组学研究院
地址：美国亚利桑那州
国籍：US
代理机构：北京安信方达知识产权代理有限公司
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第十九届矩阵与统计国际学术会议在上海金融学院召开
刘永辉;方勇
【期刊名称】《应用概率统计》
【年(卷),期】2010(26)5
【摘要】第十九届矩阵与统计国际学术会议于2010年6月5日至8日在上海金
融学院召开．本次会议的主题是“矩阵、统计及其在金融中的应用”．来自世界
23个国家和地区的186位数学家、统计学家和金融学家参加了这次大会．中国科学院石钟慈院士担任会议的科学委员会主席，新西兰原统计学会主席、数学会院士JeffreyHunter教授担任了本次会议的国际组织委员会主席．
【总页数】1页(P559-559)
【关键词】国际学术会议;中国科学院;统计学;金融学;矩阵;上海;科学委员会;国际组织
【作者】刘永辉;方勇
【作者单位】上海金融学院
【正文语种】中文
【中图分类】O212
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