Self-injective algebras and the second Hochschild cohomology group
Acknowledgements This paper is a part of my PhD. thesis at the University of Leicester. I thank Taif University in Saudi Arabia for funding my PhD. research, and Dr. Nicole Snashall for her helpful supervision and valuable suggestions. The author also thanks the referee for their helpful comments. Introduction In this paper we study the second Hochschild cohomology group HH2 (Λ) of all finite dimensional self-injective algebras Λ of finite representation type over an algebraically closed field K . In general, finite dimensional self-injective algebras of finite representation type over an algebraically closed field K were shown by Riedtmann in [9] to fall into one of the types A, D or E , depending on the tree class of the stable Auslander-Reiten quiver of the algebra. Riedtmann classified the stable equivalence representatives of these algebras of type A in [10]; Asashiba then showed that the stable equivalence classes are exactly the derived equivalence classes for types A, D in [2, Theorem 2.2]. In [1],the derived equivalence class representatives are given explicitly by quiver and relations. Happel showed in [8] that Hochschild cohomology is invariant under derived equivalence. So if A and B are derived equivalent then HH2 (A) ∼ = HH2 (B ). Hence 2 to study HH (Λ) for all finite dimensional self-injective algebras of finite representation type over an algebraically closed field K , it is enough to study HH2 (Λ) for the representatives of the derived equivalence classes. The algebras of type A fall into two types: Nakayama algebras and M¨ obius algebras, and the Hochschild cohomology of these algebras has already been studied. In [3], Erdmann and Holm give the dimension of the second Hochschild cohomology group of a Nakayama algebra. In [6], Green and Snashall find the second Hochschild cohomology group for the M¨ obius algebras. The main work of this paper is thus in determining HH2 (Λ) for the finite dimensional self-injective algebras of finite representation type of types D and E . In
SELF-INJECTIVE ALGEBRAS AND THE SECOND HOCHSCHILD COHOMOLOGY GROUP
arXiv:0709.0986v2 [math.RT] 22 Aug 2008
பைடு நூலகம்
DEENA AL-KADI Abstract. In this paper we study the second Hochschild cohomology group HH2 (Λ) of a finite dimensional algebra Λ. In particular, we determine HH2 (Λ) where Λ is a finite dimensional self-injective algebra of finite representation type over an algebraically closed field K and show that this group is zero for most such Λ; we give a basis for HH2 (Λ) in the few cases where it is not zero.
Date : August 22, 2008.
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AL-KADI
Section 1 we give a summary of [1] which gives the explicit derived equivalence representatives we consider. Section 2 gives a short description of the projective resolution of [6] which we use to find HH2 (Λ). In Section 3, we give a general theorem, Theorem 3.6, which we use to show that HH2 (Λ) = 0 for most of our algebras. This is motivated by work in [6]. The strategy of the theorem is to show that every element in Hom(Q2 , Λ) is a coboundary so that HH2 (Λ) = 0, where Q2 is the second projective in a minimal projective resolution of Λ as a Λ,Λbimodule. For all other cases which are not covered by Theorem 3.6, we determine HH2 (Λ) by direct calculation, and find a basis for HH2 (Λ) in the instances where HH2 (Λ) = 0. The standard algebras are considered in Sections 4 and 5 and the non-standard algebras in Section 6. Finally Theorem 6.5 summarises our results and describes HH2 (Λ) for all finite dimensional self-injective algebras Λ of finite representation type over an algebraically closed field. As a consequence, we show that dim HH2 (Λ) = dim HH2 (Λ′ ) for a non-standard algebra Λ and its standard form Λ′ , where Λ and Λ′ are of type (D3m , 1/3, 1). This gives an alternative proof that Λ and Λ′ are not derived equivalent. 1. The derived equivalence representatives We give here Asashiba’s full classification from [1] and [2] of the derived equivalence class representatives of the finite dimensional self-injective algebras of finite representation type over an algebraically closed field. These derived equivalence class representatives are listed according to their type. From [9], the stable Auslander Reiten quiver of a self-injective algebra Λ of finite representation type has the form Z∆/ g , where ∆ is a Dynkin graph, g = ζτ −r such that r is a natural number, ζ is an automorphism of the quiver Z∆ with a fixed vertex, and τ is the Auslander-Reiten translate. Then typ(Λ) := (∆, f, t), where t is the order of ζ and f := r/m∆ such that m∆ = n, 2n − 3, 11, 17 or 29 as ∆ = An , Dn , E6 , E7 or E8 , respectively. We take the following results from [2]. Proposition 1.1. [2, Theorem 2.2] Given Λ a self-injective algebra of finite representation type then the type typ(Λ) is an element of one of the following sets: {(An , s/n, 1)|n, s ∈ N}; {(A2p+1 , s, 2)|p, s ∈ N}; {(Dn , s, 1)|n, s ∈ N, n ≥ 4}; {(Dn , s, 2)|n, s ∈ N, n ≥ 4}; {(D4 , s, 3)|s ∈ N}; {(D3m , s/3, 1)|m, s ∈ N, m ≥ 2, 3 ∤ s}; {(En , s, 1)|n = 6, 7, 8, s ∈ N}; and {(E6 , s, 2)|s ∈ N}. Theorem 1.2. [2, Theorem 2.2] Let Λ and Π be self-injective algebras of finite representation type. (i) If Λ is standard and Π is non-standard then Λ and Π are not derived equivalent. (ii) If Λ and Π are either both standard or both non-standard then the following are equivalent: 1) Λ and Π are derived equivalent; 2) Λ and Π are stably equivalent; 3) typ(Λ) = typ(Π).
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Duality for modules over finite rings and applications to coding theory
1991 Mathematics Subject Classi cation. Primary: 94B05, 16L60, 16P10. Key words and phrases. Characters, Frobenius rings, Morita duality, equivalence, MacWilliams identities. Partially supported by NSA grants MDA904-91-H-0029, MDA904-94-H-2025, and MDA904-96-1-0067, and by Purdue University Calumet Scholarly Research Awards. Copyright c The Johns Hopkins University Press. The article rst appeared in the American Journal of Mathematics, 121 1999, 555 575. 1
Computer-Vision计算机视觉英文ppt
Its mainstream research is divided into three stages:
Stage 1: Research on the visual basic method ,which take the model world as the main object;
Stage 2: Research on visual model ,which is based on the computational theory;
the other is to rebuild the three dimensional object according to the two-dimensional projection images .
History of computer vision
1950s: in this period , statistical pattern recognition is most applied in computer vision , it mainly focuse on the analysis and identification of two-dimensional image,such as: optical character recognition, the surface of the workpiece, the analysis and interpretation of the aerial image.
students approach to learning(学习品质)
ISSN 1306-3065 Copyright © 2006-2013 by ESER, Eurasian Society of Educational Research. All Rights Reserved.Student Approaches to Learning in Physics – Validityand Exploration Using Adapted SPQManjula Devi Sharma*1, Chris Stewart 1, Rachel Wilson 1, and MuhammedSait Gökalp 2Received 22 December 2012; Accepted 9 March 2013Doi: 10.12973/ijese.2013.203aAbstract: The aim of this study was to investigate an adaptation of the Study ProcessesQuestionnaire for the discipline of physics. A total of 2030 first year physics students at an Australian metropolitan university completed the questionnaire over three different year cohorts. The resultant data has been used to explore whether the adaptation of the questionnaire is justifiable and if meaningful interpretations can be drawn for teaching and learning in the discipline. In extracting scales for deep and surface approaches to learning, we have excised several items, retaining an adequate subset. Reflecting trends in literature, our deep scale is very reliable while the surface scale is not so reliable. Our results show that the behaviour of the mean scale scores for students in different streams in first year physics is in agreement with expectations. Furthermore, different year cohort performance on the scales reflects changes in senior high school syllabus. Our experiences in adaptation, validation and checking for reliability is of potential use for others engaged in contextualising the Study Processes Questionnaire, and adds value to the use of the questionnaire for improving student learning in specific discipline areasKeywords: Student approaches to learning, learning in disciplines, university physics education1University of Sydney, Australia* Corresponding author. Sydney University Physics Education Research group, School of Physics, University of Sydney, NSW 2006, Australia. Email: sharma@.au 2Dumlupınar University, TurkeyIntroductionSince the mid 1960’s a series of inventories exploring student learning in highereducation have been developed based on learning theories, educational psychology and study strategies. For reviews of the six major inventories see Entwistle and McCune (2004)International Journal of Environmental & Science EducationVol. 8, No. 2, April 2013, 241-253242C o p y r i g h t © 2006-2013 b y E S E Rand Biggs (1993a). As can be seen from the reviews, these inventories have two common components. One of these components is related to study strategies and the other one is about cognitive processes. Moreover, these inventories usually have similar conceptual structures and include re-arrangement of the items (Christensen et al., 1991; Wilson et al., 1996).In the current study, as one of these inventories, The Study Processes Questionnaire (SPQ) has been selected to be adapted for use in physics. The SPQ is integrated with the presage-process-product model (3P model) of teaching and learning (Biggs, 1987). Several studies have successfully used the SPQ across different culture s and years to compare students’ approaches in different disciplines (Gow et al., 1994; Kember & Gow, 1990; Skogsberg & Clump, 2003; Quinnell et al., 2005; Zeegers, 2001). Moreover, several other researchers used modified version of the SPQ at their studies (Crawford et al. 1998a,b; Fox, McManus & Winder, 2001; Tooth, Tonge, & McManus, 1989; Volet, Renshaw, & Tietzel, 1994). For example, Volet et al (2001) used a shortened SPQ included 21 items to assess cross cultural differences. Fox et al (2001) modified the SPQ and tested its structure with confirmatory factor analysis. In their study the modified version of the SPQ had 18 items, and this shortened version had same factor structure as the original SPQ. In another study, Crawford et al. (1998a, b) adapted the SPQ for the discipline of mathematics. That adapted questionnaire was named as Approaches to Learning Mathematics Questionnaire.Three different approaches of the students to learning are represented in the SPQ: surface, deep, and achieving approaches. Idea of approaches to learning was presented by Marton and Säljö (1976) and further discussed by several other researchers (eg. Biggs, 1987; Entwistle & Waterston, 1988). Basically, surface approach indicates that the students’ motivation to learn is only for external consequences such as getting the appreciation of the teacher. More specifically, it is enough to fulfill course requirements for the students with surface approach. On theother hand, a deep approach to learning indicates that the motivation is intrinsic. This approach involves higher quality of learning outcomes (Marton & Säljö, 1976; Biggs, 1987). Students with deep approach to learning try to connect what they learn with daily life and they examine the content of the instruction more carefully. On the other hand, achieving approach is about excelling in a course by doing necessary things to have a good mark. However, current study is not focused on this approach. Only the first two approaches were included in the adapted SPQ.Inventories like the SPQ are used in higher education because of several reasons. Such inventories can help educators to evaluate teaching environments (Biggs, 1993b; Biggs, Kember, & Leung, 2001). Moreover, with the use of these inventories, university students often relate their intentions and study strategies for a learning context in a coherent manner. On the other hand, the SPQ is not a discipline specific inventory. It can be used across different disciplines. However, in a research study, if the research questions are related with the common features of learning and teaching within 3P model framework, the SPQ can be used satisfactorily for all disciplines. But, a discipline specific version of the SPQ is required if resolution of details specific to a discipline area is necessary for the research questions. Moreover, in order to reduce systematic error and bias that can be resulted from students in different discipline areas; a discipline specific version may be required. As a community of educators, we are aware that thinking, knowing, and learning processes can differ across discipline areas. A direct consequence of this acknowledgement is the need to understand and model learning in specific discipline areas, such as by adapting the SPQ. However, for the theoretical framework to be valid the conceptual integrity of the inventory must be maintained.This paper reports on how the SPQ has been adapted for physics. The teaching context is first year physics at a research focused Australian university where students are grouped according to differing senior243Student approaches to learning in physicsC o p y r i g h t © 2006-2013 b y E S E Rhigh school experiences into Advanced, Regular, and Fundamentals streams.We report on the selection of items for the deep and surface scales and reliability and validity analyses. A comparison of the Advanced, Regular and Fundamentals streams is carried out to ensure that interpretations associated with the deep and surface scales are meaningful. This is a stage of a large-scale project. The project aims to understand and improve student learning based on the deep and surface approaches to learning inherent in the 3P model (Marton & Säljö, 1976; Biggs, 1987).The studyAs mentioned before, The SPQ has been designed for higher education; however, this questionnaire is not discipline specific. Therefore, in this study, we adapted the SPQ to physics for the following reasons: (1) The first year students have confusions about university studies when they come to university (White et al., 1995). This can lead to misinterpretation of the items. However, specific items related to physics can reduce these misinterpretations. For example students enrolled in a general science degree would view questions related to employment differently to those in professional degrees, and the students we have surveyed are from a range of degree programs. (2) In order to compare the students from the different discipline areas, we need discipline specific inventories. (3) We believe that there are contentious items in the original SPQ and aspects that are specific to physics. For example the use of “truth” in the following item was strongly challenged by a group of physicists validating the questionnaire. While I realize that truth is forever changing as knowledge is increasing, I feel compelled to discover what appears to me to be the truth at this time (Biggs, 1987, p. 132).The item was changed to the following, more in line with the post-positivist paradigm and agreeable to physicists.While I realize that ideas are always changing as knowledge is increasing, I feel a need to discover for myselfwhat is understood about the physical world at this time.One could argue that this is an issue of clarifying the item rather than being specific to physics. However, to our knowledge the clarity of this item has not been debated in literature.Just after we commenced this study in 2001, we became aware that Biggs et al (2001) had produced a revised Study Processes Questionnaire (R-SPQ- 2F). However, it was too late for our study and we did not switch midway. There are four main differences between the SPQ and the R-SPQ-2F; first, the removal of all items on employment after graduation; second, increased emphasis on examination; third, removal of words that imply specificity; and fourth exclusion of the contentious achieving factor identified by Christensen et al., 1991. We focus on the deep and surface approaches and not on the strategy and motive sub-scales as these are not pertinent to our larger study. The SPQ deep and surface scales, in particular, have been shown to be robust (see for example Burnett & Dart, 2000).The participant of the current study was from a university in New South Wales, Australia. Students are provided three basic physics units in the School during their first semester of university: Fundamentals, Regular or Advanced. Students are divided into these three groups of physics units based on their senior high school physics backgrounds. The students from the Fundamentals unit have done no physics in senior high school or have done poorly. On the other hand, in the Regular unit, there were the students had scored high grades in senior high school physics. The last unit, the Advanced unit, is suitable for those who have done extremely well overall in physics during all their years in senior high school.The three physics units that students can register in are for the degree programs in Engineering, Medical Science and Arts. Students who intend to major in physics as well as postgraduate physics students are selected from those enrolled in all three basic physics course in their first semester at university. The largest proportion of students of physics major is from the Advanced244C o p y r i g h t © 2006-2013 b y E S E Rstream, followed by those in the Regular stream, and finally the Fundamentals stream. The data was collected from these streams from 2001 to 2004. From 2001 to 2004, the high school physics syllabi and assessment system was changed in the state of New South Wales in Australia. The details of the changes can be seen in Binnie (2004). Due to these changes, the 2004 cohort of students in this study were instructed using a different curriculum.Within the above context, we have adapted the SPQ to generate a Study Processes Questionnaire for Physics (SPQP). The research questions addressed in this paper are as follows.(a) How do the factor solutions for the SPQP compare with those of the SPQ? (b) Is the SPQP reliable and valid?(c) Are the scales robust enough to reflect detail in senior high school syllabus change? The answers to the research questions will determine if the SPQP is a reliable and valid measure of student approaches to learning physics in our context.MethodRevising the items for the SPQPWe have adapted the SPQ by simply inserting the word “physics” in some items and making substantial changes to others. The adaptations are based on our experiences of student responses to open-ended questions and discipline knowledge, and have been extensively discussed amongst a group of physics educators. The adaptations are of the types listed below. (See appendix A for all items and the types of adaptations.) Type 0: No changeType 1: A simple insertion of terms such as “physics”, “studying physics”.I find that at times studying gives me afeeling of deep personal satisfaction. I find that at times studying physics gives me a feeling of deep personal satisfaction.Type 2: A substantial change in wording that can change the meaning, without intending to.I usually become increasingly absorbed in my work the more I do. When studying physics, I become increasingly absorbed in my work the more I do.Type 3: An intentional change in meaning.My studies have changed my views about such things as politics, my religion, and my philosophy of life. My studies in physics have challenged my views about the way the world works.The number of items corresponding to each Type of change is displayed in Table 1, as are the number of items selected from each Type for inclusion in the SPQP. Type 1 items were more useful in generating the items used in the SPQP.Administering the SPQPThe SPQP was administered at the beginning of the first semester to students in the Advanced, Regular and Fundamentals streams in 2001, 2002 and 2004, respectively. On the questionnaire, the students were requested to indicate their level of agreement with each item on a Likert scale with the options of Strongly Disagree, Disagree, Neutral, Agree, and Strongly Agree . Response rates of 2001, 2002, and 2004 cohorts was 95%, 65%, and 85%, respectively. Except 2002 cohort, the response rates were satisfactory. The main reasons of the lower response rate of 2002 cohort were the changes in class organization and questionnaire administration. Over these245Student approaches to learning in physicsC o p y r i g h t © 2006-2013 b y E S E Rthree years, a total of 2030 first year physics student were responded the SPQP: 63 percent of students in the Fundamentals stream was female, and about 30 percent of them was females in the Regular and Advanced streams. Nevertheless, the three streams are similar in other respects. The sample size of 2030 is large enough to access the natural variance within the diverse population. However, due to missing answers some of the cases were excluded from the analysis. These exclusions were only about 3% of the whole sample. Therefore, we can say that this missing data did not affect the overall results. Data Analysis MethodsFollowing analyses were carried out to answer research questions.(a) Both exploratory and confirmatory factor analyses were performed to validate the two-factor solution: the deep and surface scales. (b) Cronbach’s alpha coefficients were calculated to determine the reliability for the deep and surface scales for the complete data set and for each stream.(c) ANOVA and boxplots were used to determine if the SPQP is able to differentiate between the three streams and changes in syllabus.ResultsFactor analysisIn order to gain construct related evidence for the validity of the SPQP, exploratory and confirmatory factor analysis were conducted. Exploratory factor analysis (EFA) was carried out by using principal components as factor extraction method with quartimax, an orthogonal rotation. The complete data set was included in this analysis. Before proceeding to interpret the results, each item was checked for normality and sphericity. In order to check multicollinearity, the correlation matrix was examined. In terms of multicollinearity, we expect the items to be intercorrelated; however, these correlations should not be so high (0.90 or higher), whichcauses to multicollinearity and singularity. The intercorrelation was checked by Bartlett’s test of sphericity. This test showed that the correlation matrix is not an identity matrix. Moreover, multicollinearity was checked with the determinant of the correlation matrix. The determinant was more than 0. This showed that there is no multicollinearity (Field, 2000). Extraction of factors was based on two criteria: Scree test and Kaiser criterion (eigen values). Based on eigen values and the Scree test, two factors, which accounts for 48% of the variance, were extracted. The items with factor loadings of less than .4 were excluded from the further analyses (Field, 2000). Appendix A shows the two-factor solution for all items including loadings. Those that were retained for the SPQP are starred – 10 items form the deep scale and 6 items the surface scale.According to the results of the EFA, we note that the deep scale is in better agreement with Biggs ’s deep scale than the surface scale - there are more “usable” items on the deep scale than on the surface scale.After having results of the EFA, confirmatory factor analysis (CFA) was performed. This second step of the factor analysis helped us to ensure the factor structure of the SPQP (see Figure 1). Maximum likelihood (ML) was used as the method of estimation at the CFA. The results of the study showed that relative chi-square, which is the “chi square/degree of freedom” ratio is 3.1. Moreover, RMSEA and CFI were found to be 0.07 and 0.69 respectively. According to Browne and Cudeck (1992), RMSEA values less than 0.05 indicate close fit and models with values greater than 0.10 should not be employed. Here, RMSEA indicates moderate fit of the model whereas relative chi-square indicates good fit. However, the CFI should be over 0.90 to have good fit. Nonetheless, we can say that the first two indices support this two-factor model of the SPQP and indicate moderate fit.246C o p y r i g h t © 2006-2013 b y E S E RFigure 1. Validated two-factor structure of the SPQP.Reliability of the SPQPCronbach alpha coefficients of each scale were calculated for each stream and whole data. The results are shown in Table 2. It is apparent that the surface scale has the lowest Cronbach alpha coefficients at each stream. Similar findings were also reported at other studies (Biggs, 1987; Biggs et al, 2001; Wilson and Fowler, 2005). The foundational efficacy of these scales, given such low reliability, is questionable. However, in ourstudy higher levels of internal consistency were apparent (lowest α =.61).Comparing reliabilities across streams, students who have less experience with physics report surface approaches more reliably than students with more experience. On the other hand, students who have more experience with physics report deep approaches more reliably than those who have less experience. Considering reliabilities within streams, the Fundamentals students report deep approaches as reliably as surface247Student approaches to learning in physicsC o p y r i g h t © 2006-2013 b y E S E Rapproaches with values greater than 0.80, while the Advanced students report very different reliabilities for the two scales. The trends are not surprising since Advanced students would tend to be more confident in content and study strategies.The above trends also raise the question: Are the persistently low reliabilities noted for the su rface scale due to student ‘behaviours’ or poor items on the inventory? An adequate reliability measure for the surface scale for the Fundamentals stream α > .80, one that is similar in magnitude to that of the deep scale, implies that there is internal consistency amongst the items for each scale for this group of students. We note that the Fundamentals students have experienced junior high school physics, and are doing concurrent science and mathematics subjects at university. University students tend to have high internal coherence among learning components, intentions and study strategies and are able to adapt their ideas of knowledge and study methods to their expectations of studying in a particular context. The internal coherence is demonstrated in the reliability scores. So why are the reliabilities for the surface scale as low as 0.61 for the Advanced stream? Is it because the nature of the surface approach is different for the Advanced and Fundamentals streams, requiring possibly different items? Or is it because the Advanced students adapt their surface approaches in diverse ways, hence report this scale less reliably? Answers to such questions will indeed add to our understanding of student learning.ANOVA and BoxplotsTo determine if the SPQP is able to differentiate between the three streams, item and scale means were compared using one-way ANOVA.When comparing the means of the three streams for each item on the SPQP, the assumption of homogeneity of variances underpinning ANOVA was tested using Mulachy’s test of Sphericity. Items A5, A13 and A25 were excluded from ANOVA because they violated the assumption of sphericity. This does not affect their use on the SPQP scales. The results of ANOVA showed that there is a significant differenceamong the SPQP scores of the students from Fundamentals, Regular, and Advanced streams for both surface and deep scales (p < .05).There is a debate among the researchers to use ANOVA with the ordinal data mainly because of the normality concern. As stated in Glass, Peckham, and Sanders (1972), violation of normality is not fatal to the ANOVA. Moreover, performing ANOVA with the ordinal data, likert type items in this case, is a controversial issue among the researchers. Here, we have a big sample and this surely increases the power of the test. Therefore, failure to meet normality assumption will not affect the results of the ANOVA. Moreover, we performed Kruskal Wallis test as non-parametric alternative of the ANOVA. The results of that test supported the results of the ANOVA given above.Moreover, in order to investigate if SPQP is robust enough to be able to differentiate changes in syllabus even when sum of the items is used instead of factor scores, boxplots were checked (see Figure 2). The boxplots show a sequence for each scale with the first panel representing the factor scores, the second panel the simple sums of the items scores for the SPQP and the third panel the simple sums of all 16 item scores that should have loaded on each scale. We note two important features. First, the three panels representing the deep scale are sufficiently similar implying that if an adaptation such as that in this study is made, the sums of the 10 SPQP item scores, and indeed all 16 items scores provide a reasonable measure of deep approaches to learning. However, this is not so for the surface scale, while the sum of the 6 SPQP item scores provides a reasonable measure of surface approaches to learning, the sum of all 16 items does not do so. This raises concerns regarding the surface scale and is a reflection of the low reliabilities for the scale.DiscussionAs we are particularly interested in issues to do with learning physics, the rationale and manner in which items were modified and the SPQ adapted are discussed in detail. The advantages of adapting an established, well248C o p y r i g h t © 2006-2013 b y E S E RFigure 2. A comparison by stream and year of the deep and surface scales. The boxplots show a vertical sequence for each scale with panel (a) representing the factor scores for the SPQP deep scale and panel (d) those for the SPQP surface scale. Panel (b) represents the simple sum of item scores for the SPQP deep scale and panel (e) those for the SPQP surface scale. Panel (c) represents the simple sum of all 14 item scores that were intended to load on the deep scale and panel (f) those for the surface scale.implemented inventory with a sound theoretical framework, both for itsdevelopment and for its practical use in a teaching environment, are evident in the249Student approaches to learning in physicsC o p y r i g h t © 2006-2013 b y E S E Rmeaningful interpretations of our results summarized below.1. The SPQ items were modified for our context based on our experiences and any changes were extensively discussed amongst a group of physics educators. Ten items were retained for the deep scale and six for the surface. The rejection of items that had anomalous factor loadings could be conceptually justified. This two-factor solution of the SPQP confirmed with the EFA and CFA and supported the factor structure of the original SPQ (Biggs, 1987).2. The trends in reliabilities according to streams are as expected, with students with less experience in physics reporting less reliably on the deep scale and more reliably on the surface scale and vice versa. The issue of low reliabilities of the surface scale for the Advanced stream raises the question of whether Advanced students have more diverse forms of exhibiting surface approaches to learning. Moreover, the issue with the surface scale coincides with the previous studies (Biggs, 1987; Biggs et al, 2001; Wilson and Fowler, 2005).3. Comparisons of deep factor scores, simple sums of the 10 SPQP items and all 16 items suggest that the deep scale is reliable and particularly robust, see Figure 2. The surface factor scores compare well with the simple sums of the 6 SPQP items, but not with all 16 items, suggesting that reliability and validity checks are particularly important for the surface scale. The implication is twofold: first the SPQ is robust when contextualised as shown by reliability scores; and second, the contextualisation did not alter the overall coherency of the inventory as shown by the meaningful interpretations across streams and years. This, together with the conceptual meanings associated with the items, provides confidence that the SPQP is consistent with the theoretical framework of the SPQ.4. Changes in senior high school physics syllabus have impacted on approaches to study in the cohorts sampled in this study. The SPQP can illustrate differences between streams and years. From our study we are confident that the SPQP is a reliable andvalid measure of approaches to learning physics in our context.5. The adaptation of the SPQ into physics adds value to our project findings as it allows us to illustrate physics’ specific detail between the streams. We are confident that features that could have systematically biased the findings have been minimized. Lastly the ways of thinking, learning and knowing in physics are embedded in the larger context of intentions and study methods in higher education.ConclusionWe have adapted the Study Processes Questionnaire into physics and confirmed that a two-factor solution provides two subsets of selected items representing deep and surface approaches to learning. The resulting inventory is called the Study Processes Questionnaire for Physics, or SPQP. Further reliability and validation checks demonstrate that the two-scale SPQP is a useable inventory for our context. Reliabilities for the Advanced, Regular and Fundamentals streams are adequate and the behaviour of the mean scale scores for the three streams is not contradictory to expected student behaviours.The process of adapting the SPQ has provided useful insights into the way physicists interpret the items, and how deep and surface approaches can be conceptualised in physics. The sound theoretical framework and research underpinning the SPQ has added value to the use of questionnaires for understanding student learning in our project. Such contextualised inventories have the potential to provide context-specific understandings of teaching and learning issues and for improving student learning.AcknowledgementsThe authors acknowledge Science Faculty Education Research (SciFER) grants, University of Sydney, and support from staff and students. The authors are grateful to Professor David Boud for his constructive feedback on this paper.。
《神经网络与深度学习综述DeepLearning15May2014
Draft:Deep Learning in Neural Networks:An OverviewTechnical Report IDSIA-03-14/arXiv:1404.7828(v1.5)[cs.NE]J¨u rgen SchmidhuberThe Swiss AI Lab IDSIAIstituto Dalle Molle di Studi sull’Intelligenza ArtificialeUniversity of Lugano&SUPSIGalleria2,6928Manno-LuganoSwitzerland15May2014AbstractIn recent years,deep artificial neural networks(including recurrent ones)have won numerous con-tests in pattern recognition and machine learning.This historical survey compactly summarises relevantwork,much of it from the previous millennium.Shallow and deep learners are distinguished by thedepth of their credit assignment paths,which are chains of possibly learnable,causal links between ac-tions and effects.I review deep supervised learning(also recapitulating the history of backpropagation),unsupervised learning,reinforcement learning&evolutionary computation,and indirect search for shortprograms encoding deep and large networks.PDF of earlier draft(v1):http://www.idsia.ch/∼juergen/DeepLearning30April2014.pdfLATEX source:http://www.idsia.ch/∼juergen/DeepLearning30April2014.texComplete BIBTEXfile:http://www.idsia.ch/∼juergen/bib.bibPrefaceThis is the draft of an invited Deep Learning(DL)overview.One of its goals is to assign credit to those who contributed to the present state of the art.I acknowledge the limitations of attempting to achieve this goal.The DL research community itself may be viewed as a continually evolving,deep network of scientists who have influenced each other in complex ways.Starting from recent DL results,I tried to trace back the origins of relevant ideas through the past half century and beyond,sometimes using“local search”to follow citations of citations backwards in time.Since not all DL publications properly acknowledge earlier relevant work,additional global search strategies were employed,aided by consulting numerous neural network experts.As a result,the present draft mostly consists of references(about800entries so far).Nevertheless,through an expert selection bias I may have missed important work.A related bias was surely introduced by my special familiarity with the work of my own DL research group in the past quarter-century.For these reasons,the present draft should be viewed as merely a snapshot of an ongoing credit assignment process.To help improve it,please do not hesitate to send corrections and suggestions to juergen@idsia.ch.Contents1Introduction to Deep Learning(DL)in Neural Networks(NNs)3 2Event-Oriented Notation for Activation Spreading in FNNs/RNNs3 3Depth of Credit Assignment Paths(CAPs)and of Problems4 4Recurring Themes of Deep Learning54.1Dynamic Programming(DP)for DL (5)4.2Unsupervised Learning(UL)Facilitating Supervised Learning(SL)and RL (6)4.3Occam’s Razor:Compression and Minimum Description Length(MDL) (6)4.4Learning Hierarchical Representations Through Deep SL,UL,RL (6)4.5Fast Graphics Processing Units(GPUs)for DL in NNs (6)5Supervised NNs,Some Helped by Unsupervised NNs75.11940s and Earlier (7)5.2Around1960:More Neurobiological Inspiration for DL (7)5.31965:Deep Networks Based on the Group Method of Data Handling(GMDH) (8)5.41979:Convolution+Weight Replication+Winner-Take-All(WTA) (8)5.51960-1981and Beyond:Development of Backpropagation(BP)for NNs (8)5.5.1BP for Weight-Sharing Feedforward NNs(FNNs)and Recurrent NNs(RNNs)..95.6Late1980s-2000:Numerous Improvements of NNs (9)5.6.1Ideas for Dealing with Long Time Lags and Deep CAPs (10)5.6.2Better BP Through Advanced Gradient Descent (10)5.6.3Discovering Low-Complexity,Problem-Solving NNs (11)5.6.4Potential Benefits of UL for SL (11)5.71987:UL Through Autoencoder(AE)Hierarchies (12)5.81989:BP for Convolutional NNs(CNNs) (13)5.91991:Fundamental Deep Learning Problem of Gradient Descent (13)5.101991:UL-Based History Compression Through a Deep Hierarchy of RNNs (14)5.111992:Max-Pooling(MP):Towards MPCNNs (14)5.121994:Contest-Winning Not So Deep NNs (15)5.131995:Supervised Recurrent Very Deep Learner(LSTM RNN) (15)5.142003:More Contest-Winning/Record-Setting,Often Not So Deep NNs (16)5.152006/7:Deep Belief Networks(DBNs)&AE Stacks Fine-Tuned by BP (17)5.162006/7:Improved CNNs/GPU-CNNs/BP-Trained MPCNNs (17)5.172009:First Official Competitions Won by RNNs,and with MPCNNs (18)5.182010:Plain Backprop(+Distortions)on GPU Yields Excellent Results (18)5.192011:MPCNNs on GPU Achieve Superhuman Vision Performance (18)5.202011:Hessian-Free Optimization for RNNs (19)5.212012:First Contests Won on ImageNet&Object Detection&Segmentation (19)5.222013-:More Contests and Benchmark Records (20)5.22.1Currently Successful Supervised Techniques:LSTM RNNs/GPU-MPCNNs (21)5.23Recent Tricks for Improving SL Deep NNs(Compare Sec.5.6.2,5.6.3) (21)5.24Consequences for Neuroscience (22)5.25DL with Spiking Neurons? (22)6DL in FNNs and RNNs for Reinforcement Learning(RL)236.1RL Through NN World Models Yields RNNs With Deep CAPs (23)6.2Deep FNNs for Traditional RL and Markov Decision Processes(MDPs) (24)6.3Deep RL RNNs for Partially Observable MDPs(POMDPs) (24)6.4RL Facilitated by Deep UL in FNNs and RNNs (25)6.5Deep Hierarchical RL(HRL)and Subgoal Learning with FNNs and RNNs (25)6.6Deep RL by Direct NN Search/Policy Gradients/Evolution (25)6.7Deep RL by Indirect Policy Search/Compressed NN Search (26)6.8Universal RL (27)7Conclusion271Introduction to Deep Learning(DL)in Neural Networks(NNs) Which modifiable components of a learning system are responsible for its success or failure?What changes to them improve performance?This has been called the fundamental credit assignment problem(Minsky, 1963).There are general credit assignment methods for universal problem solvers that are time-optimal in various theoretical senses(Sec.6.8).The present survey,however,will focus on the narrower,but now commercially important,subfield of Deep Learning(DL)in Artificial Neural Networks(NNs).We are interested in accurate credit assignment across possibly many,often nonlinear,computational stages of NNs.Shallow NN-like models have been around for many decades if not centuries(Sec.5.1).Models with several successive nonlinear layers of neurons date back at least to the1960s(Sec.5.3)and1970s(Sec.5.5). An efficient gradient descent method for teacher-based Supervised Learning(SL)in discrete,differentiable networks of arbitrary depth called backpropagation(BP)was developed in the1960s and1970s,and ap-plied to NNs in1981(Sec.5.5).BP-based training of deep NNs with many layers,however,had been found to be difficult in practice by the late1980s(Sec.5.6),and had become an explicit research subject by the early1990s(Sec.5.9).DL became practically feasible to some extent through the help of Unsupervised Learning(UL)(e.g.,Sec.5.10,5.15).The1990s and2000s also saw many improvements of purely super-vised DL(Sec.5).In the new millennium,deep NNs havefinally attracted wide-spread attention,mainly by outperforming alternative machine learning methods such as kernel machines(Vapnik,1995;Sch¨o lkopf et al.,1998)in numerous important applications.In fact,supervised deep NNs have won numerous of-ficial international pattern recognition competitions(e.g.,Sec.5.17,5.19,5.21,5.22),achieving thefirst superhuman visual pattern recognition results in limited domains(Sec.5.19).Deep NNs also have become relevant for the more generalfield of Reinforcement Learning(RL)where there is no supervising teacher (Sec.6).Both feedforward(acyclic)NNs(FNNs)and recurrent(cyclic)NNs(RNNs)have won contests(Sec.5.12,5.14,5.17,5.19,5.21,5.22).In a sense,RNNs are the deepest of all NNs(Sec.3)—they are general computers more powerful than FNNs,and can in principle create and process memories of ar-bitrary sequences of input patterns(e.g.,Siegelmann and Sontag,1991;Schmidhuber,1990a).Unlike traditional methods for automatic sequential program synthesis(e.g.,Waldinger and Lee,1969;Balzer, 1985;Soloway,1986;Deville and Lau,1994),RNNs can learn programs that mix sequential and parallel information processing in a natural and efficient way,exploiting the massive parallelism viewed as crucial for sustaining the rapid decline of computation cost observed over the past75years.The rest of this paper is structured as follows.Sec.2introduces a compact,event-oriented notation that is simple yet general enough to accommodate both FNNs and RNNs.Sec.3introduces the concept of Credit Assignment Paths(CAPs)to measure whether learning in a given NN application is of the deep or shallow type.Sec.4lists recurring themes of DL in SL,UL,and RL.Sec.5focuses on SL and UL,and on how UL can facilitate SL,although pure SL has become dominant in recent competitions(Sec.5.17-5.22). Sec.5is arranged in a historical timeline format with subsections on important inspirations and technical contributions.Sec.6on deep RL discusses traditional Dynamic Programming(DP)-based RL combined with gradient-based search techniques for SL or UL in deep NNs,as well as general methods for direct and indirect search in the weight space of deep FNNs and RNNs,including successful policy gradient and evolutionary methods.2Event-Oriented Notation for Activation Spreading in FNNs/RNNs Throughout this paper,let i,j,k,t,p,q,r denote positive integer variables assuming ranges implicit in the given contexts.Let n,m,T denote positive integer constants.An NN’s topology may change over time(e.g.,Fahlman,1991;Ring,1991;Weng et al.,1992;Fritzke, 1994).At any given moment,it can be described as afinite subset of units(or nodes or neurons)N= {u1,u2,...,}and afinite set H⊆N×N of directed edges or connections between nodes.FNNs are acyclic graphs,RNNs cyclic.Thefirst(input)layer is the set of input units,a subset of N.In FNNs,the k-th layer(k>1)is the set of all nodes u∈N such that there is an edge path of length k−1(but no longer path)between some input unit and u.There may be shortcut connections between distant layers.The NN’s behavior or program is determined by a set of real-valued,possibly modifiable,parameters or weights w i(i=1,...,n).We now focus on a singlefinite episode or epoch of information processing and activation spreading,without learning through weight changes.The following slightly unconventional notation is designed to compactly describe what is happening during the runtime of the system.During an episode,there is a partially causal sequence x t(t=1,...,T)of real values that I call events.Each x t is either an input set by the environment,or the activation of a unit that may directly depend on other x k(k<t)through a current NN topology-dependent set in t of indices k representing incoming causal connections or links.Let the function v encode topology information and map such event index pairs(k,t)to weight indices.For example,in the non-input case we may have x t=f t(net t)with real-valued net t= k∈in t x k w v(k,t)(additive case)or net t= k∈in t x k w v(k,t)(multiplicative case), where f t is a typically nonlinear real-valued activation function such as tanh.In many recent competition-winning NNs(Sec.5.19,5.21,5.22)there also are events of the type x t=max k∈int (x k);some networktypes may also use complex polynomial activation functions(Sec.5.3).x t may directly affect certain x k(k>t)through outgoing connections or links represented through a current set out t of indices k with t∈in k.Some non-input events are called output events.Note that many of the x t may refer to different,time-varying activations of the same unit in sequence-processing RNNs(e.g.,Williams,1989,“unfolding in time”),or also in FNNs sequentially exposed to time-varying input patterns of a large training set encoded as input events.During an episode,the same weight may get reused over and over again in topology-dependent ways,e.g.,in RNNs,or in convolutional NNs(Sec.5.4,5.8).I call this weight sharing across space and/or time.Weight sharing may greatly reduce the NN’s descriptive complexity,which is the number of bits of information required to describe the NN (Sec.4.3).In Supervised Learning(SL),certain NN output events x t may be associated with teacher-given,real-valued labels or targets d t yielding errors e t,e.g.,e t=1/2(x t−d t)2.A typical goal of supervised NN training is tofind weights that yield episodes with small total error E,the sum of all such e t.The hope is that the NN will generalize well in later episodes,causing only small errors on previously unseen sequences of input events.Many alternative error functions for SL and UL are possible.SL assumes that input events are independent of earlier output events(which may affect the environ-ment through actions causing subsequent perceptions).This assumption does not hold in the broaderfields of Sequential Decision Making and Reinforcement Learning(RL)(Kaelbling et al.,1996;Sutton and Barto, 1998;Hutter,2005)(Sec.6).In RL,some of the input events may encode real-valued reward signals given by the environment,and a typical goal is tofind weights that yield episodes with a high sum of reward signals,through sequences of appropriate output actions.Sec.5.5will use the notation above to compactly describe a central algorithm of DL,namely,back-propagation(BP)for supervised weight-sharing FNNs and RNNs.(FNNs may be viewed as RNNs with certainfixed zero weights.)Sec.6will address the more general RL case.3Depth of Credit Assignment Paths(CAPs)and of ProblemsTo measure whether credit assignment in a given NN application is of the deep or shallow type,I introduce the concept of Credit Assignment Paths or CAPs,which are chains of possibly causal links between events.Let usfirst focus on SL.Consider two events x p and x q(1≤p<q≤T).Depending on the appli-cation,they may have a Potential Direct Causal Connection(PDCC)expressed by the Boolean predicate pdcc(p,q),which is true if and only if p∈in q.Then the2-element list(p,q)is defined to be a CAP from p to q(a minimal one).A learning algorithm may be allowed to change w v(p,q)to improve performance in future episodes.More general,possibly indirect,Potential Causal Connections(PCC)are expressed by the recursively defined Boolean predicate pcc(p,q),which in the SL case is true only if pdcc(p,q),or if pcc(p,k)for some k and pdcc(k,q).In the latter case,appending q to any CAP from p to k yields a CAP from p to q(this is a recursive definition,too).The set of such CAPs may be large but isfinite.Note that the same weight may affect many different PDCCs between successive events listed by a given CAP,e.g.,in the case of RNNs, or weight-sharing FNNs.Suppose a CAP has the form(...,k,t,...,q),where k and t(possibly t=q)are thefirst successive elements with modifiable w v(k,t).Then the length of the suffix list(t,...,q)is called the CAP’s depth (which is0if there are no modifiable links at all).This depth limits how far backwards credit assignment can move down the causal chain tofind a modifiable weight.1Suppose an episode and its event sequence x1,...,x T satisfy a computable criterion used to decide whether a given problem has been solved(e.g.,total error E below some threshold).Then the set of used weights is called a solution to the problem,and the depth of the deepest CAP within the sequence is called the solution’s depth.There may be other solutions(yielding different event sequences)with different depths.Given somefixed NN topology,the smallest depth of any solution is called the problem’s depth.Sometimes we also speak of the depth of an architecture:SL FNNs withfixed topology imply a problem-independent maximal problem depth bounded by the number of non-input layers.Certain SL RNNs withfixed weights for all connections except those to output units(Jaeger,2001;Maass et al.,2002; Jaeger,2004;Schrauwen et al.,2007)have a maximal problem depth of1,because only thefinal links in the corresponding CAPs are modifiable.In general,however,RNNs may learn to solve problems of potentially unlimited depth.Note that the definitions above are solely based on the depths of causal chains,and agnostic of the temporal distance between events.For example,shallow FNNs perceiving large“time windows”of in-put events may correctly classify long input sequences through appropriate output events,and thus solve shallow problems involving long time lags between relevant events.At which problem depth does Shallow Learning end,and Deep Learning begin?Discussions with DL experts have not yet yielded a conclusive response to this question.Instead of committing myself to a precise answer,let me just define for the purposes of this overview:problems of depth>10require Very Deep Learning.The difficulty of a problem may have little to do with its depth.Some NNs can quickly learn to solve certain deep problems,e.g.,through random weight guessing(Sec.5.9)or other types of direct search (Sec.6.6)or indirect search(Sec.6.7)in weight space,or through training an NNfirst on shallow problems whose solutions may then generalize to deep problems,or through collapsing sequences of(non)linear operations into a single(non)linear operation—but see an analysis of non-trivial aspects of deep linear networks(Baldi and Hornik,1994,Section B).In general,however,finding an NN that precisely models a given training set is an NP-complete problem(Judd,1990;Blum and Rivest,1992),also in the case of deep NNs(S´ıma,1994;de Souto et al.,1999;Windisch,2005);compare a survey of negative results(S´ıma, 2002,Section1).Above we have focused on SL.In the more general case of RL in unknown environments,pcc(p,q) is also true if x p is an output event and x q any later input event—any action may affect the environment and thus any later perception.(In the real world,the environment may even influence non-input events computed on a physical hardware entangled with the entire universe,but this is ignored here.)It is possible to model and replace such unmodifiable environmental PCCs through a part of the NN that has already learned to predict(through some of its units)input events(including reward signals)from former input events and actions(Sec.6.1).Its weights are frozen,but can help to assign credit to other,still modifiable weights used to compute actions(Sec.6.1).This approach may lead to very deep CAPs though.Some DL research is about automatically rephrasing problems such that their depth is reduced(Sec.4). In particular,sometimes UL is used to make SL problems less deep,e.g.,Sec.5.10.Often Dynamic Programming(Sec.4.1)is used to facilitate certain traditional RL problems,e.g.,Sec.6.2.Sec.5focuses on CAPs for SL,Sec.6on the more complex case of RL.4Recurring Themes of Deep Learning4.1Dynamic Programming(DP)for DLOne recurring theme of DL is Dynamic Programming(DP)(Bellman,1957),which can help to facili-tate credit assignment under certain assumptions.For example,in SL NNs,backpropagation itself can 1An alternative would be to count only modifiable links when measuring depth.In many typical NN applications this would not make a difference,but in some it would,e.g.,Sec.6.1.be viewed as a DP-derived method(Sec.5.5).In traditional RL based on strong Markovian assumptions, DP-derived methods can help to greatly reduce problem depth(Sec.6.2).DP algorithms are also essen-tial for systems that combine concepts of NNs and graphical models,such as Hidden Markov Models (HMMs)(Stratonovich,1960;Baum and Petrie,1966)and Expectation Maximization(EM)(Dempster et al.,1977),e.g.,(Bottou,1991;Bengio,1991;Bourlard and Morgan,1994;Baldi and Chauvin,1996; Jordan and Sejnowski,2001;Bishop,2006;Poon and Domingos,2011;Dahl et al.,2012;Hinton et al., 2012a).4.2Unsupervised Learning(UL)Facilitating Supervised Learning(SL)and RL Another recurring theme is how UL can facilitate both SL(Sec.5)and RL(Sec.6).UL(Sec.5.6.4) is normally used to encode raw incoming data such as video or speech streams in a form that is more convenient for subsequent goal-directed learning.In particular,codes that describe the original data in a less redundant or more compact way can be fed into SL(Sec.5.10,5.15)or RL machines(Sec.6.4),whose search spaces may thus become smaller(and whose CAPs shallower)than those necessary for dealing with the raw data.UL is closely connected to the topics of regularization and compression(Sec.4.3,5.6.3). 4.3Occam’s Razor:Compression and Minimum Description Length(MDL) Occam’s razor favors simple solutions over complex ones.Given some programming language,the prin-ciple of Minimum Description Length(MDL)can be used to measure the complexity of a solution candi-date by the length of the shortest program that computes it(e.g.,Solomonoff,1964;Kolmogorov,1965b; Chaitin,1966;Wallace and Boulton,1968;Levin,1973a;Rissanen,1986;Blumer et al.,1987;Li and Vit´a nyi,1997;Gr¨u nwald et al.,2005).Some methods explicitly take into account program runtime(Al-lender,1992;Watanabe,1992;Schmidhuber,2002,1995);many consider only programs with constant runtime,written in non-universal programming languages(e.g.,Rissanen,1986;Hinton and van Camp, 1993).In the NN case,the MDL principle suggests that low NN weight complexity corresponds to high NN probability in the Bayesian view(e.g.,MacKay,1992;Buntine and Weigend,1991;De Freitas,2003), and to high generalization performance(e.g.,Baum and Haussler,1989),without overfitting the training data.Many methods have been proposed for regularizing NNs,that is,searching for solution-computing, low-complexity SL NNs(Sec.5.6.3)and RL NNs(Sec.6.7).This is closely related to certain UL methods (Sec.4.2,5.6.4).4.4Learning Hierarchical Representations Through Deep SL,UL,RLMany methods of Good Old-Fashioned Artificial Intelligence(GOFAI)(Nilsson,1980)as well as more recent approaches to AI(Russell et al.,1995)and Machine Learning(Mitchell,1997)learn hierarchies of more and more abstract data representations.For example,certain methods of syntactic pattern recog-nition(Fu,1977)such as grammar induction discover hierarchies of formal rules to model observations. The partially(un)supervised Automated Mathematician/EURISKO(Lenat,1983;Lenat and Brown,1984) continually learns concepts by combining previously learnt concepts.Such hierarchical representation learning(Ring,1994;Bengio et al.,2013;Deng and Yu,2014)is also a recurring theme of DL NNs for SL (Sec.5),UL-aided SL(Sec.5.7,5.10,5.15),and hierarchical RL(Sec.6.5).Often,abstract hierarchical representations are natural by-products of data compression(Sec.4.3),e.g.,Sec.5.10.4.5Fast Graphics Processing Units(GPUs)for DL in NNsWhile the previous millennium saw several attempts at creating fast NN-specific hardware(e.g.,Jackel et al.,1990;Faggin,1992;Ramacher et al.,1993;Widrow et al.,1994;Heemskerk,1995;Korkin et al., 1997;Urlbe,1999),and at exploiting standard hardware(e.g.,Anguita et al.,1994;Muller et al.,1995; Anguita and Gomes,1996),the new millennium brought a DL breakthrough in form of cheap,multi-processor graphics cards or GPUs.GPUs are widely used for video games,a huge and competitive market that has driven down hardware prices.GPUs excel at fast matrix and vector multiplications required not only for convincing virtual realities but also for NN training,where they can speed up learning by a factorof50and more.Some of the GPU-based FNN implementations(Sec.5.16-5.19)have greatly contributed to recent successes in contests for pattern recognition(Sec.5.19-5.22),image segmentation(Sec.5.21), and object detection(Sec.5.21-5.22).5Supervised NNs,Some Helped by Unsupervised NNsThe main focus of current practical applications is on Supervised Learning(SL),which has dominated re-cent pattern recognition contests(Sec.5.17-5.22).Several methods,however,use additional Unsupervised Learning(UL)to facilitate SL(Sec.5.7,5.10,5.15).It does make sense to treat SL and UL in the same section:often gradient-based methods,such as BP(Sec.5.5.1),are used to optimize objective functions of both UL and SL,and the boundary between SL and UL may blur,for example,when it comes to time series prediction and sequence classification,e.g.,Sec.5.10,5.12.A historical timeline format will help to arrange subsections on important inspirations and techni-cal contributions(although such a subsection may span a time interval of many years).Sec.5.1briefly mentions early,shallow NN models since the1940s,Sec.5.2additional early neurobiological inspiration relevant for modern Deep Learning(DL).Sec.5.3is about GMDH networks(since1965),perhaps thefirst (feedforward)DL systems.Sec.5.4is about the relatively deep Neocognitron NN(1979)which is similar to certain modern deep FNN architectures,as it combines convolutional NNs(CNNs),weight pattern repli-cation,and winner-take-all(WTA)mechanisms.Sec.5.5uses the notation of Sec.2to compactly describe a central algorithm of DL,namely,backpropagation(BP)for supervised weight-sharing FNNs and RNNs. It also summarizes the history of BP1960-1981and beyond.Sec.5.6describes problems encountered in the late1980s with BP for deep NNs,and mentions several ideas from the previous millennium to overcome them.Sec.5.7discusses afirst hierarchical stack of coupled UL-based Autoencoders(AEs)—this concept resurfaced in the new millennium(Sec.5.15).Sec.5.8is about applying BP to CNNs,which is important for today’s DL applications.Sec.5.9explains BP’s Fundamental DL Problem(of vanishing/exploding gradients)discovered in1991.Sec.5.10explains how a deep RNN stack of1991(the History Compressor) pre-trained by UL helped to solve previously unlearnable DL benchmarks requiring Credit Assignment Paths(CAPs,Sec.3)of depth1000and more.Sec.5.11discusses a particular WTA method called Max-Pooling(MP)important in today’s DL FNNs.Sec.5.12mentions afirst important contest won by SL NNs in1994.Sec.5.13describes a purely supervised DL RNN(Long Short-Term Memory,LSTM)for problems of depth1000and more.Sec.5.14mentions an early contest of2003won by an ensemble of shallow NNs, as well as good pattern recognition results with CNNs and LSTM RNNs(2003).Sec.5.15is mostly about Deep Belief Networks(DBNs,2006)and related stacks of Autoencoders(AEs,Sec.5.7)pre-trained by UL to facilitate BP-based SL.Sec.5.16mentions thefirst BP-trained MPCNNs(2007)and GPU-CNNs(2006). Sec.5.17-5.22focus on official competitions with secret test sets won by(mostly purely supervised)DL NNs since2009,in sequence recognition,image classification,image segmentation,and object detection. Many RNN results depended on LSTM(Sec.5.13);many FNN results depended on GPU-based FNN code developed since2004(Sec.5.16,5.17,5.18,5.19),in particular,GPU-MPCNNs(Sec.5.19).5.11940s and EarlierNN research started in the1940s(e.g.,McCulloch and Pitts,1943;Hebb,1949);compare also later work on learning NNs(Rosenblatt,1958,1962;Widrow and Hoff,1962;Grossberg,1969;Kohonen,1972; von der Malsburg,1973;Narendra and Thathatchar,1974;Willshaw and von der Malsburg,1976;Palm, 1980;Hopfield,1982).In a sense NNs have been around even longer,since early supervised NNs were essentially variants of linear regression methods going back at least to the early1800s(e.g.,Legendre, 1805;Gauss,1809,1821).Early NNs had a maximal CAP depth of1(Sec.3).5.2Around1960:More Neurobiological Inspiration for DLSimple cells and complex cells were found in the cat’s visual cortex(e.g.,Hubel and Wiesel,1962;Wiesel and Hubel,1959).These cellsfire in response to certain properties of visual sensory inputs,such as theorientation of plex cells exhibit more spatial invariance than simple cells.This inspired later deep NN architectures(Sec.5.4)used in certain modern award-winning Deep Learners(Sec.5.19-5.22).5.31965:Deep Networks Based on the Group Method of Data Handling(GMDH) Networks trained by the Group Method of Data Handling(GMDH)(Ivakhnenko and Lapa,1965; Ivakhnenko et al.,1967;Ivakhnenko,1968,1971)were perhaps thefirst DL systems of the Feedforward Multilayer Perceptron type.The units of GMDH nets may have polynomial activation functions imple-menting Kolmogorov-Gabor polynomials(more general than traditional NN activation functions).Given a training set,layers are incrementally grown and trained by regression analysis,then pruned with the help of a separate validation set(using today’s terminology),where Decision Regularisation is used to weed out superfluous units.The numbers of layers and units per layer can be learned in problem-dependent fashion. This is a good example of hierarchical representation learning(Sec.4.4).There have been numerous ap-plications of GMDH-style networks,e.g.(Ikeda et al.,1976;Farlow,1984;Madala and Ivakhnenko,1994; Ivakhnenko,1995;Kondo,1998;Kord´ık et al.,2003;Witczak et al.,2006;Kondo and Ueno,2008).5.41979:Convolution+Weight Replication+Winner-Take-All(WTA)Apart from deep GMDH networks(Sec.5.3),the Neocognitron(Fukushima,1979,1980,2013a)was per-haps thefirst artificial NN that deserved the attribute deep,and thefirst to incorporate the neurophysiolog-ical insights of Sec.5.2.It introduced convolutional NNs(today often called CNNs or convnets),where the(typically rectangular)receptivefield of a convolutional unit with given weight vector is shifted step by step across a2-dimensional array of input values,such as the pixels of an image.The resulting2D array of subsequent activation events of this unit can then provide inputs to higher-level units,and so on.Due to massive weight replication(Sec.2),relatively few parameters may be necessary to describe the behavior of such a convolutional layer.Competition layers have WTA subsets whose maximally active units are the only ones to adopt non-zero activation values.They essentially“down-sample”the competition layer’s input.This helps to create units whose responses are insensitive to small image shifts(compare Sec.5.2).The Neocognitron is very similar to the architecture of modern,contest-winning,purely super-vised,feedforward,gradient-based Deep Learners with alternating convolutional and competition lay-ers(e.g.,Sec.5.19-5.22).Fukushima,however,did not set the weights by supervised backpropagation (Sec.5.5,5.8),but by local un supervised learning rules(e.g.,Fukushima,2013b),or by pre-wiring.In that sense he did not care for the DL problem(Sec.5.9),although his architecture was comparatively deep indeed.He also used Spatial Averaging(Fukushima,1980,2011)instead of Max-Pooling(MP,Sec.5.11), currently a particularly convenient and popular WTA mechanism.Today’s CNN-based DL machines profita lot from later CNN work(e.g.,LeCun et al.,1989;Ranzato et al.,2007)(Sec.5.8,5.16,5.19).5.51960-1981and Beyond:Development of Backpropagation(BP)for NNsThe minimisation of errors through gradient descent(Hadamard,1908)in the parameter space of com-plex,nonlinear,differentiable,multi-stage,NN-related systems has been discussed at least since the early 1960s(e.g.,Kelley,1960;Bryson,1961;Bryson and Denham,1961;Pontryagin et al.,1961;Dreyfus,1962; Wilkinson,1965;Amari,1967;Bryson and Ho,1969;Director and Rohrer,1969;Griewank,2012),ini-tially within the framework of Euler-LaGrange equations in the Calculus of Variations(e.g.,Euler,1744). Steepest descent in such systems can be performed(Bryson,1961;Kelley,1960;Bryson and Ho,1969)by iterating the ancient chain rule(Leibniz,1676;L’Hˆo pital,1696)in Dynamic Programming(DP)style(Bell-man,1957).A simplified derivation of the method uses the chain rule only(Dreyfus,1962).The methods of the1960s were already efficient in the DP sense.However,they backpropagated derivative information through standard Jacobian matrix calculations from one“layer”to the previous one, explicitly addressing neither direct links across several layers nor potential additional efficiency gains due to network sparsity(but perhaps such enhancements seemed obvious to the authors).。
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Artificial Intelligence in Education
Toward supporting hypothesis formation andtesting in an interpretive domainVincent Aleven 1and Kevin Ashley 21Human-Computer Interaction InstituteCarnegie Mellon University2Intelligent Systems Program,Learning Research and Development CenterUniversity of PittsburghAbstr act .The resea rch field of AI &Educa tion has long been inter ested in cognitiveproce sses in which stude nts formulate and test hypotheses by considering them inlight of specific cases.However,few if any of the syste ms that have been built targe tdomains which are ill-structured and in which deter mining whether a hypothesizedrule and proposed outcome are consistent with past decisions is a matte r ofinter pretation,rathe r than deduc tive infer ence.The goals of our proje ct are to (1)develop an AI model of hypothesis forma tion and testing in an inter pretive domain,US Supre me Court oral arguments and (2)to use it in an intelligent tutor ing syste mto guide law stude nts in learning that proce ss.As a first step towar d these goals wewill conduct an exper iment to evaluate whether self-explanation prompts facilitatelearning by studying argument transcripts.IntroductionThe research field of AI &Education has long been interested in processes of inquiry learning in which students formulate hypotheses and test them against specific cases ([1-5]).These processes as they occur in ill-structured domains,however,have received little attention, one exception being the work on CATO [6-8].Oral arguments before the United States Supreme Court offer prime examples of hypothesis formulation and testing in an ill-structured domain. In these arguments, advocates frame hypotheses for deciding a case and the Justices challenge them,often by posing hypothetical scenarios that test the hypotheses’limits.While these processes in the legal domain bear some resemblance to the corresponding processes in science or mathematics, in the legal domain determining whether a hypothesized rule and proposed outcome are consistent with past decisions and plausible hypotheticals is much more a matter of interpretation.The goals of our project are (1)to develop a computational model of the reasoning processes exemplified in US Supreme Court oral argument and (2)to use the model as the basis for an intelligent tutoring system that will engage students in an appropriately simplified version of these processes.A planned experimentAs a first step,we will run an experiment to find out whether specific prompts for self-explanation help students gain a deeper understanding,as they study transcripts of Supreme Court oral arguments.The cognitive science literature supports that studying examples is an effective learning strategy at the early stages of acquiring a cognitive skill [9]and that self-explanation prompts can help students gain a deeper understanding of the subject matter ([10-12]).However,the effectiveness of prompts has not yet been shown in ill-structured domains as complex as the legal reasoning exemplified in Supreme Court oralArtificial Intelligence in EducationC.-K. Looi et al. (Eds.)IOS Press, 2005© 2005 The authors. All rights reserved.732Table1:Excerpt of transcript of oral argument made before the US Supreme Court in California v.Carney,105S.Ct.2066(1985),with self-explanation prompts addedArgument Transcript Self-Explanation PromptsQUESTION:Well,what if the vehicle is inone of these mobile home parks and hookedup to water and electricity but still has itswheels on?MR.HANOIAN:[*9]If it still has itswheels and it still has its engine,it iscapable of movement and it is capable ofmovement very quickly.1.Do you think H's response is effective?QUESTION:Even though the people areliving in it as a home and are paying rent forthe trailer space,and so forth?2.Why are the Justices adding these features to the hypothetical?QUESTION: Well, there are places where people can plug into water,and electricity, and do.There are many places,for example, in the state I came from where people go and spend the winter in a mobile home.And you think there would be no expectation of privacy in such circumstances?3.Why does it matter whether there would be expectations ofprivacy?4.If it was clear that there is,or should be,a high expectation ofprivacy in the current fact situation,would that favor H'sposition?5.Nothing is said from which we can infer how this particularhypothetical should be decided.Does that matter?That is,what good is it to use hypotheticals whose outcome is unknown?Wouldn't it be better to cite past cases,whose outcome we do know?MR.HANOIAN:Well,I am not suggesting that there is no expectation of privacy in those circumstances,Your Honor.6.By conceding that there are expectations of privacy in thehypothetical scenarios sketched by the judges,does H notreduce his chances of winning the case at hand?7.Does H concede that the mobile home park hypothetical shouldhave the opposite result as the case at hand?8.How would H distinguish the current case from the mobilehome park hypothetical?arguments.In light of the evidence that prompts do not benefit all students equally([10, 11]), it is important to ask how effective prompts are in such challenging domains.Table1shows excerpts from oral arguments made in the case of California v. Carney,105S.Ct.2066(1985),with self-explanation prompts inserted.This case involved the legality under the4th Amendment of the US Constitution of a warrantless search of a motor home located in a downtown San Diego parking lot.Police suspected defendant Carney of trading marijuana for sex acts.After they questioned a boy leaving Carney’s motor home,agents entered the motor home without a warrant or Carney’s consent, observed marijuana,and arrested Carney.The case pitted two conflicting principles:the State’s right to deal effectively with the exigent possibility that evidence of a crime will disappear versus the citizen’s constitutionally protected expectation of autonomy and privacy in his home.In the oral argument,the State’s attorney,Mr.Hanoian,proposed a bright line test: if the vehicle/home is capable of self-locomotion, then no warrant is required to search it.As shown in Table1,he then has to respond to the Justice’s challenge hypothetical:what result would his test produce when applied to a summer motor home with wheels that is hooked up to utilities?Mr.Hanoian responds that such a vehicle still might be moved in a hurry,but concedes the owners would have some expectation of privacy. Some of the self-explanation prompts focus on the effectiveness of that response. Others focus on the Justices’strategies and possible reasons for posing hypotheticals. DiscussionIn order to evaluate the effect of the self-explanation prompts,the study will compare the learning results of students studying argument transcripts with and without self-explanation V.Aleven and K.Ashley/Toward Supporting Hypothesis Formation and Testing733prompts.A pilot study involving two law students,a first-year student and a second-year student,provided some evidence that the prompts are useful.The students went through the Carney transcript twice,the first time without self-explanation prompts,the second time with.Each time,they were asked to answer a number of questions about the argument exchange they had just studied.We saw a difference in the quality of the answers between the first-year and the second-year student,indicating that the material is challenging.Further,we saw that the answers of the first-year student improved,after studying the transcript with the self-explanation prompts.Of course,such evidence is preliminary,due to the “small N”.Also,the improvement in the answers could be attributed simply to the fact that the student went through the transcript twice.This confound will be avoided in the actual experiment by having a control group.We are currently working on developing a suitable task by which we can measure any improvement in students’argument-making capabilities, a preliminary challenge for any research in an ill-structured domain.We expect the study to yield information about how students understand and make arguments. This information will help us start to build an argument model and develop an intelligent tutoring system.The study will also contribute to cognitive science by testing whether specific self-explanation prompts can help students to learn to engage in a process of hypothesis formation and testing in an ill-structured domain.AcknowledgementsThis research is sponsored by NSF Award IIS-0412830.The contents of the paper are solely the responsibility of the authors and do not necessarily represent the official views of the NSF.References[1].Bunt,A.,C.Conati,and K.Muldner,Scaffolding Self -Explanation to Impr ove Learning in Exploratory Learning Environments,in Proce edings of the 7th International Conference on Inte lligent TutoringSyste ms,ITS 2004,J.Lester,R.M.Vicar io,and F.Paraguaçu,Editors.2004,Springer:Berlin.[2].Collins,A.and A.L.Stevens,Goals and Strategies of I nquiry Tea chers,in Advances in InstructionalPsychology,R.G laser,Editor.1982,Lawrence Erlbaum:H illsdale,NJ.p.65-119.[3].Murra y,T.,L.Winship,and N.Stillings,Evaluation of the SimForest Inquiry Learning Environment:Inquiry Cycles and Collaborative Te aching Pra ctices,in AERA.2004:San Diego,CA.[4].Shute ,V.J.and R.Glaser ,A Large-Scale Evaluation of an Intelligent Discovery World:Smithtow n.Interactive Learning Environments,1990.1:p.51-77.[5].Woolf ,B.P.,et al.,Trac king Stude nt Propositions in a n Inquiry System,in Proce edings of the 11thInternational Conference on Artific ial Intelligence in Education,AIED 2003,U.H oppe,F.Verde jo,and J.Ka y,Editors.2003,IO S Press:A msterdam.p.21-28.[6].Aleve n,V.,Using Background Knowle dge in Case-Based Le gal Reasoning:A Computationl Model andan Intelligent Learning Environment.Artificial Inte lligence,2003.150:p.183-238.[7].Aleve n,V.and K.D.Ashle y,Teaching Case-Based Argumentation Thr ough a Model and Exa mples:Empir ical Evaluation of a n Intelligent Learning Environment,in Proce edings of the 8th InternationalConfe rence on Artificial Intelligence and Education,AI -ED '97,B.du Boula y and R.Mizoguchi,Editors.1997,IOS Press:Amsterda m.p.87-94.[8].Ashle y,K.D.,R.Desai,a nd J.Levine,Teaching Case-Ba sed Argume ntation Concepts Using DialecticArguments vs.D idactic Explanations,in Proce edings of the Sixth International Conference on Intelligent Tutoring System s,ITS 2002,S.A.Cerri,G.Gouar dères,and F.Paraguaçu,Editors.2002,Spinger:Berlin.p.585-595.[9].Atkinson,R.K.,et al.,Learning from e xamples:I nstructional principles from the worked examplesresea rch.Revie w of Educational Research,2000.70(2):p.181-214.[10].Chi,M.T.H.,et al.,Elic iting Self -Explanations Improves Understanding.Cognitive Science,1994.18:p.439-477.[11].Renkl,A.,et a l.,Learning from Worked-Out Examples:the Effects of Example Variability and ElicitedSelf-Explanations.Conte mporary Educational Psychology ,1998.23:p.90-108.[12].Schworm,S.and A.Renkl,Learning by solved example problems:I nstructional explana tions reduce self-expla nation activity,in Proce eding of the 24th Annual Confe rence of the Cognitive Science Society,W.D.Gray and C.D.Schunn,Editors.2002,La wrence Erlbaum:Mahw ah,NJ.p.816-821.V .Aleven and K.Ashley /Toward Supporting Hypothesis Formation and Testing734。
the rising sea 代数几何 发表
the rising sea 代数几何发表The Rising Sea: Algebraic Geometry in Modern ApplicationsIn the field of mathematics, algebraic geometry is a vibrant and rapidly growing discipline. It merges geometry and algebra, creating a powerful toolset for understanding and manipulating complex mathematical structures. The Rising Sea: Algebraic Geometry in Modern Applications is a groundbreaking textbook that delves into the depth and breadth of this fascinating field.The book opens with a discussion of the fundamental principles and building blocks of algebraic geometry. It introduces the reader to the basic concepts of polynomial rings, ideals, and varieties. The explanations are clear and accessible, laying a solid foundation for the more advanced topics that follow.The Rising Sea then delves into more advanced topics, such as abstract algebra, commutative algebra, and scheme theory. The text carefully explains these concepts, providing numerous examples and exercises to aid comprehension. It also introduces the reader to the powerful tools of homological algebra and algebraic geometry, further enriching their understanding of the subject.The book’s scope extends beyond the theoretical aspects of algebraic geometry. It also includes chapters on the applications of algebraic geometry in various fields, such as computer science, physics, and economics. These applications demonstrate the practical relevance and impact of algebraic geometry in modern society.The Rising Sea is an essential resource for students and researchers in mathematics, computer science, and related fields. Its clear and comprehensive approach makes it an excellent textbook for undergraduate and graduate courses on algebraic geometry. It will also serve as a valuable reference for professionals in industry and academia who seek a deeper understanding of this rapidly developing field.。
Linear_differential_equations_pdf_
Pre-requisites:throughout this chapter,the following basic properties and notions are assumed to be known:∙derivatives and antiderivatives;∙usual functions and in particular,their derivatives;∙complex numbers;roots of a second degree polynomial.Contents1General definitions and standard vocabulary2 2First order linear differential equations32.1Exponential functions and their characterization (4)2.2Structure of the set of solutions and sum principle (6)2.3Solutions to the associated homogeneous equation (7)2.4Variation of parameters (8)2.5Solution satisfying a given initial condition:existence and uniqueness (14)3Second order linear differential equations with constant coefficients153.1Definition and structure of the set of solutions (15)3.2Solutions to the associated homogeneous equation (16)3.3Finding a particular solution when the second member is of type exponential-polynomial (20)3.4Variation of parameters (22)3.5Solution satisfying a given initial condition:existence and uniqueness (24)1General definitions and standard vocabularyThroughout this chapter,unless specified otherwise, will be used to state results that are valid with either =ℝor =ℂ.Definition 1.0.1Let be a positive integer.A differential equation of order is an equation in which the unknown is a function (with domain (to be determined)and (at least) times differentiable on )and of the form:( ): ( , , ′,⋅⋅⋅, ( −1), ( ))=0where for each ∈[∣0, ]∣, ( )is the -th derivative of and is a function of ( +2)variables,and is not constant with respect to the last variable.Example 1.0.1You have probably already studied a number of differential equations,in particular during your physics class.Here a few examples of differential equations:∙ ′+2 =0is a (linear)differential equation of order 1(setting ( , , ′)= ′+2 );∙ ′( )− ( )− 2=0is a (linear)differential equation or order 1;∙ ′=1+ 2is a also a differential equation of order 1(but non linear);∙ ′′+ 2 =0is a (linear)differential equation of order 2;∙ (6)− (3)+2 2=cos( )is a (non linear)differential equation of order 6.Remark:the condition “which is not constant with respect to the last variable”is a technical condition.It is there to guarantee that the equation is really of order .For instance,if we set: ( , , , )= + then the differential equation ( , , ′, ′′)=0is + ′=0,which is really of order 1and not 2.Definition 1.0.2We say a function with domain (a non trivial interval)is a solution to ( )if is at least times differentiable on and:∀ ∈ , ( , ( ), ′( ),⋅⋅⋅, ( )( ))=0Therefore,a solution to the differential equation ( )is really a couple ( , ),where is a real interval and a function at least times differentiable on .Example 1.0.2The function with domain [0;1]given by ( )=exp( )for all ∈[0;1]is a solution of ′= .However,it is clearly not the only one as we could define for any interval , ( )=exp( )for ∈ which would also be a solution to the same differential equation,but on .This is why,in our search for solutions,we focus on solutions with maximal domain.Definition 1.0.3To solve the differential equation ( )is to find all maximal solutions,i.e.find all solutions whose domain’s are maximal (that is,there is no solution to ( )with domain such that ⊊ and ∣ = ).Example 1.0.3In our previous example,the maximal solutions of ′= are the functions with domain ℝgiven by ( )= exp( )for all real number .Obviously,there are a number of differential equations we do not know how to solve explicitly.This is where numerical analysis can be very convenient:there are many methods to give approximations of solutions,even though we do not know an exact form.We will study this later in the year.However,there are one kind of differential equations for which,under“reasonable conditions”,we always know how to solve explicitly:namely,linear equations.The object of the next sections is to study these equations,but only when their order is either1or2.Before we do so,we will give a formal definition of what a linear differential equation actually is.Definition1.0.4A differential equation is said to be linear if it is of the form: ( )− =0,where is the unknown, is a function and is a linear map,i.e. satisfies:( 1+ 2)= ( 1)+ ( 2)and ( 1)= ( 1)for all“unknowns” 1and 2and all scalar constants ∈ .In that case,we say that:∙ is the second member of( );∙the differential equation( ): ( )=0is the homogeneous equation associated to( ).Remark:it is often convenient to group both properties in the definition of a linear map.More specifically, is a linear map if and only if,for all unknowns 1, 2and all , ∈ , ( 1+ 2)= 1+ 2.Example1.0.4Setting ( )= ′− ,we have for all unknowns 1and 2,and all , ∈ :( 1+ 2)=( 1+ 2)′−( 1+ 2)= ′1+ ′2− 1− 2= ( ′1− 1)+ ( ′2− 2)= ( 1)+ ( 2)Therefore, ′− =0is indeed a linear differential equation.Example1.0.5We stated earlier that ′=1+ 2is not a linear differential equation.Indeed,we have here: ′− 2=1 so that ( )= ′− 2and =1(constant function).However, (2 )=2 ′−4 2=2( ′− 2)if =0.Therefore, is not a linear map(the condition must be verified for all unknowns).2First order linear differential equationsDefinition2.0.5A linear differential equation of order1is a differential equation of the form:( ) ′( )+ ( ) ( )= ( )where , and are three functions defined on a real interval .We will only work withfirst order linear differential equations which are“resolved”in ′,i.e.of the form:′( )+ ( ) ( )= ( )Remark:often,we write ′+ ( ) = ( )which theoretically is abusive.The proper forms would either be ′( )+ ( ) ( )= ( )for all in a certain interval or ′+ = (equality amongst functions).However,this abuse is currently accepted and we will always write the equations in that form.Proposition2.0.1Afirst order linear differential equation is indeed a linear differential equation.Hence,the name.Proof:To begin,we observe that the equation can be written in the form: ( )= ,where =and ( )= ( ) ′+ ( ) (again,we should write ( ): −→ ( ) ′( )+ ( ) ( )).Now,let 1and 2be two unknowns and let , be two scalars in .We then have:( 1+ 2)= ( )( 1+ 2)′+ ( )( 1+ 2)= ( )( ′1+ ′2)+ ( )( 1+ 2)= ( ( ) ′1+ ( ) 1)+ ( ( ) ′2+ ( ) 2)= ( 1)+ ( 2)Thus, is a linear map and the equation is indeed linear.⊠2.1Exponential functions and their characterizationTheorem2.1.1Let ∈ .Then the function :ℝ−→−→exp( )is the only(maximal)solution of ′=that satisfies the initial condition (0)=1.Proof:Suppose that is a solution of ′= that verifies (0)=1and let ( )= ( )exp(− ).Then is differentiable onℝ(product of two functions that are)and for all real number ,′( )= ′( )exp(− )− ( )exp(− )=exp(− )( ′( )− ( ))=0Hence, is constant onℝ.Furthermore, (0)= (0)exp(0)=1×1=1so that for all ∈ℝ,( )exp(− )=1.Multiplying both sides by exp( )=0,wefind that: ( )=exp( )forall real number .This proves that if is a solution of ′= that satisfies (0)=1then= .Conversely,one can easily check that satisfies ′ = and (0)=1.⊠Remark:this is one way of characterizing the exponential functions(real or complex)as the only solutions of ′= with initial condition (0)=1.Theorem2.1.2(Second characterization)The only functions differentiable onℝthat verify:∀ , ∈ℝ, ( + )= ( )× ( )(∗)are either exponential functions with ∈ℂor identically zero.Proof:⇐=The identically zero function and exponential functions obviously are differentiable onℝand verify(∗).=⇒Let be a function which is differentiable onℝand verifies the functional equation(∗).Next,let ∈ℝand consider::ℝ−→−→ ( + )− ( ) ( )As is differentiable on ℝ, is also.Furthermore,by hypothesis, and hence, ′are identically zero.Therefore,by differentiating:∀ ∈ℝ, ′( + )= ′( ) ( )Setting =0,we have: ′( )= ′(0) ( ),i.e. ′= with = ′(0).We now consider two possibilities:∙either is identically zero (and the result follows);∙else there is at least one real number 0such that ( 0)=0.Setting = 0and =0in (∗),we have: ( 0)= ( 0+0)= (0)× ( 0).However,by hypothesis, ( 0)=0;hence, (0)=1.We have therefore shown that satisfies ′= and (0)=1.According to our last theorem, = for some ∈ℂ.⊠Remark:one can actually show that the conditions we imposed are a bit strong.Indeed,we need not suppose differentiable.The hypothesis continuous on ℝis sufficient,and actually,even continuous at one point is enough.Exercise 2.1.1Let be a function that verifies (∗).Prove that:(1) is identically zero or does not vanish at all and (0)=1;(2)if is continuous at (with ∈ℝ),then is continuous on ℝ;(3)if is not identically zero,then there exists >0such that∫( )d =0and that therefore, is differentiableon ℝ.Remark:this second characterization of exponential functions is very useful.For instance,this is how one can prove that continuous random variable without memory: ( > + ∣ > )= ( > )for all , ⩾0has necessarily an exponential distribution.Remark:we can also mention a few applications of these theorems to Physics:∙radioactive decay:take for instance plutonium isotope Pu-239.Physics laws state that if ( )is the number ofradioactive atoms (or otherwise the mass (multiplying by the molar mass)),then the activity -given by =−dd-is proportional to .This leads to the relation −d d = .Here,ln 2is what we call the half-life,i.e.thetime required for half the atoms in a sample of radioactive material to decay.∙Newton’s law of cooling states that the rate of change in the temperature of an object is proportional to the difference between the object’s temperature and the temperature of the surrounding medium,i.e.dd=ℎ( 0− )where is the temperature of the object, 0the temperature of the surrounding medium,andℎis what we call the Newton coefficient.2.2Structure of the set of solutions and sum principleTheorem2.2.1Let( )be a linear differential equation,( ): ( )= with a linear map.Assume that is a particular solution of( ).Then for any differentiable function , is a solution of( )if and only if − is a solution of( ).In other words,setting (respectively( ))the set of solutions of( )(respectively( )),one has:= +Proof:Let be any differentiable function.Then,is a solution of( )⇐⇒ ( )=⇐⇒ ( )= ( )(because is a solution of(E))⇐⇒ ( )− ( )=0⇐⇒ ( − )=0(because is linear)⇐⇒ − is a solution of( )⊠this fundamental theorem has a very important practical signification.In order to solve alinear differential equation,one mustfind one solution(what we call a particular solution)and then add all the solutions to the homogeneous equation.Remark:this is why in the next paragraphs,we will focus on how to solve the homogeneous equation and then how to find a particular solution.Remark:for your information,we will say that is an affine space-that is the sum of“a point”( )and a“vector space”( ).Affine spaces and vector spaces are fundamental algebraic structures which we will study next year. Proposition2.2.1(Sum principle)Let be a linear map, 1be a solution of ( )= 1and 2a solution of ( )= 2,where 1and 2are two functions.Then:(i) 1+ 2is a solution of ( )= 1+ 2;(ii)for all ∈ , 1is a solution of ( )= 1.Proof:All these properties are direct consequences of the fact that is linear.Indeed,( 1+ 2)= ( 1)+ ( 2)= 1+ 2and( 1)= ( 1)= 1⊠Example2.2.1Say we want to solve the differential equation( ): ′− =cos( )+3 2 .We then proceed in three steps:∙Step1:wefind a particular solution of ′− =cos( ).∙Step2:wefind a particular solution of ′− = 2 .∙Step3:we solve the homogeneous equation(here the solutions are −→ )We can therefore conclude that the solutions of( )are the functions:= 1+3 2particular solutionby the sum principle +solutions of( )There are now two questions that arise naturally:1.How do we solve the homogeneous equation?2.How do wefind a particular solution if there is no evident one?That’s what we will now focus on.2.3Solutions to the associated homogeneous equationReminder(which we have already reviewed in the chapter on usual functions)Definition2.3.1Let be a function.We say that a function is an antiderivative of on the real interval if is differentiable on and ′( )= ( )for all in .Theorem2.3.1Any continuous function on an interval has antiderivatives.Furthermore,if ∈ ,then −→∫( )d is the only antiderivative of on which vanishes at .Proposition2.3.1Let be a continuous function on the real interval (with values in )and consider the differential equation( ): ′+ ( ) =0.Let be any antiderivative of on .Then the solutions of( )are the functions with domain given by:∀ ∈ , ( )= exp(− ( )),with ∈Proof:First,we notice that since we assumed to be continuous on , does indeed have an-tiderivatives.Therefore, exists.Next,let be any function and set = exp( ).As theexponential function(real or complex)does not vanish,we have:= exp( )⇐⇒ = exp(− )Hence, is differentiable on if and only if is.We then have:∈ ⇐⇒∀ ∈ , ′( )+ ( ) ( )=0⇐⇒∀ ∈ , ′( )exp(− ( ))− ( ) ( )exp−( ( ))+ ( ) ( )exp(− ( ))=0⇐⇒∀ ∈ , ′( )exp(− ( ))=0⇐⇒∀ ∈ , ′( )=0⇐⇒∃ ∈ ,∀ ∈ , ( )=⇐⇒∃ ∈ ,∀ ∈ , ( )= exp(− ( ))⊠Remark:please note that this result holds for =ℂas well;and also,that this proposition states that the maximal solutions are all defined on !Example2.3.1Consider the following differential equations:∙ ′= where ∈ℂ.We already know from the characterization of exponential functions that the solutions are ( )= (0)exp( ).Setting this aside,( )can be written as ′− =0and if we apply the previous proposition,we know that −→− is continuous onℝand −→ is an antiderivative onℝ.Therefore,the solutions are of the form:( )= exp( ),with ∈We do indeedfind the same solutions,which is reassuring!∙( ): ′+11+ 2=0. −→11+ 2is continuous onℝand has −→arctan( )for antiderivative.Therefore,the solutions of( )are the functions defined on by:∀ ∈ , ( )= exp(−arctan( )), ∈Remark:solving a homogeneous differential equation of order one basically boils down tofinding an antiderivative of a given function.Proposition2.3.2With the previous notations,(i)solutions of( ),other than zero,do not vanish on ;(ii) ={ − , ∈ }is a“one-dimensional vector space”or a“line”:all elements of are“proportional”(“collinear”)to − .Proof:(ii)is simply a formal way of writing the set of solutions.As for( ),let be a solution of( ).Then by the previous proposition,there is a constant ∈ such that = exp(− ).Moreover,we know that the exponential never vanishes;therefore,if ( 0)=0for some0∈ ,then =0and =0(identically zero).⊠2.4Variation of parametersNow we consider the full equation( ): ′+ ( ) = ( ).Let 0be a non-zero solution of( )(( 0)is a basis of ).From the previous paragraph,we know that the solutions of the homogeneous equation( )are of the form 0,with ∈ .We are therefore going to try tofind a particular solution of the same form,but by“variation of the parameter ”,i.e.of the form:( )= ( ) 0( )By( )in the previous proposition,we know that for all ∈ , 0( )=0;hence,= 0⇐⇒ = 0and therefore, is differentiable on if and only if is.We then deduce that for any differentiable function : = 0; ′= ′ 0+ ′0et ′+ = ′ 0+( ′+ )0( 0solution of( ))= ′ 0Hence,is a solution of( )⇐⇒ ′ 0= ⇐⇒ ′= 0We can now state our result formally:Theorem 2.4.1Consider the differential equation ( ): ′+ ( ) = ( ),where and are two continuous functions on .Let be an antiderivative of and an antiderivative of → ( ) ( ).Then a particular solution of ( )is given by:∀ ∈ , ( )= ( ) − ( )In particular,if 0∈ ,then the solutions of ( )aregiven by:∀ ∈ , ( )=(∫( ) ( )d ) − ( )+ − ( )Proof:We choose 0=exp(− ).The previous calculations show that:is a solution of ( )⇐⇒ ′= exp( )⇐⇒= +Substituting in = exp(− )yields the results.⊠△!Caution:beware!All the theorems we stated apply for equations which are “resolved”in ′.If they are not,i.e.of the form ( ) ′+ ( ) = ( ),then one most solve the equation on intervals on which does not vanish,wherewe can write ′+ ( ) ( ) = ( )( ).Example 2.4.1Consider a resistor and a capacity mounted in series with a generator delivering a constant tension .We know that the tension (or voltage)then satisfies the following differential equation:( 1):dd+ = As = =0,( 1)is equivalent tod d + =.We now apply our method.∙Solutions to the associated homogeneous equation:−→1 is continuous on ℝand has −→for antiderivative.Therefore,the solutions to ( 1, )are thefunctions given by:∀ ∈ℝ, ( )= exp(−), ∈ℝ∙Search for a particular solution:we could simply apply the variation of parameters method.However,it is much simpler here to search for a trivial solution.The second member is constant:we therefore try to find a constant solution,i.e.such that dd =0.This yields: = .∙ConclusionThis proves that the tension is of the form:∀ ∈[0;+∞[, ( )= + −.We can then determine the value of by using the initial condition.Example2.4.2Consider the equation( 2): ′− =cos( ).∙Solutions to the associated homogeneous equation:It is clear that the solutions to the homogeneous equation are given by: ( )= for all ∈ℝ,with ∈ℝ.∙Search for a particular solution:–First method:variation of parametersApplying the variation of parameters,we set for all inℝ, ( )= ( ) where is a differentiable function onℝ.Then:is a solution of( 2)⇐⇒∀ ∈ℝ, ′ ( ) =cos( )⇐⇒∀ ∈ℝ, ( )= − cos( )We must therefore determine an antiderivative onℝof −→ − cos( ).To do this,we can,for example, use integration by parts twice:∫ 0 − cos( )d =[− − cos( )]−∫(− − )(−sin( ))d=− − cos( )+1−[− − sin( )]−∫− cos( )dThus,2∫− cos( )d =− − cos( )+ − sin( )+1So that −→ −2(sin( )−cos( ))is an antiderivative of −→ − cos( ).Hence,one particular solution of( 2)is given by:∀ ∈ℝ, ( )=12(sin( )−cos( ))–Second method:using complex numbersConsider the new equation:( ): ′− = .If wefind a particular solution ,thenℜ ( )will be a particular solution of( 2)1.Next,we try tofind a solution of the form = with ∈ℂ(for more details,see section3.3).We then have the following equivalences:solution of( )⇐⇒∀ ∈ℝ, − =⇐⇒( −1) =1⇐⇒ =1 −1⇐⇒ =− −12Thus,a particular solution of( 2)is given by:for all ∈ℝ,( )=ℜ (−1+2)=−12cos( )+12sin( )–Third method:trying tofind a trivial solutionGiven the second member,it might seem reasonable to look for a solution of the form: −→ cos( )+ sin( ).∙ConclusionWe can now conclude that the solutions to( 2)are given by:∀ ∈ℝ, ( )= −2sin( )− −2cos( )+ − , ∈ℝ1Careful!This method works here because the coefficients are all real!Otherwise,we could not say thatℜ ( ′− )=ℜ ( )′−ℜ ( ).Example2.4.3Consider( 3): ′+1+ 2=11+ 2.∙Step1:solutions to the associated homogeneous equation−→1+ 2is continuous onℝand has −→12ln(1+ 2)for antiderivative.Thus, ={ℝ−→ℝ−→ exp(−12ln(1+ 2)), ∈ℝ}={−→√1+ 2, ∈ℝ}∙Step2:search for a particular solutionApplying the previous theorem,we know that a particular solution is of the form: −→( )√1+ 2where is anantiderivative of −→ ( ) − ( )=1√1+ 2.Here,we can choose =Argsh which leads to:∀ ∈ℝ, ( )=Argsh( )√1+ 2∙ConclusionThis shows that the solutions to the equation( 3)are the functions given by:∀ ∈ℝ, ( )=Argsh( )+√1+ 2,with ∈ℝExercise2.4.1Solve the differential equation: ′+ = 3.Finally,to conclude this paragraph,we will study one example of equation which is not“resolved”in ′and see how one goes about determining maximal solutions.Example2.4.4We wish to solve the differential equation:( ):(1− 2) ′−2 =sin( ).First,notice that1− 2=0if and only if =±1.Therefore,even though( )is defined onℝ,we cannot simply apply our theorems onℝ.We must solve the equation on each interval where1− 2does not vanish,i.e.on 1=]−∞;−1[, 2=]−1;1[and 3=]1;+∞[,and then determine whether or nor,there are any solutions onℝ.∙Step1:Solutions to( )on each interval where1− 2does not vanishLet ∈{1,2,3}.We then have:solution of( )on ⇐⇒ ′+22−1=sin( )1− 2–Solutions to the associated homogeneous equation−→22−1is continuous on and −→ln∣ 2−1∣is an antiderivative of that function on .Therefore,the solutions to( )(the homogeneous equation)are given by:∀ ∈ , ( )= exp (−ln∣ 2−1∣)=∣ 2−1∣, ∈ℝFurthermore,on the interval , 2−1has constant sign.We can thus remove the absolute value and incorporate the sign of 2−1in the sign of the constant .In other words,the solutions to( )are ofthe form:∀ ∈ , ( )=1− 2, ∈ℝ△!Caution:it is extremely important to realize that the parameter that wefind solving the equation depends of the interval on which we are solving the equation.Thus,the notation to indicate this fact.–Search for a particular solutionApplying the variation of parameter method,we set ( )= ( )1−for in ,with a differentiablefunction on .Then,is a solution of( )on ⇐⇒∀ ∈ , ′( )1−=sin( )1−⇐⇒∀ ∈ , ′ ( )=sin( ) Hence,a particular solution of( )on is given by:∀ ∈ , ( )=−cos( ) 1− 2–ConclusionThis proves that the solution of( )on are the functions of the form:∀ ∈ , ( )= −cos( )1− 2,with ∈ℝRemark:we actually could have solved( )much faster by noticing that(1− 2) ′−2 =((1− 2) )′so that( )is equivalent to(1− 2) =−cos( )+ .∙Solving( )completely–Necessary conditionsSuppose is a solution of( )(i.e.a solution onℝ).Then,its restriction to each interval (for ∈{1,2,3}) is a solution of( )on .According to what we have just shown,there are three real constants 1, 2and3such that:( )=⎧⎨⎩1−cos( )1− 2if ∈]−∞;−1[ 2−cos( )1−if ∈]−1;1[ 3−cos( )1− 2if ∈]1;+∞[Also,setting =±1in the equation,we see that (±1)=−sin(1)2so that in fact,if is solution of( )onℝ,then necessarily:( )=⎧⎨⎩1−cos( )1−if ∈]−∞;−1[−sin(1)2if =−12−cos( )1− 2if ∈]−1;1[−sin(1)2if =13−cos( )1− 2if ∈]1;+∞[Furthermore,if is a solution of( ),we know that necessarily, is differentiable at±1,hence continuous.Moreover,lim→−1<−1(1− 2)=0−and lim→−1<−1( 1−cos( ))= 1−cos(1).Therefore,for to be continuousfrom the left at−1,it is necessary that 1−cos(1)=0,i.e. 1=cos(1).Proceeding in the same way,one shows that for to be continuous(period)at−1and1,one must have 1= 2= 3=cos(1).Therefore, we have shown that if a solution to( )exists,it is necessarily given by:( )=⎧⎨⎩cos(1)−cos( )1− 2if =±1−sin(1)2if =±1–Search for sufficient conditionsAssume that is defined onℝby the previous piecewise expression.We wish to determine whether or not is differentiable onℝand solution of( ).First,it is clear that is differentiable onℝ∖{−1,1}and solution of( )on each interval]−∞;−1[,]−1;1[ and]1;+∞[.We must therefore study the differentiability at±1:by construction,if is differentiable at ±1,then will automatically satisfy the equation for =±1.Next,let be any real number such that∣ ∣=1.Then,( )=cos(1)−cos( )(1− )(1+ )=−12×sin(−12)−12×sin(+12)+12Setting ( )=⎧⎨⎩sin( )if =01if =0,we see that in fact,for all real number (including =±1):( )=−12(−12)× (+12)Thus,to prove the differentiability of at ±1,it is sufficient to prove that is differentiable on ℝ.I leave that fact as an exercise for you to do 2.∙ConclusionWe have proved that the differential equation ( )has one and only one solution on ℝwhich is given by:∀ ∈ℝ, ( )=⎧ ⎨ ⎩cos(1)−cos( )1− 2if =±1−sin(1)2if =±1Important facts to remember from this example:(1)when solving a first order differentiable equation which isn’t “resolved”in ′,we cannot directly apply ourtheorems (notice that we found only one solution as opposed to an infinite number of solutions);(2)in this case,it is important to remember to solve on each interval where the function (coefficient in frontof ′)does not vanish,and that the constants (or parameters)we find depend on the interval;(3)then comes the big piece of work:determining necessary conditions on the constants so that a function canbe a solution on the full interval,and then,conversely,checking that such a function is indeed differentiable and a solution to the differentiable equation.(4)note that these equations require a whole lot more work then the other form....2.5Solution satisfying a given initial condition:existence and uniquenessTheorem 2.5.1Consider the differential equation ( ): ′+ ( ) = ( ),where and are two continuous functions on .Let 0∈ and let 0be any scalar in .Then the equation ( )has one and only one solution satisfying the initial condition ( 0)= 0.Furthermore,we can give an exact expression for this solution:∀ ∈ , ( )=( 0+∫ 0( ) ∫ 0 ( )d d )−∫ 0 ( )d Proof:Let be a solution of ( ).Consider : −→ given by ( )=∫( )d so that isthe only antiderivative of on that vanishes at 0.By theorem 2.4.1,we know that thereexists a scalar ∈ such that:∀ ∈ , ( )=(∫ 0 ( ) ( )d ) − ( )+ − ( )Thus,( 0)= 0⇐⇒(∫ 0 0 ( ) ( )d ) − ( 0)+ − ( 0)= 0⇐⇒= 0⊠2hint:start by proving that for any non-negative , − 36⩽sin( )⩽ .3Second order linear differential equations with constant coefficients3.1Definition and structure of the set of solutionsDefinition3.1.1A second order linear differential equation with constant coefficients is by definition a differential equation of the form:( ): ′′+ ′+ = ( )where , , are three scalars in such that =0and a function.Proposition3.1.1A second order linear differential equation with constant coefficients is indeed a linear differ-ential equation.Proof:Set ( )= ′′+ ′+ ;then( )is equivalent to ( )= .Now we show that is alinear map.Let 1, 2be two functions twice differentiable on a real interval and let ,[]be two scalars in .Then:( 1+ 2)= ( 1+ 2)′′+ ( 1+ 2)′+ ( 1+ 2)= ( ′′1+ ′′2)+ ( ′1+ ′2)+ ( 1+ 2)= ( ′′1+ ′1+ 1)+ ( ′′2+ ′2+ 2)= ( 1)+ ( 2)This proves that is a linear map and therefore,that( )is a linear differential equation.⊠As for the structure of the set of solutions,if you read the proof of theorem2.2.1and proposition2.2.1,you can easily see that they do not depend on the order of the differential equation,but solely on the fact that it is linear.Hence, they also apply for second order linear differential equations:Theorem3.1.1Let( )be a linear differential equation,( ): ( )= with a linear map.Assume that is a particular solution of( ).Then for any differentiable function , is a solution of( )if and only if − is a solution of( ).In other words,setting (respectively( ))the set of solutions of( )(respectively( )),one has:= +Proposition3.1.2(Sum principle)Let be a linear map, 1be a solution of ( )= 1and 2a solution of ( )= 2,where 1and 2are two functions.Then:(i) 1+ 2is a solution of ( )= 1+ 2;(ii)for all ∈ , 1is a solution of ( )= 1.Once again,we are now left with two problems:finding the solutions to the homogeneous equation andfinding a particular solution.3.2Solutions to the associated homogeneous equationWe now focus on the associated homogeneous equation:( ): ′′+ ′+ =0We know that for linear differential equations of order 1with constant coefficients,the solutions of the homogeneous equation are exponentials.Also,we know that when differentiating an exponential −→exp( )(for ∈ ),we get a function which is proportional to the same exponential.Therefore,it is perfectly logical to try to find solutions to the homogeneous equation of the form: ( )= with ∈ .Let ∈ .The function is at least twice differentiable on ℝ, ′ = and ′′ = 2 .Thus,the followingpropositions are equivalent:is a solution of ( )⇐⇒′′ + ′ + =0⇐⇒2 + + =0⇐⇒ ×( 2+ + )=0Considering the fact that the exponential function is never zero (or does not vanish),we find that:is a solution of ( )⇐⇒ 2+ + =0Definition 3.2.1The characteristic polynomial associated to ( )(or to ( ))is the second degree polynomial:( )= 2+ +One also says that 2+ + =0is the characteristic equation of ( )or of ( ).Our previous results show that is a solution of ( )if and only if ( )=0.Therefore,the number of different exponentials which will be solutions depends of the number of roots that has in .Hence,one must distinguish two possibilities.∙First case: has at least one root inLet 1be a root of in .Relations between roots and coefficients show that has another root 2(eventually 2= 1is a root with multiplicity 2)that verifies 1+ 2=−.Next,let be any function twice differentiable on ℝ.Copying our variation of parameter,we set: ( )= ( ) − 1 ,or equivalently, ( )= ( ) 1 .Then is twice differentiable on ℝas well and for all real number ,′( )= ′( ) 1( )+ ( ) 1 1( )′′( )= ′′( ) 1( )+2 1 ′( ) 1( )+ 21 ( ) 1( )Consequently,collecting terms and replacing in the equation,we find that:is a solution of ( )⇐⇒ 1×( ′′+2 1 ′+ 21+ ′+ 1 + )=0⇐⇒ ′′+(2 1+ ) ′+( 21+ 1+ ) =0Furthermore,by definition, 1is a root of so that ( 1)=0,i.e 21+ 1+ =0.Also,substituting =− ( 1+ 2)we find that 2 1+ = ( 1− 2)and thus:is solution of ( )⇐⇒∀ ∈ℝ, ′′+ ( 1− 2) ′=0⇐⇒∀ ∈ℝ, ′′+( 1− 2) ′=0⇐⇒∀ ∈ℝ, ′( )= ( 2− 1) ,for some ∈We must now find ,i.e.find antiderivatives of −→ ( 2− 1) .which again introduces two separate cases:First sub-case: 2= 1(i.e. has two distinct roots in )。