over the past few years , there has been remarkable research interest in analysis of interconnected uncertain systems , see for example , , , , , , , , and .robustness analysis of large - scale interconnected uncertain systems poses different computational challenges mainly due to their large state , uncertainty and input - output dimensions .there are different approaches for analyzing robustness of interconnected uncertain systems , among which , iqc analysis is one of most general ones which provides a unified framework for analyzing uncertain systems with different types of uncertainties, , and . as an example of large - scale interconnected systems, one can refer to power grids .these networks can be viewed as an interconnection of of huge number of subsystems or components . due to the continuing deregulation in generation of power and distribution sectors in the u.s . andeurope , the number of players involved in the generation and distribution of power has increased significantly .this can in turn the introduce uncertainties within the system , e.g. , in generation and consumption .due to this intrinsic uncertainty their robustness analysis is put in new demand .in , the authors proposed algorithms to alleviate computational issues with conventional robustness analysis techniques when applied to large - scale interconnected uncertain systems . due to a common structure in such systems , that is sparsity of interconnections among subsystems , it is possible to devise efficient centralized or distributed algorithms for solving the corresponding analysis problems , see e.g. , , .this is done by exploiting the sparsity in the interconnections which then makes it possible to either use efficient sparse solvers or decompose the problem and use distributed computational methods .such efficient analysis algorithms rely on a particular model description .consequently , for analyzing large - scale interconnected uncertain systems with sparse interconnections , it is necessary to provide the algorithms with such model descriptions .however , generally the provided models are not in the suitable format and are expressed using nested block - diagram - based representations within commonly used simulation softwares , such as simulink and modelica .this description allows for natural modelling of such systems . considering the sheer size of such models then, it is essential to devise automated approaches for extracting the model required for the analysis algorithms from such descriptions .  in this paper, we first put forth a general description of interconnected uncertain systems .however initially this description differs from the standard representation of uncertain systems , i.e. , lft representation , and hence we are prohibited from using efficient analysis tools discussed in for analyzing such systems . in order to solve this issue, we show how to convert the aforementioned mathematical description to that of used in . to this end , we have taken extra care to preserve the sparsity in the interconnections in the modified description . in order to make the conversion to the standard form , we present a two stage procedure .this procedure is also illustrated using an example .in this section , we put forth a mathematical description for networks of dynamical systems in the presence of uncertainty .figure [ fig:2 ] illustrates one such system .as shown in figure [ fig:2 ] , main building blocks for describing such systems are dynamic interconnected systems , and uncertainty blocks .generally , the block - diagram - based description of interconnected uncertain systems can also include nested structure . that is certain blocksare themselves indirectly described using these fundamental blocks . due to this, the description of the total system in its most basic form also relies on these two types of blocks .first we provide a formal description of these fundamental blocks .let us assume that the interconnected system includes dynamic system blocks .a dynamic system block is defined using a system transfer function matrix with and , as its -dimensional and - dimensional input and output vectors , respectively , and is given as for .the uncertainty blocks represent causal , bounded operators , , with -dimensional input and output vectors and , respectively , and is given as for . here is the number of such blocks in the interconnected uncertain system representation .the uncertainty blocks are commonly described using iqcs , .let us now summarize all the input - output relationship of the dynamic system and uncertainty blocks of interconnected system as  [ eq__5ab _ ]   where , , and .moreover , and are block - diagonal matrices , with and as their diagonal elements , respectively , for and .this means that , with , and : with .what remains now is to describe the interconnection matrix among the aforementioned fundamental blocks .having described the building blocks of the system , we now discuss the interconnection among these blocks .this can be done using a so - called interconnection constraint . to this end, we use a 0 - 1 matrix as below where and and is a matrix that .the structure of can be described in more detail as and it basically describes how outputs of each of the blocks are connected to inputs of other blocks .given or the description of interconnections together with completes our description of interconnected uncertain systems .it is also possible to accommodate external inputs and outputs to and from an interconnected uncertain system using a similar approach .let us define and , where and are vectors of the external inputs and outputs , respectively .then we can incorporate the external inputs and outputs by modifying the interconnection description as where with the description of the interconnections in place we can now summarize the interconnected uncertain systems description as  [ eq_10abc _ ]   next we discuss how this description of interconnected uncertain systems in figure [ fig:5 ] can be transformed to a linear fractional representation used within scalable analysis algorithm .at this point we have defined a general mathematical description for interconnected uncertain systems . for the sake of simplicity and brevitywe assume that we do not have any external inputs and outputs .this is particularly the case when we are concerned with analysis only . in this sectionwe describe how to convert the description presented above , also illustrated as in figure [ fig:5 ] , to a standard lft representation . to this end , let us first review this standard model description for interconnected uncertain systems . the system description considered in scalable analysis algorithmsare given in the form e.g. , see .now let us discuss how the representation discussed in the previous section can be written in the same format as above .let us first define then it is possible to rewrite the interconnected system description in section [ sec:2 ] as in , also shown in figure [ fig:6 ] .moreover , notice that .it is then possible to identify different elements in as consequently this description is now in lft format .so far , we have described how given a general mathematical description of an interconnected uncertain system , it is possible to convert this description to a standard lft representation . however , this mathematical description , particularly the interconnection description , is not generally given and it needs to be extracted from a nested block - diagram schematic of such systems .for large - scale interconnected uncertain systems , this extraction can not be done manually and needs to be automated .next we describe our proposed automated approach to extract this mathematical description .in order to generate a model for a large - scale interconnected uncertain system in the from of , we need to extract the interconnection matrix from a nested description of the system , as shown in figure [ fig:7 ] . in order to be able to collapse the nested structure and recover the description in ( [ eq_10abc _ ] ), we first need to describe this structure in the system .this can commonly be done using a graphical representation , particularly a tree . herewe assume that this representation is given .this can be provided using different approaches such as regional or adaptive clustering .next , we describe this graphical description , and then we discuss how it can be used to recover the model as in .     as was mentioned before the provided models for interconnected uncertain systems have a nested structure .this is mainly to make descriptions of interconnected uncertain systems more intuitive and comprehensible .in order to describe the nested structure in the system we can use a tree .each node in this tree corresponds to a nested sub - block , nsb , in the system .in such a description the node at the root of the tree corresponds to the nsb that does not appear in the description of any other nsb .the children for each node correspond to the nsbs that appear in its description . within this treethen the blocks at the leaves of the tree correspond to ones that does not include any other nsbs .consequently , the description for these blocks only relies on the fundamental building blocks .we refer to this description as the fundamental description .    at this point ,let us consider an example .consider the interconnected uncertain system depicted in figure [ fig:10 ] .as can be seen from the figure , in this system description there are 5 nsbs , namely . hence the tree that describes the nested structure in this systemis comprised of 5 nodes .this tree is illustrated in figure [ fig:9 ] .as can be seen from the figure , since does not appear in the description of any other nsbs it is assigned to the root of the tree . also in this tree , for instance , is child of since appears in the description of .this tree - based description of the nested structure in the system is at the heart of our approach in retrieving the interconnection matrix of the system .before we present our algorithm for this purpose , we first need to provide a formal description for each of the nsbs .let us consider nsb and assume that the dynamic system blocks and uncertainty blocks in its description are given by and which are both block diagonal and each diagonal entry corresponds to each of such blocks in this nsb .also let us denote the inputs and outputs vectors from the nsbs within nsb description as and , i.e. , these vectors correspond to the inputs and outputs from the underlying nsbs stacked .we can then summarize the mathematical description of the nsb as  [ eq_14abc _ ]   where and with and denoting the external input and output for the nsb .similar to the matrix can be given as here we assume that .this assumption implies that there is no direct link between the external inputs and outputs .notice that this assumption is not restrictive and can be imposed by simply adding dummy blocks and input - outputs . as was mentioned before, the description for each nsb can be expressed in terms of fundamental building blocks . in order to see this , notice that the description of nsbs at the leaves of the tree are given in terms of fundamental blocks .this means that the fundamental description for these blocks are readily available .for instance for the example in figure [ fig:10 ] the description of nsb , that is a leaf of the tree , is given as    which is in the same form as and only depends on fundamental building blocks .the only difference from is that in , the inputs and outputs concerning the underlying nsbs are missing as they do not exist .particularly , and , and consequently , based on the description of in figure [ fig:10 ] , having described the nsbs at the leaves of the tree , it is then possible for each parent to the leaves to merge their description with that of its own .for instance let an nsb be a parent to only leaf nodes , indexed as and with descriptions given as in .the nsb can then merge the description of its children ( the leaves ) nsbs with that of its own as  [ eq : merging ]   at this point , it is also possible to write the description of nsb only in terms of the fundamental building blocks as for the leaves .to this end , we need to eliminate the external inputs and outputs for nsbs .this can be done using the interconnection descriptions in .let us illustrate this using the example in figure [ fig:10 ] .as for nsb , we can also describe nsb as    with at this point we can describe nsb as in with  [ eq : mergeexample ]   where and .let us now stack the interconnection constraints in as let us reorder blocks in this equation and rewrite it as notice that in order to convert this description to a fundamental one , we need to eliminate or the last three block equations from this description .this can be done by firstly noting that recall that we have and combining equations in will result in this then allows us to rewrite the set of equations in that we want to keep as   and notice that this description is in the same format as the description given for the leaves and in fact can be written as    by partitioning the defined matrices in accordingly , which is in fundamental form .the same procedure can be generalized for all problems , that is for an nsb with all its children descriptions given in fundamental form , it is possible to recover the fundamental description for this nsb by simply forming where are the children of nsb , and through equations . following the same procedure upwards the tree , the fundamental description at the root will correspond to the description of the interconnected system as in .in this paper we put forth an automated framework for extracting standard description of large - scale interconnected uncertain systems used in scalable analysis algorithms .particularly we showed that given a nested block - diagram - based description of an interconnected uncertain system , it is possible to first describe the nested structure in the system using a tree , and how based on this it is possible to recover a suitable description of the system .  1 c.  langbort , r.s. chandra , and r. dandrea , distributed control design for systems interconnected over an arbitrary graph , _ ieee transactions on automatic control _ , 49:(9 ) , 15021519 , 2004 .r. andrea and g. dullerud , distributed control design for spatially interconnected systems , in _ ieee transactions on automatic control _ , 48:(9 ) , 1478-1485 , 2003 . g. dullerud and r. d. andrea , distributed control of heterogeneous systems , in _ ieee transactions on automatic control _ , 49:(12 ) , 2113-2128 , 2004 . c. langbort , v. gupta and r. murray , distributed control over failing channels , in _ lecture notes in control and information sciences series _springer berlin heidelberg , 2006 . c. langbort and r.d .andrea , distributed control of heterogeneous systems interconnected over an arbitrary graph , in proceedings of _42nd ieee conference on decision and control _, 28352840 , 2003 . v. a. ugrinovskii , i. peterson , a. savkin and e. ugrinovskaya , decentralized state - feedback stabilization and robust control of uncertain large - scale systems with integrally constrained interconnections , in _ system and control letters _ , 40:(2 ) ,107119 , 2000 . h. fang and p. j. antsaklis , distributed control with integral quadratic constraints , _ isis technical report isis-2007 - 001 _ , university of notre dame , 2007 .n. motee , ; a. jadbabaie , receding horizon control of spatially distributed systems over arbitrary graphs , in proceedings of _ 45th ieee conference on decision and control _ , 34673472 , 2006 . b. chen and s. lall , dissipation inequlaities for distributed systems on graph , in proceedings of _42nd ieee conference on decision and control _, 30843090 , 2003 . g. scorletti and g. due , an lmi approach to decentralized control , in _ international journal of control _ , 74:(3 ) , 211224 , 2001 .a. megretski and a. rantzer , system analysis via integral quadratic constraints , _ ieee transactions on automatic control _ , 42:(6 ) , 819830 , 1997 .u. jonsson , lecture notes on integral quadratic constraints , kth , 2001 .k. zhou , j. c. doyle and k. glover , robust and optimal control , prentice hall , 1997 .m. s. andersen , s. khoshfetrat pakazad , a. hansson , and a. rantzer , robust stability analysis of sparsely interconnected uncertain systems , _ ieee transactions on automatic control _ , 59:(8 ) , 21512156 , 2014 .m. s. andersen , a. hansson , s. khoshfetrat pakazad , and a. rantzer . distributed robust stability analysis of interconnected uncertain systems . in proceedings of _ the 51st ieee conference on decision and control _ , 1543 - 1548 , maui , hawai , 2012 . f.  demourant , new algorithmic approach based on integral quadratic constraintsfor stability analysis of high order models , in proceedings of _european control conference _, 359364 , 2013 . s. khoshfetrat pakazad , a. hansson , m. s. andersen and a. rantzer , distributed robustness analysis of interconnected uncertain systems using chordal decomposition , in proceedings of__the 19th ifac world congress _ _ , 25942599 , 2014 .a. megretsk , c. kao , u. jonsson , a. rantzer , a guide to iqc for robustness analysis , massachussetts institute of technolgy , 1998 .a. kahrobaeian and y. a. i. mohamed , suppression of interaction dynamics in dg converter - based microgrids via robust system - oriented control approach , _ ieee transactions on smart grid _ , 3:(4 ) , 18001811 , 2012 .n. pogaku , m. prodanovic , t . c. green , modeling , analysis and testing of autonomous operation of an inverter - based microgrid , _ ieee transactions on power electronics _ , 22:(2 ) , 613625 , 2007 .c. ragusa , a. liotta , and g. pavlou , an adaptive clustering approach for the management of dynamic systems , _ ieee journal on selected areas in communications _, 23:(12 ) , 22232235 , 2005 .

Abstract: the first challenge in robustness analysis of large - scale interconnected uncertain systems is to provide a model of such systems in a standard - form that is required within different analysis frameworks . this becomes particularly important for large - scale systems , as analysis tools that can handle such systems heavily rely on the special structure within such model descriptions . we here propose an automated framework for providing such models of large - scale interconnected uncertain systems that are used in integral quadratic constraint ( iqc ) analysis . specifically , in this paper we put forth a methodological way to provide such models from a block - diagram and nested description of interconnected uncertain systems . we describe the details of this automated framework using an example .


how to best spend lecture time to promote student learning in introductory courses is a topic that we often debate with ourselves and with our colleagues .recent research suggests that actively engaging students in learning activities results in better conceptual understanding compared to passively listening to lectures .there is also evidence that students are better prepared to learn difficult concepts from lectures or lecture demonstrations when they are aware of what they do nt know and do nt understand .the common thread running through this diverse set of studies is that there is more and deeper learning when students come to class with some basic exposure to the content so that they are prepared to learn from the experiences in lecture . reading the textbook priorto coming to class is one way of providing the necessary basic exposure to content , but getting students to read the textbook is a difficult goal to achieve .  data from a survey administered to students at the university of illinois at urbana - champaign ( uiuc ) over three consecutive years , shown in table [ textbooksurvey ] , indicate that both life - science majors and engineering majors seldom , if ever , open the textbook prior to coming to class .about 70% of both groups admit they `` rarely '' or `` never '' read the textbook ; 69% of both groups claim that the textbook was either `` useless '' or `` not very useful '' in helping them understand the course material .what makes these findings surprising is that at uiuc we attempt to enforce textbook reading prior to class time by administering just - in - time teaching ( jitt ) `` preflight '' questions crafted from the readings .these data indicate that students answer the preflight questions based on intuition , or merely to fulfill a requirement , rather than from an informed perspective . based on the comparability of uiuc students to students nationwide , and on the frequent complaints voiced by physics professors across the country that students come to class unprepared , we believe that students `` textbook irrelevance '' attitude is quite prevalent .these circumstances create a dilemma . on the one hand, research suggests that we should actively engage our students during class and that instructors expertise is best used to coach students in learning the subtleties of physics concepts and problem solving ; but this requires that students come to class with some exposure to basic content and with some idea of what they do and do not understand something that we know students are not doing . on the other hand , recognizing that students come to class unprepared to participate in meaningful activities suggests that we should spend class time covering basic content from the textbook ; but this would leave little time for coaching students in developing conceptual understanding and problem solving skills .one possible solution to this dilemma is to use multimedia presentations to help students learn basic content prior to coming to class .there are three reasons why multimedia presentations might prove effective in this context . first , today s generation of students has grown up in a multimedia age , and they not only know how to use multimedia better than the current generation of physics professors , but they also like using multimedia . second , presenting basic physics content via asynchronous , web - based multimedia presentations can be monitored and enforced better than policing textbook reading .multimedia modules can be designed with embedded assessments , and a modest amount of course credit can be assigned to viewing the materials ( so long as some minimum time is spent on them ) and to performance on the embedded assessments .the third reason why multimedia presentations of basic content might provide a viable solution to the textbook - irrelevance problem is that a considerable body of research exists on how to design multimedia materials to improve learning .this is in stark contrast to the situation with textbooks , where enhancements continue to be added ( e.g. , addition of color ; boxes containing problem solving tips , reflection questions , and questions to quiz oneself ) with no evidence on whether they add , or perhaps detract , from student learning .principles for designing multimedia materials draw from both cognitive science and empirical studies .for example , work on cognitive load over the last twenty years by sweller and his collaborators indicates that if the student is using valuable working memory resources in order to simply coordinate images and words , s / he has less capacity for actual learning .the expanding research base in the field of multimedia learning has led to the following conclusions : people have separate audio and visual channels , _ b ) _ these channels have limited capacity , and _ c ) _ learning involves the active selection , organization and integration of the information presented via the auditory and visual channels .several design principles have emerged from multimedia learning research findings .for example , students learn better _( i ) _ from words and pictures than from words alone , _( ii ) _ when words and pictures are presented simultaneously rather than successively , _ ( iii ) _ when extraneous words , pictures and sounds are excluded , and _( iv ) _ from animation and narration than from animation and on - screen text .although most of these are intuitive , one principle that is perhaps somewhat counterintuitive is that students learn better from animation and narration than from animation , narration and on - screen text .overall , this body of research suggests that in designing multimedia learning materials the goal should be to stay on - message with respect to the learning goals and to use different input channels ( visual , auditory ) to provide complementary but necessary information to help the learner build meaning and understanding .the current situation can be summarized as follows : students do nt read the textbook as much as we would like and textbook reading has proven difficult to enforce . on the other hand , viewing and answering embedded questions in multimedia presentations delivered via the web could be more closely monitored and awarded credit , encouraging students to conduct the requisite pre - lecture preparation .there is , however , no research at the university level examining whether or not students learn basic physics content better from multi - media presentations compared to reading the textbook . beforeany major effort is devoted to designing and implementing multimedia presentations in large enrollment courses , we need to know if they are at least as effective as reading a textbook in terms of imparting basic content knowledge .  thus , an obvious question to ask is : how well do textbook presentations fare compared to multimedia presentations in teaching introductory physics content ?the study reported here is an attempt to answer this question .more specifically we compared three different modes of presenting the content of the first two weeks of a typical calculus - based introductory electricity and magnetism course : 1 ) paper presentations based on a popular introductory textbook , 2 ) web - based multimedia presentations designed according to principles from multimedia learning , and 3 ) paper presentations consisting of the script from the web - based presentation in item 2 .we were interested in comparing these three modes of presentation in terms of assessments of learning both immediately following the presentations , as well as after an extended period of time as a measure of retention .in addition , we asked students two attitudinal questions regarding their use of textbooks in the course , and their preference for the materials used in this study compared to their current textbook .we specifically chose the material for our study from the first two weeks of an introductory , calculus - based electricity & magnetism ( e&m ) course .topics included coulomb s law , superposition , electric fields from point charges and continuous distributions , electric flux , and gauss law .students find e&m concepts very abstract and difficult , making them ideal for investigating which type of presentation mode works best for helping students learn basic abstract content .four pilot _ multimedia learning modules _ ( _ mlms _ ) , were developed : _ 1 ) _ coulomb s law , _ 2 ) _ electric fields , _ 3 ) _ electric flux , and _ 4 ) _ gauss law .the _ mlms _ drew heavily on research related to multimedia learning summarized above ._ covered one lecture s worth of course content , and was divided into approximately 10 scenes .each scene was implemented as a flash movie containing dynamic animations synchronized with an audio narration that was controlled by the student ( pause , play , rewind , and position ) .embedded formative assessments were included in two or three of the scenes for each module , which consist of questions that must be answered correctly before the student can move to the next scene .the average narration time for an _ mlm _ was 12 minutes ; students typically took about 17 minutes to complete a single _ mlm _ including the embedded assessment .one of the two paper - presentation modes consisted of scripts of the _ mlm _ narrations , ( heretofore referred to as _ mlm - script _ ) together with accompanying static pictures from the _ mlms_ animations .each _ mlm - script _ was about 7 pages long and contained 7 figures .the second paper - presentation mode consisted of verbatim material from a traditional textbook ( heretofore referred to as the _ textbook _ group ) subject to the constraint that the same worked out examples covered in the _ mlms _ had to also be covered in the textbook - based materials .the reason for this constraint is that the cognitive research literature suggests that students often focus on worked out examples to learn how concepts are instantiated , and so we wanted to keep the worked out examples in all presentations equivalent .we selected materials from the tipler and mosca textbook , largely because the material and worked - out examples in this textbook paralleled those in the _mlms_. we deleted a total of six worked examples from the four textbook units that were not covered in the _mlms_. we also added one example from the halliday , resnick and walker textbook to match the one example in _ mlm - script _ that had no counterpart in the tipler & mosca materials .each edited textbook unit was about 7 pages long and contained 8 figures and 2 examples .volunteers were solicited from students taking the first semester calculus - based mechanics course ( physics 211 ) at uiuc and offered compensation in exchange for participation in the study .students were paid 20 study - completion bonus .students were randomly assigned to three groups , each receiving a different treatment ( _ mlm _ , _ mlm - script _ , or _ textbook _ ) .the four units were delivered within two 90-minute sessions distributed over two days in the same week .each 90-minute session consisted of two `` lessons , '' each lasting about 45 minutes .each lesson began with a presentation ( the _ mlm _ group sat at computers to receive the multimedia presentation ; the other two groups received paper booklets containing either the _ mlm - script _ , or _ textbook _presentation ) and was followed by a post - lesson assessment .two weeks later students returned for a retention test , which lasted about 1 hour .the post - lesson assessments consisted of multiple - choice and free - response questions ; to streamline this article and discussion of results , we will only present results from the multiple choice questions .the multiple - choice questions were largely conceptual ; some were taken from the conceptual survey or electricity and magnetism and some were constructed by us .the 43 multiple choice questions on the post - lesson assessments may be obtained from the authors .the retention test administered about two weeks following the lessons had 37 questions ( both multiple choice and free - response ) , 33 of which were either identical or very similar ( e.g. , surface feature changes , change in order of answers ) to questions on the post - lesson assessments , and 32 of which were multiple choice . only results from the multiple choice questions will be presented here .the 32 multiple choice questions from the retention test may be obtained from the authors .   at the end of the third `` retention '' testing session ,the following two survey questions were administered to all three groups :  1 .compare the presentation of the material in this study , with that of your current textbook 1 .strongly prefer current textbook 2 .somewhat prefer current textbook 3 .about the same 4 .  somewhat prefer these study materials 5 .strongly prefer these study materials 2 .how often do you now read your physics textbook ? 1 .two or more times per week 2 .  about once per week 3 .about 1 to 3 times per month 4 .just to cram for exams 5 .  nevera total of 63 students who were enrolled in the calculus based mechanics course volunteered for the study and were randomly assigned to one of the three groups .the results reported here are based on the 45 students who completed the entire study ( 16 _ mlm _ , 13 _ mlm - script _ , 16 _ textbook _ ) .the average physics 211 exam scores ( based on combining scores on the three course hour exams ) for each of the three groups , along with its uncertainty , are shown in the second column of table [ basicresults ] . the spread is about what one would anticipate statistically , with a slight advantage to the _ mlm - script _ group. we will provide raw results , and results that use the covariance to adjust for differences between the groups as determined by these mechanics exam scores .first we will address how the three groups compare in terms of initial learning immediately following each lesson by analyzing the students performance on the 43 multiple - choice questions of the four post - lesson assessments .then we will compare the three groups on the retention test administered two weeks following the lessons by analyzing the students performance on the of 32 multiple choice questions from the retention test .the combined results from the post - lesson assessments and associated error are given in column 3 of table [ basicresults ] .the _ mlm _group had an average score of 75% compared with the _ textbook _ group s score of 63% .the standard deviation of the post - lesson assessments scores is 13.5% , which means students in the _ mlm _ group scored nearly a full standard deviation higher on the exams than their peers who used the textbook ( effect size = 11.5/13.5 = 0.85 ) .since the textbook group s mechanics exam scores were slightly lower ( 1.5% ) than those of the _ mlm _ group we can use the correlation between the mechanics exam scores and the post - lesson assessment scores to estimate an adjusted score for the textbook group of 65% as shown in column 4 . including this adjustment the _ mlm _ group still scored nearly 10 percentage points higher than the textbook group giving an effect size of about 0.7 . performing an analysis of covariance based on the mechanics exam scores and the post - lesson assessment scores , we obtain the result p 0.01 . hence students using the _ mlms _ demonstrated a large and statistically significant advantage in performance on the post - lesson assessment items compared to students using the textbook .we now address the question of whether or not this advantage perseveres by looking at the retention test .the retention test was given approximately two weeks after the lessons , and the raw results are shown in column 4 of table [ basicresults ] .once again the _ mlm _ group performed about 13% higher than the textbook group , with an average score of 70% compared with the textbook group s raw average of 57% , which gets adjusted to 58% based on their mechanics hour exam performance .the adjusted effect size is 0.7 with p 0.01 .once again we have a large , statistically significant result showing that students using the _ mlms _ not only learn more immediately following the lessons , but also retain that information better than the students using the textbook presentation two weeks following exposure .figure [ stelzerfig1 ] illustrates the significance of this effect .each dot represents a student ; on the horizontal axis is their average mechanics hour exam score , and on the vertical axis is their average combined score including both the post - lesson assessments and the retention test ( labeled study scores on the graph ) .the line in the figure represents a linear fit to all of the data .notice that 75% of the _ mlm _ group scores are above this line , while 75% of the textbook group scores are below it .hence , independent of the students ability ( based on their mechanics score ) , more learning and better retention resulted from the _ mlm _ presentations versus the _ textbook _ presentation .   according to multimedia learning theory , the large differential in performance in favor of the _ mlms _ over the _ textbook _ is likely attributable to two factors : 1 ) _ mlms _dispense information via _ dual channels _( visual and auditory ) , and 2 ) _ mlms _ create less memory load compared to the textbook .the motivation for including the _ mlm - script _ group in our study was to test the multimedia learning theory further by comparing its effectiveness to both the _ mlm _ and the _ textbook_. the _ mlm - script _ presentation and the _ textbook _ presentation differed in one important way according to multimedia learning theory : multimedia learning theory states that presentation layout should minimize cognitive load by excluding extraneous words and pictures ( the `` coherence principle '' ) ; in short , a presentation should stay strictly `` on - message '' with little else to distract the learner from that message .this principle , coupled with the `` dual channel '' principle , would predict that : 1 ) the _ mlm - script _ group should outperform the _ textbook _ group ( not only because the `` enhancements '' in textbooks serve to distract students from staying `` on - message , '' but also because the _ mlm - scripts _ were `` spartan '' presentations , with a tight story - line , augmented by the few crucial figures needed to illustrate the ideas in the text ) , but 2 ) the _ mlm - script _ group should perform lower than the _ mlm _ group ( because , although both were `` on - message , '' the _ mlm - scripts _ lacked the dual channel processing available in the mlms ) .  our findings were in accord with these predictions , as shown in table [ basicresults ] and fig .[ stelzerfig2 ] .the _ mlm - script _ group s scores are above those of the _ textbook _ group , but below those of the _ mlm _ group .table [ stats ] shows the effect size and the p - value for all possible comparisons of the three groups for both the post - lesson assessments and the retention test .the p values were computing using ancova with the physics 211 exam scores as the covariant .table [ surveyresults ] and fig . [ stelzerfig3 ] summarize the results for the two survey questions .we report the results of the first question for each group separately , and the second question for all students together ( this second question did not depend on the treatment received ; it only asked students to report on their use of the current textbook in their course ) .as can be seen from the results of the second question summarized in fig .[ stelzerfig3 ] , students in this study reported similar usage of their textbooks as those in table [ textbooksurvey ] .only 16% of all students who finished the study reported that they read their textbooks once per week or more and 50% claim to never have used the textbook the entire semester !  the results from the first question support the statement made earlier , namely that today s generation of students prefer to learn from multimedia materials compared to textbooks .the survey results for question 1 indicate that the _ mlm _ group showed a greater preference for the study materials encountered during the intervention over their traditional textbook as compared to the _ textbook _ and the _ mlm - script _ groups ( p 0.02 , fisher exact test ) .( for this comparison the two text - based presentations were combined ) . the response pattern between the _ textbook _ and _ mlm - script_ groups did not significantly differ from each other ( p = 0.4 fisher exact test ) .our study has shown that learning of basic physics content by the current generation of undergraduates from a typical modern introductory textbook fared poorly when compared to learning from multimedia modules that were designed based on principles derived from research into multimedia learning .the effect sizes obtained in this study are consistent with those from numerous other multi - media learning studies . for example ,mayer and collaborators have performed a series of 39 experiments designed to compare the retention performance of groups using presentations that either were , or were not prepared according to the seven design principles they developed and they found positive effects in 33 of them with the average effect size being 0.8 . we conclude that multimedia learning modules represent a viable , and more easily enforceable alternative means of pre - lecture preparation for introductory physics students .we wish to thank adam feil for his help in preparing this manuscript .[ cols="<,<,<,<",options="header " , ]   cccc + response & _ mlm _ & _ mlm - script _ & _ textbook _ + a ) , or b ) , or c ) & 3 & 8 & 8 + d ) , or e ) & 13 & 5 & 8 +figure 1 : a scatter plot of a student s average mechanics exam score vs his / her combined post - learning and retention scores for both _ mlm _ and _ textbook _ groups .figure 2 : average adjusted scores on the combined post - learning assessments and the retention test for all three groups ( _ mlm _ , _ mlm - script _ , and _ textbook _ )  figure 3 : survey results for frequency that student uses textbook in current physics course . since exams occur about once per month , we have grouped choices c ) 1 - 3 / month and d ) cram for exams into one category labeled every month ., width=566 ]

Abstract: a clinical study was performed comparing the efficacy of multimedia learning modules with traditional textbooks for the first few topics of a calculus based introductory electricity and magnetism course . students were randomly assigned to three different groups experiencing different presentations of the material ; one group received the multimedia learning module presentations and the other two received the presentations via written text . all students were then tested on their learning immediately following the presentations as well as two weeks later . the students receiving the multimedia learning modules performed significantly better than the students experiencing the text - based presentations on both tests .


in these days many experts and also the public more and more talk about unexpected fluctuations of different economic phenomena and about impacts of these fluctuations on economics . in this studywe focused on unexpected fluctuation of long - term real interest rate .we try to explain these unexpected fluctuations and show that these fluctuations can be only seemingly unexpected .+ for description of the economics , we use fundamental macroeconomic is - lm model .this model explain aggregate macroeconomic equilibrium , i.e. goods market equilibrium ( represented by the curve is ) and money market or financial assets market equilibrium ( represented by the curve lm ) simultaneously .many experts researched the problem of macroeconomic ( un)stability and also the is - lm model . already in 1937 , j. r. hicks in published the original is - lm model as mathematical expression of j. m. keynes s theory .this original is - lm model has two main deficiencies .both the price level and exogenous money supply are assumed to be constant .many versions of this model and related problems were presented in several works , see e.g. , , , , , , but we created our own new version of is - lm model which eliminates above - mentioned deficiencies .we modified the model and named it as new is - lm model .+ the mainly criticized problem is assumption of strictly exogenous money supply , i.e. supply of money is certain constant money stock determined by central bank . todays economists can not find consensus in the problem of endogenity or exogenity of money supply , see e.g. .endogenous money supply means that money are generated in economics by credit creation , see e.g. .we resolve this dilemma including some money supply function , which represents conjunction of endogenous and exogenous conception of money supply , to this model .the second deficiency is missing modeling inflation effect in original model .that s why we include also modelling inflation .the is - lm model with this two changes is new is - lm model .+ in these days , the real presumption is that the money ( financial assets ) market is faster than the goods market , i.e. the subjects and their reactions on money ( financial assets ) market are faster than on goods market . in the is - lm model, we work with a demand for money function and money supply function . these two functions and theirs equalitydescribe the money or financial assets market .if we assume money market faster than goods market and unusual ( precisely reverse than usual ) behaviour of these two functions , then there will exist special type of cycle resembled limit cycle called relaxation oscillation on money market .many experts were modelling different type of oscillations in various branch of science ( physics , ecology , biology , economy ) for several centuries , see e.g. , , , , , . todays situation and unexpected fluctuations in the economics confirm the necessity of modelling different types of oscillations occurring in the economics .the relaxation oscillations are the fluctuations of the main variables caused by different velocity of the motion along the trajectory and by the ( un)stability of the arcs of the relevant isoclines .relaxation oscillations may explain the situation when one unstable regime is located between two stable regimes .+ relaxation oscillations include some short parts looking like `` jumps '' .these `` jumps '' can be interpreted like unexpected .the precisely reversed behaviour of money demand and money supply function correspond to special economic situation called liquidity trap . if there exists only one part with liquidity trap then there will be one s - bend of the curve lm and so there will be only one cycle - one relaxation oscillation. if there exists more parts with liquidity trap then there will be more s - bends of the curve lm and so there will be more cycles - more relaxation oscillations and theirs conjunctions .+ the relaxation oscillations on money market are caused by changing of aggregate income .aggregate income can be changed in consequence of a fiscal policy . in other wordsthe cycle , which originates on money market , is caused by the shifting of the curve is and this shifting can be explained like the demonstration of the fiscal policy .so , this new is - lm model with relaxation oscillations on money market can be the estimation of the fiscal policy impacts .+ with monetary policy it is not so easy .if the central bank try to control real interest rate , then only the short - term real interest rate is influenced .but we are interested in the long - term real interest rate .central bank can influence this long - term real interest rate only indirectly by inflation targeting .the households and firms adapt the expected inflation rate according to prediction of central bank .this is the principle how the expected inflation rate turns to the real inflation rate . in accordance with this ,we demonstrate monetary policy by changing of ( expected ) inflation rate in our models .another tool of monetary policy to influence money market is determination of the constant part of money supply by central bank , i.e. determination of the part of money stock .both of these changes causes the shift of the curve lm up or down . + the real interest rate can be also negative because of inflation ratethe ( expected ) inflation rate can be such that there can emerge the fluctuation of long - term real interest rate from positive to negative level due to relaxation oscillations and the other way around .this phenomenon is very interesting and can explain seemingly unexpected fluctuation from positive to negative level of long - term real interest rate .+ last main section of this paper aims at interaction between fiscal and monetary policy in our new is - lm model with relaxation oscillations . the long - term real interest rate can quickly jump or fall as a consequence of fiscal policy .we suggest how central bank can soften these emerging oscillations by appropriate decrease or increase of the ( expected ) inflation rate or of the constant part of money stock .+ in this paper , we try to find context and explanation of the `` strange '' behaviour of the long - term real interest rate .we found the possible reason of this unexpected fluctuation and we model this situation using new version of is - lm model , theory of relaxation oscillation and description of the special behaviour of money demand and supply . at the endwe suggest a way how soften originating relaxation oscillations .1 .  basic notations : + + [ cols= "< , < " , ]  2.  the economic assumptions of the original is - lm model are the following : * modelled two - sectors economy ( households - firms ) , * it is a demand - oriented model ( the supply is fully adapted to demand ) , * , * a price level is constant ( i.e. absence of inflation effect ) , * a money supply is an exogenous quantity ( means that money supply is some money stock determined by the central bank ) .3 .  according to e.g. the original is - lm model is the system of two algebraic equations : , where , in the static form and the system of two differential equations \\ \frac{d r}{d t } & = & \beta [ l(y , r)-m_s ] , \end{array}\ ] ] where , in the dynamic form .the curve is is a set of ordered pairs  ] which satisfy the equation .the one of the main outcomes of is - lm model is intersection point ( one or more ) of the curves is and lm representing aggregate macroeconomic equilibrium ( i.e. the equilibrium on goods market and on money market , or on financial assets market , simultaneously ) . in dynamic version we research the ( un)stability of this macroeconomic system , i.e. ( un)stability of the equilibrium on goods market and on money market ( or on financial assets market ) simultaneously .we suppose that all of considering functions hereinafter are differentiable .economic properties of functions , and are the following :  this section , we present several models .the first model is new is - lm model on which another models are based on .the other models are the models of seemingly unexpected fluctuation of long - term real interest rate .there we assume money market faster than goods market .so , the second model is model with one s - bend of the curve lm . using this model ,we show how relaxation oscillations can arise on money market .these relaxation oscillations are conditioned by special bend of the curve lm in the form of the letter s. another model is model with twice s - bent curve lm .there originate two different cycles of relaxation oscillations and their conjunction .the last subsection briefly outline the problems of the models with more times s - bent curve lm .our models of seemingly unexpected fluctuation of long - term real interest rate are based on our own new version of is - lm model .the economic assumptions of the new is - lm model are the following :  * an economy consists of two sectors ( households and firms ) , * a demand - oriented model ( the supply is fully adapted to demand ) , * and can be positive , zero and also negative , * a price level is variable ( i.e. included inflation effect ) , * a conjunction of endogenous and exogenous money supply ( means that money are generated by credit creation with some certain intervention of central bank ) .we add to the original is - lm model a floating price level , i.e. inflation effect .so , we need to distinguish two type of interest rate - a long - term real interest rate and a short - term nominal interest rate in addition to the original is - lm model .there is the long - term real interest rate on goods market and the short - term nominal interest rate on money market ( or financial assets market ) .the well - known relation holds , where is a maturity premium and is an expected inflation rate .while and are constants , it holds   then , we consider that the money supply is not strictly exogenous quantity , but the supply of money is endogenous quantity ( money are generated in economics by credit creation ) with some exogenous part ( certain money stock determined by the central bank ) .  [ definition_m ] we define the _ supply of money _ by the formula where represents the endogenous part of money supply and represents the exogenous part of money supply .so , there are the functions and on the money market. it also holds and because of and are constant .+ now , we can define the new is - lm model .[ definition_new_is - lm_model ] the _ new is - lm model _ is given by the following system of two algebraic equations where , in the static form and by the system of two differential equations \\ \textrm{lm : } & \frac{d r}{d t } = \beta [ l(y , r- mp + \pi^e)-m(y , r - mp + \pi^e)-m_s ] , \end{array}\ ] ] where are parameters of dynamics , in the dynamic form .the investment , saving and money demand function have the economic properties ( [ economic_i ] ) , ( [ economic_s ] ) and ( [ economic_l ] ) .we have to put on our new function of money supply the standard economic properties :  the formula ( [ economic_m_y ] ) means that the relation between supply of money and aggregate income is positive and that the rate of increase of money supply depending on aggregate income is smaller than the rate of increase of money demand depending on aggregate income because the banks are more cautious than another subjects .and the formula ( [ economic_m_r ] ) means that the relation between supply of money and interest rate is positive .+ it is easy to see ( using the implicit function theorem ) that the course of the curve lm is increasing in this new is - lm model with properties ( [ economic_l ] ) , ( [ economic_m_y ] ) and ( [ economic_m_r ] ) . if it also holds in additions to the properties ( [ economic_i ] ) and ( [ economic_s ] ) , then the course of the curve is will be decreasing .now , we denote the function , whose graph is the curve is , as and the function , whose graph is the curve lm , as .these functions exist because of the implicit function theorem .now , if we assume then there will exist at least one intersection point of the curve is and lm .+ these presented model is is - lm model without two main deficiencies of the original model and moreover we are certain that there exists at least one aggregate macroeconomic equilibrium .as mentioned above we assume the real presumption that the money ( or financial assets ) market is faster than the goods market .we present sufficient condition for existence relaxation oscillations on money market which is once s - bent curve lm and show where the seemingly unexpected fluctuation of the long - term real interest rate can originate in our situation .+ the relaxation oscillations can originate in the model described by the system of two differential equations when we suppose that first main state variable is changing very slowly in time in proportion to the second variable .so , the derivation of this variable by time is approaching to zero . in our model , we consider that the first main variable is the aggregate income and the second main variable is the interest rate .so , there can exist relaxation oscillations on the money market ( or financial assets market ) .that is related to assumption that the subjects and their reactions on money market are faster than on goods market .so , we suppose that aggregate income is changing very slowly in time in proportion to the interest rate .we can describe this situation by following equations  \\\frac{d r}{d t } & = & \beta [ l(y , r - mp + \pi^e)-m(y , r - mp + \pi^e)-m_s ]   \end{array}\ ] ] where is some very small positive parameter .if the parameter is very small , then we can consider and we rewrite previous equations in the following form .  \end{array}\ ] ]  then , there remains only the curve lm , called isocline of the equation lm .the curve is does not exist .the whole curve lm consists of singular points .+ now , we need to formulate new conditions fulfilled for demand for money and supply of money function .these conditions will be sufficient for existence of relaxation oscillations on money or financial assets market ( it is described below ) .we suppose some unusual behaviour of the demand for money and supply of money . we assume three phases of the course of money demand and money supply function depending on for some fixed . in the first phase , for , these subjects on the money ( or financial assets )market behave usual and the course of the money demand and money supply function is standard how we describe in the condition ( [ economic_l ] ) and ( [ economic_m_r ] ) . in the second phase , for , these subjects behave unusual , precisely reversely .we can describe this behaviour using formulas  these properties correspond to unusual economic situation called liquidity trap .this means that the subjects on money or financial assets market ( demand side ) prefer liquidity despite relatively high level of ( short - term nominal ) interest rate .they own money rather than stocks , although they could have bigger gain because of relative high level of ( short - term nominal ) interest rate .this `` irrational '' behaviour of these subjects can be caused by big risk of holding these stocks and by small willingness to undergo this risk .the supply of money fully adapts to money demand ( we assume demand - oriented model ) . in the third phase , for ,the subjects behave usual as in the first phase .we can see the graphs of money demand and money supply function describing this behaviour on the following figure [ fig : l(i_s)_and_m(i_s)_3_phases ] .   and,height=188 ]  in the points and it holds and .  in the textbelow , we will call this behaviour of money demand and money supply like _ three phases money demand and money supply_.  it is easy to see ( using the implicit function theorem ) that isoclines ( curve is and curve lm ) of the system ( [ dynamic_model_lm_parameter ] ) without requirement of and with properties ( [ economic_i ] ) , ( [ economic_s ] ) , ( [ economic_i_s ] ) and three phases money demand and money supply are such as displayed on the following figure [ fig : curve_is_lm ] .  )( excluding ),height=188 ]  there can occur at least one and the most three singular points .we can see this using the shifting of the curve is or lm downwards or upwards .+ now , we consider the system ( [ dynamic_model_lm_parameter ] ) with requirement of , so we can consider the system ( [ dynamic_model_lm_parameter_2 ] ) in the simplified way .there we can see that the variable is a parameter in equations  ] in this situation ( where is considered as a parameter ) is displayed in the following figure [ fig : curve_lm ] .   as a parameter , height=188 ]  there , every points of the curve lm are singular points .we need to distinguish the stable and unstable arcs of the isocline lm .the arcs and are stable arcs and the arc is unstable arc . this ( un)stability follows from the qualification of possible singular points emerging in the system [ dynamic_model_lm_parameter ] excluding .( on existence of relaxation oscillations on money market ) + let us consider the new is - lm model with very slow changes of variable in time ( [ dynamic_model_lm_parameter ] ) ( including ) with economic properties ( [ economic_i ] ) , ( [ economic_s ] ) , ( [ economic_i_s ] ) and three phases money demand and money supply .then there exist the counterclockwise relaxation oscillations .if we have new is - lm model with very slow changes of variable in time ( [ dynamic_model_lm_parameter ] ) considering , then we can consider the system ( [ dynamic_model_lm_parameter_2 ] ) instead the system ( [ dynamic_model_lm_parameter ] ) .+ three phases money demand and money supply and the economic property ( [ economic_m_y ] ) of the money demand and money supply function ensure the form of the curve lm using the implicit function theorem , see figure [ fig : curve_lm ] .+ the trajectories of the system ( [ dynamic_model_lm_parameter_2 ] ) are directed almost vertically downwards or upwards ( parallel to axis ) considering , see the figure [ fig : one_s - bend_rel_osc ] . up or down direction of the trajectoriesis given by the sign of the function  ] may change along the curve lm in such a way , that these trajectories form some path resembled limit cycle called relaxation oscillation .this oscillation contains the trajectories described by parts of stable arcs and with finite velocity and the trajectories described by vertical segments ( between points and and also between points and ) with infinite velocity ( looking like `` jump '' ) , see the figure [ fig : one_s - bend_rel_osc ] .these trajectories form counterclockwise cycle because of the form of the curve is and stability or unstability of the relevant arcs .thus , we have just constructed the counterclockwise cycle which is one vibration of the the relaxation oscillations .let see the quick `` jumps '' which are parts of our cycles .these quick `` jumps '' seem to be unexpected . andthe existence of relaxation oscillations on money market describing by previous system under certain conditions can explain the ( previously ) unexpected fluctuations of long - term real interest rate .so , this model is the model of seemingly unexpected fluctuations of the long - term real interest rate .   in this subsection, we briefly present how the model with twice s - bent curve lm can look like .presented model is only illustrative .there exist more models depending on concrete bends of the curve lm , but the principle of these models is similar as of this presented model . + we assume the same system ( [ dynamic_model_lm_parameter ] ) , or rather ( [ dynamic_model_lm_parameter_2 ] ) describing the economic situation as in the previous subsection. the economic properties of the demand for money function and money supply function are only changed by the following way .+ according to the name of this illustrative model , the curve lm is twice s - bent .these bends follows from the count of the parts where the demand for money function and money supply function depending on short - term nominal interest rate for some fixed aggregate income behave unusual , i.e. count of the liquidity trap parts .the illustration of this money demand and supply behaviour is displayed on the following figure [ fig : two_s - bends_l(i_s)_and_m(i_s ) ] .   and with two liquidity trap parts , height=226 ]  it is easy to see ( using the implicit function theorem ) that if there are two liquidity trap parts of the money demand and money supply function depending on for some fixed than there will be two s - bends of the curve lm . on the next figure [ fig : two_s - bends_rel_osc ] , we can see the dynamical behaviour in this case , i.e. the relaxation oscillations emerging in this model with twice s - bent curve lm .the arcs , and are stable and arcs and are unstable .the trajectories are directed almost vertically downwards or upwards . up or down direction of the trajectories is given by the stability or unstability of the arcs .the trajectories are attracted to the stable arcs and are drove away the unstable arc .the velocity of the motion along the trajectory is finite near the isocline lm and nearness of the curve lm the trajectories go along the curve lm .the velocity of the motion along the trajectory is infinite large elsewhere .the direction of the path of the moving point is depend on its current position and on the changing of the parameter .+ if the moving point is near the stable arc and if we are changing parameter from the level to , then this moving point will go along the stable arc and then the moving point is attracted from point  ] .so the long - term real interest rate `` jumps '' from the level to .then if we are changing parameter from the level to , the moving point will go along the stable arc and then the moving point is attracted from point  ] .so the long - term real interest rate `` falls '' from the level to .so , we have similar counterclockwise cycle as in previous model ( with once s - bent curve lm ) which consists of two segments looking like `` jumps '' with infinite velocity and two segments corresponding to the part of the stable arc and . +but if the moving point is near the stable arc and we are changing parameter from the level for example to , the moving point will go along the stable arc and then the moving point is attracted from the point  ] .so the long - term real interest rate `` falls '' from the level to .after that the moving point will join to previous described cycle . and if the moving point is near the stable arc and we are changing parameter from the level for example to , the moving point will go along the stable arc and then the moving point is attracted from the point  ] .so the long - term real interest rate `` jumps '' from the level to .after that the moving point will join to previous described cycle .+ we can see that the dynamical behaviour of the moving point is strictly depended on its current position and on the changing of the parameter . in our model , there can be four seemingly unexpected fluctuations of the long - term real interest rate , from the level to , from the level to , from the level to and from the level to .+ at the end of this subsection we have to remind that we illustrate this situation on the concrete model with these concrete bends of the curve lm . if the curve lm will be twice s - bent in a different way , than the trajectories can be different but similar in principle .   in this section ,we illustrate the models with more times s - bent curve lm on the one concrete phase portrait .this concrete curve lm is three times s - bent .we consider all of the previous assumptions .three bends follows from the three parts where the demand for money function and money supply function depending on for some fixed behave unusual , i.e. from the three parts of the liquidity trap .we can see this dynamical behaviour in this concrete situation on the next figure [ fig : three_s - bends_rel_osc ] .+ the trajectories are attracted to the stable arcs , , and and are drove away the unstable arcs , and . in our situation thereexist two counterclockwise cycles .the first cycle consists of two vertical segments with infinite velocity , i.e. segment from the point  ] and from the point  ] , and of two parts of the stable arcs with finite velocity , i.e. parts of and .the second cycle ( one vibration of this relaxation oscillations ) consists of two vertical segments with infinite velocity , i.e. segment from the point  ] and from the point  ] , and of two parts of the stable arcs with finite velocity , i.e. parts of and . +but if the moving point is near the stable arc and if the parameter is increasing from the level then the moving point will go along the stable arc and then the moving point will join to the second mentioned cycle .similarly if the moving point is near the stable arc and if the parameter is decreasing from the level then the moving point will go along the stable arc and then the moving point will be attracted from the point  ] ( with infinite velocity ) and then the moving point will join to the first mentioned cycle .analogously if the moving point is near the stable arc and if the parameter is decreasing from the level then the moving point will go along the stable arc and then the moving point will join to the second mentioned cycle .similarly if the moving point is near the stable arc and if the parameter is increasing from the level then the moving point will go along the stable arc and then the moving point will be attracted from the point  ] ( with infinite velocity ) and then the moving point will join to the first mentioned cycle .+ summary , there can be six vertical segments with infinite velocity which seem to be unexpected , i.e. segment from the point  ] , from the point  ] , from the point  ] , from the point  ] , from the point  ] and from the point  ] , see the figure [ fig : three_s - bends_rel_osc ] .+ we can see that we can expected several fluctuations of the long - term real interest rate depending on the concrete bends of the curve lm , i.e. on the count and type of the liquidity trap parts .fiscal policy can influence the behaviour of the long - term real interest rate and origination of the mentioned quick `` jumps '' or `` falls '' of this variable in the following way .fiscal policy can change the level of aggregate income using its own tools for do this . and this is the core of the government influence .we illustrate the impact of this government decision to change level of aggregate income on the model with once s - bent curve lm ( see the section [ model_once_s - bent_curve_lm ] ) using the next figure [ fig : rel_osc_fiscal_policy ] .if the government decide to increase aggregate income from the level to the level , then the long term real interest rate will be increasing slowly ( according to the stable arc ) , then in the point the long - term real interest rate will jump from the level to the level and then the long - term real interest rate will be increasing very slowly again until aggregate income will be on the level .there can be similar situation if the government decide to decrease aggregate income .the long - term real interest rate would decrease very slowly excluding the point where the long - term real interest rate would fall from the level to .+  in the model with several times s - bent curve lm there can be several jumps or falls of the long - term real interest rate , for example in our presented model with three times s - bent curve lm in section [ models_more_s - bent_curve_lm ] , see figure [ fig : three_s - bends_rel_osc ] .in this situation the economics is represented for example by the point on the stable arc for the level of the aggregate income .then government decide to increase aggregate income to the level for some reasons .then the long - term real interest rate will be growing slowly due to this change according to the stable arc , in the point the long - term real interest rate will jump from the level to the level , then it will be growing slowly along the stable arc until the point where the long - term real interest rate will jump from the level to .+ the fiscal policy use to be demonstrate by shifting of the curve is in the is - lm model .if the government increases the aggregate income the curve is will shift upwards and vice versa .so , we can show emerging relaxation oscillations using the both isoclines is and lm with shifting curve is downwards and upwards in the case with once s - bent curve lm , see the figure [ fig : demonstration_fiscal_policy ] .    in this figure, we can see three singular points , and ( intersection points of is and lm ) . and are stable singular points , is unstable saddle point .then the curve is shifts upwards or downwards according to the changing of the aggregate income until there are two singular points and ( intersection points of is and lm ) , or and ( intersection points of is and lm ) .the point and is stable node or stable focus and the point and is unstable .+ let us remind the economic sense of this model in general . is - lm model describes the equilibrium on the goods market via curve is and the equilibrium on the money market via curve lm and simultaneous equilibrium via theirs intersection points .we can present the cycle which is part of appropriate relaxation oscillations by proper interpretation of the figure [ fig : demonstration_fiscal_policy ] . if the government increases the aggregate income ,the curve is is shifted upwards .the general equilibrium point is shifted slowly due to the change of along the curve lm for example from the the point upwards until the general equilibrium point is the point . then the general equilibrium point is quick shifted from the point to the point along the moved curve is ( here figured as is ) .similarly if the government decreases the aggregate income , the curve is is shifted downwards .the general equilibrium point is shifted slowly due to the change of along the curve lm for example from the the point downwards until the general equilibrium point is the point . then the general equilibrium point is quick shifted from the point to the point along the moved curve is ( here figured as is ) .the central bank realize monetary policy .if the central bank try to control real interest rate , then only the short - term real interest rate is influenced .but we are interested in the long - term real interest rate .central bank can influence this long - term real interest rate only indirectly by inflation targeting .the households and firms adapt their expected inflation rate according to prediction of central bank .this is the principle how the expected inflation rate turns to real inflation rate . in accordance with this, we demonstrate monetary policy by changing of ( expected ) inflation rate in our models .another tool of monetary policy to influence money market is determination of the constant part of the money stock by central bank .both of these changes causes the shift of the curve lm up or down .+ if the central bank increases the inflation rate , then the long - term real interest rate will decrease because of the relation [ relation_i_s_r ] .then the curve lm will be shifted downwards .if the central bank increases controlled money stock , then the curve lm will also be shifted downwards because of the formula of the equation lm ( see the definition [ definition_new_is - lm_model ] ) .this mechanism works also vice versa , i.e. the curve lm is shifted upwards in the case of the decreasing inflation rate or controlled money stock .we can illustrate this mechanism on model with once s - bent curve lm ( see the section [ model_once_s - bent_curve_lm ] ) by the following figure [ fig : demonstration_monetary_policy ] .    now , let focus on situation where the inflation rate or constant part of money stock is on such level when the part of the curve lm exists for negative values of the long - term real interest rate , see the shifted curve lm on the figure [ fig : demonstration_monetary_policy ] . in this casethe emerging relaxation oscillations cause the quick jump of long - term real interest rate from negative to positive value and vice versa the quick fall from positive to negative value .this fact can be very interesting for economic prediction , interpretation etc .   in this sectionwe suggest how this interaction between fiscal and monetary policy can look like . the long - term real interest rate can quickly jump or fall as a consequence of fiscal policy .monetary policy can soften these emerging oscillations by appropriate decrease or increase of ( expected ) inflation rate or constant part of money stock .our aim is elimination of the negative influence of fiscal policy which is seemingly unexpected fluctuation of the long - term real interest rate .if we identify the factors of this problem then we could reduce this fluctuation by the controlled change of money stock or inflation rate .+ we demonstrate this interaction on model with once s - bent curve lm ( see the section [ model_once_s - bent_curve_lm ] ) by the following figure [ fig : interaction_fiscal_monetary_policy ] .we consider the case of fiscal expansion , means the increase of aggregate income .let start on the stable arc of the curve lm for the level of the aggregate income .government increases aggregate income from to , so the moving point is going along the stable arc to the point  ] , denoted by .so the long - term real interest rate is forced to jump to the level to the part of the stable arc . in this momentthe central bank could increase constant part of the money stock or ( expected ) inflation rate to shift the curve lm downwards such that the point would be the point $ ] .the shifted curve lm is figured as the curve lm. now , the moving is still on the required stable arc ( now figured ) but the long - term real interest rate does not jump to the level and is still on the level .+ the defence contra seemingly unexpected falling of long - term real interest rate is similar . at the right momentthe central bank should decrease or to shift curve lm upwards .we created several models based on new version of is - lm model and theory of relaxation oscillations to explain seemingly unexpected fluctuation of long - term real interest rate .is - lm model describes the equilibrium on the goods market via curve is and the equilibrium on the money market via curve lm and simultaneous equilibrium via theirs intersection points .new version of is - lm model eliminates two main deficiencies of original model which are assumptions of constant price level and strictly exogenous money supply .we present and illustrate model with once , twice and more times s - bent curve lm .+ in the second main part of this paper we pointed out the impacts of the fiscal and monetary policy on these models .fiscal policy can cause the relaxation oscillations on money market and monetary policy can soften the forced falls or jumps of the long - term real interest rate .in these days , many experts and also the public more and more talk about unexpected fluctuation of different phenomena in economics and about impact of these fluctuations on economics . in this paper , we show one point of view on this problem .we try to model some unexpected fluctuation of the long - term real interest rate on money or financial assets market using own new version of is - lm model and theory of relaxation oscillations .+ we show several models strictly depended on the behaviour of money demand and money supply function . if there exist one or more unusual parts of these functions describing economic situation called liquidity trap , then mentioned seemingly unexpected fluctuations of long - term real interest rate will arise on money market . in these days ,existence of liquidity traps is not so uncommon .the observation of actual economic situation supports this idea .+ every possible considering model can be a little bit different .it depends on concrete situation . in every model ,the trajectories are little bit different but the same in principle .there can also originate controversial situations where we can not determinate the path of the moving point .for example the situation where the s - bends of the curve lm is exactly the same .but these situations are not very probable from the economic sense. any economic quantity does not behave still exactly in the same way .there is always any at least small difference .the research was supported , in part , by the student grant competition of silesian university in opava , grant no .sgs/19/2010 .  99 barkov , l. : _ asymptotic behaviour of solutions of differential equations and its applications in economics _ , dissertation thesis , masaryk university , brno , 2004 .bogoliubov , n. n. ; mitropolsky , y. a. : _ asymptotic methods in the theory of non - 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Abstract: in this paper , we create a model of unexpected fluctuation of the long - term real interest rate . + this model is based on our new version of is - lm model . the new is - lm model eliminates two main deficiencies of the original model : assumptions of constant price level and of strictly exogenous money supply . the unexpected fluctuations of the long - term real interest rate can be explained by existence of special type of cycle called relaxation oscillation on money ( or financial assets ) market . relaxation oscillations include some short parts looking like `` jumps '' . these `` jumps '' can be interpreted like unexpected . in other words , we try to explain these `` unexpected '' fluctuations of long - term real interest rate and show that these fluctuations can be only seemingly unexpected . + last but not least , we show some impacts of the government intervention by fiscal or monetary policy on economics using this models . then we suggest possible interaction between fiscal and monetary policy .