many complex systems exhibit some degree of organization at different physical scales .often , the organization is hierarchical .there exist examples of this fact in variegated fields , like biological , social and technological systems . among the former , and starting from complex molecules ( such as lipids , proteins , rna or dna ) while increasing the scale of observation , new levels of organizationare found : organelles , cells , tissues , organs , anatomical systems , organisms , populations and ecosystems . in the social context ,human societies organize from the level of individuals , groups , cities , up to the global scale of countries or continents .finally , among technological systems , computer networks are also arranged at different scales from the local network level up to the domain level routing systems that constitute the backbone of internet .hierarchical organizations seem ubiquitous in complex systems and , despite the early interest of the scientific community about the subject  , it is far from being fully understood .the description of hierarchical organization of complex systems remains , to a great extent , at the semantic level .this is mainly because the following difficulties : the existence of several relevant physical scales , the existence of a variety of organizing principles , the large number of components , and the lack of a generally enough and well defined formal theory for the identification of hierarchies .the study of complex networks  plays a central role in the characterization of the organization of complex systems .in essence , networks are used to represent the structure of the interactions between the components of the system under consideration .therefore , it is reasonable to assume that some complex networks have hierarchically organized topologies , reflecting the underlying hierarchical organization of the associated complex systems . a natural way of thinking about hierarchical network topologies is that of hierarchical community structures ; i.e. communities within communities of nodes  . typically , the identification of the communities of a network is computationally intensive and a statistically difficult problem  .although a large number of community detection methods have been developed already  including methods for the identification of hierarchical community structures   not all methods provide comparable results .this is true , specially for hierarchical community structures .therefore , similarity measures for the comparison of hierarchical community structures are of crucial importance .the aim of this paper is to introduce an information - theoretic tool which can be used to compare hierarchies , or trees , which might be composed of network communities .we further show that this tool can be employed to trace the evolution of hierarchies when temporal networks are analyzed .  a standard way to quantify the similarity of two community structures is to compute the mutual information between the associated node partitions  . extending the idea , the present paper introduces a _ hierarchical mutual information _ , generalizing the traditional mutual information to work with hierarchical partitions . in principle, there might be different ways in which the mutual information can be generalized into a hierarchical mutual information . in this work , hierarchies are considered to be of divisive nature ; i.e. the whole is divided into parts , each of which is sub - divided into sub - parts , an so on , following a top - down approach . as a consequence , in thiscontext hierarchies are represented by trees with branches of varying length .other possible generalization approaches might exist .for example , generalizations that consider agglomerative hierarchies i.e. bottom up approaches or overlapping communities. alternatively , related methods exists for the comparison of phylogenetic trees  .recently , a method to compare hierarchies was introduced ; the method follows a combinatorial approach  .however , to the best of our knowledge , no previous method based on information - theoretic measures , exists for the comparison of hierarchies .these alternative methods , and the previously mentioned generalization approaches , are not discussed further in this paper , but can be considered in future works .the outline of the paper is the following . in section[ sec : theo ] , the hierarchical mutual information is motivated and introduced . in section [ sec : resu ] , this measure is tested on different synthetic setups .more specifically , in sub - section [ sec : resu : arthier ] , the behavior of the hierarchical mutual information is tested in artificial hierarchies ; while in sub - section [ sec : resu : artnets ] , the hierarchical mutual information is used to analyze the hierarchical community structure of artificial networks , or network models .a similar procedure is performed on empirical networks in section [ sec : resu : empnets ] , including the case of a temporal one .finally , the discussion and conclusions are summarized in section [ sec : conclu ] .a hierarchical partition is a generalization of the traditional concept of partition . here, each element of the partition can be recursively partitioned into others , yielding a hierarchy .the formal definition is as follows .consider a set of elements , or universe , denoted by .an element in is denoted by .the set splits into a hierarchy of sub - sets , denoted by .the number of elements in the sub - set is written as . the _ hierarchical partition _ , or simply hierarchy ,is represented by a tree denoted by .the root is the `` oldest ancestor '' of the various vertices , or descendants in the tree . as a sub - set, the root contains the whole set of elements , i.e. . for any sub - set , denotes the set of direct descendants of .a sub - set is at the -th level ( or depth ) of the hierarchy if is the topological distance from to .when there is no confusion , we simplify the notation to , i.e. by omitting the reference to .consider a network of nodes and links ( weighted or not ) . here , the terms _ elements _ and _ nodes _ are used interchangeably ; both , referring to the entities denoted by .traditionally , the community structure of a network is represented by a node partition . in many cases ,these communities present a hierarchical organization .in particular , if the hierarchy is constituted by sub - communities within communities , then the structure can be mapped to a hierarchical partition .depending on the context , is referred to as a tree , as a hierarchical community structure , or simply as a hierarchy ; i.e. the terms are used interchangeably .each sub - set corresponds to one and only one sub - community of the network hierarchical community structure ( see fig .[ fig:1]a ) .the root represents the set of all nodes in the network .the children correspond to a partition of the sub - community into sub - communities .the leaves of are the smallest sub - communities of the network .finally , each sub - community has an associated sub - network with links , between the pair of nodes .   in this section , the definition of the _hierarchical mutual information _ is motivated .only shannon - based information measures are used throughout the rest of the paper  .consider how the uncertainty about the identification of a specific node is reduced when going down a tree . as the root represents the set of all nodes , to look for a specific node requires checking binary choices .in other words , the uncertainty is reduced by nats when a node is unequivocally identified ( a nats is a unit of information equals to bits ) , and there is no uncertainty left . sometimes the information pointing towards a specific node is not precise , and the uncertainty reduction is not complete .for example , if node is specified to be in the sub - community , the uncertainty reduction is nats , and nats of uncertainty still remains .traversing a hierarchy along descendants is similar to a sequential reduction of uncertainty .more specifically , it is possible to write where is the deepest level at which node can be found . each term can be considered ed a conditional uncertainty reduction .specifically , how much the uncertainty is reduced when new information is gained ( that ) , given that some other information was already available ( that ) .it is possible to average over nodes using an appropriate weighted version of the expression in eq .( [ eq:1 ] ) .more specifically , the average uncertainty reduction along the tree is defined as where , for simplicity , we wrote the equation for the particular case in which all branches of the tree have the same length , i.e. for all .the general case is introduced later in sub - section [ ss : theo : hmi : def ] . in eq .[ eq:2 ] , every reduction step is weighted by the fraction of nodes that are found by following the corresponding branch of the tree . using similar ideas ,the hierarchical mutual information is defined in the next section .   in community detection problems, it is customary to quantify the similarity between two inferred community structures using the mutual information between the corresponding node partitions  . here , the goal is to introduce the hierarchical mutual information to quantify the similarity between two hierarchical partitions , or trees , associated to corresponding hierarchical community structures .consider two trees and and two sub - communities and , both at the same topological distance , or level , from the roots of their corresponding trees .it is not necessary for the trees and , nor the sub - communities and to contain the same elements .let represent the sub - tree of root obtained from .the analogous holds for . the hierarchical mutual information between the sub - trees and is denoted by . by definition, it is assumed that if either or is a leaf of the corresponding tree .otherwise , is recursively defined by the formula in eq .[ eq:3 ] , the first term of the r.h.s .is called the _ one step mutual information _ , and is defined as where represents the shannon entropy .these terms are computed as and in all cases , the convention is adopted . finally , the hierarchical mutual information of two full trees and is denoted and defined by where and are the roots of and , respectively .each term involved in is non - negative , and thus , the hierarchical mutual information is a non - negative quantity . also , , i.e. it is a symmetric function of its arguments .when the trees and are just stars , i.e. a root plus one generation of descendants , it is possible to think of them as standard partitions . in this case , the hierarchical mutual information reduces to the standard mutual information .note , the hierarchical mutual information is not a measure of the similarity between the corresponding final partitions of the nodes at the leaves of the trees ( except when both trees are stars ) .rather , it is a summation of weighted local one - step contributions , measuring how similar the partitions are at each corresponding point in both trees .for example , if two nodes and are separated at level in tree and at level in tree then , the separation of and contributes with zero to the value of the hierarchical mutual information .  for practical purposes ,a _ normalized _ hierarchical mutual information is defined as we would like to notice the reader that there exists more than one way to normalize the mutual information . here , we work with one that takes inspiration from the cauchy - schwarz inequality but , future experiments may prove other normalization methods to be more convenient depending on the particular context in which the hierarchical mutual information is used .the value of lays in the interval  ] links , and its average degree when . to make the analysis more interesting, a fraction of the links in the sierpinski networks are randomly rewired .the rewiring procedure is well - known  .essentially , successive pairs of links , each of which is chosen at random , swap the nodes at their extremes until a fraction of the links is affected . in this way , there is a well - defined hierarchy of communities for , which is progressively blurred out as increases .  in the following sections ,the different community detection methods , and these two network models are combined into a set of experiments analyzed using the normalized hierarchical mutual information .   and .the network contains nodes , 4 big communities of 128 nodes each , and 16 small communities of 32 nodes .the darkest links connect pairs of nodes sharing the same small community at level , the links of intermediate brightness connect nodes sharing intermediate communities at level , but not sharing the same small communities . finally , the brightest links connect pairs of nodes at level , but not nodes sharing communities at levels and . ]each network model has an associated natural or _hierarchy , denoted as .specifically , the reference hierarchy for the sierpinski and hpm models are shown in figs .[ fig:1]a and [ fig:4 ] , respectively .the hierarchies identified by the community detection methods are not necessarily equal to the reference ones , and in some cases , they do nt even resemble it .the degree of fidelity of the community detection method measures how similar are the identified communities to the reference ones .formally , given a community detection method , a network sample and a reference hierarchy , the _ hierarchical - fidelity _ or simply , _fidelity _ of the community method is defined as the average normalized hierarchical mutual information , .the average is computed summing over an ensemble of hierarchies , obtained by repeatedly identifying the hierarchical community structure of the network , using the chosen community detection method . in the results shown ,the ensemble was composed by 100 hierarchies .furthermore , the fidelity is averaged by sampling 100 networks from each network model .the procedure is repeated for different values of the network models parameters , and using the different community detection methods .  for the case of the hpm ,two different model re - parameterizations are used . in one case the whole network structure change simultaneously , while in the other case , only one level is affected  .more specifically , in case 1 all parameters , and are linearly re - parameterized by  ] , while diagonal entries are equal to one . to simplify the analysis ,the correlation matrices are transformed according to the expression  , .the transformation returns a weighted network of non - negative entries and zero diagonals .the transformed networks are the ones used for the computation of the hierarchies . even though more sophisticated approaches exist( see for example ref . ) , for the sake on simplicity the approach taken is the one described above .  to perform a temporal analysis , different correlation matrices , or weighted networks ,are computed by processing the data over different time windows  ] , and the other set from the time window defined days after .we analyze the auto - similarity varying for fixed , and varying for fixed . in the first case ,we study how the time separation affects the hierarchy , and in the second case we compare hierarchies corresponding to consecutive time windows as time evolves . in fig .[ fig:12]b , both quantities are plotted . on the one hand, the auto - similarity decays as the time separation grows ( circles ) , i.e. the hierarchy drift away from the initial structure . on the other hand, the auto - similarity fluctuates around ( triangles ) , indicating that the hierarchies of consecutive time windows always share a significant amount of information ., determines how the hierarchies change with the time length of the time window over which the data is processed . in b ) , two different comparisons are presented using the auto - similarity . with circles , determines how similar are the hierarchies at day one , with the hierarchies days after . with triangles, determines how similar are the hierarchies of consecutive time windows , separated by 100 days , as time evolves . in all cases ,the bars represent standard - deviations around the mean ., title="fig : " ] + , determines how the hierarchies change with the time length of the time window over which the data is processed . in b ) , two different comparisons are presented using the auto - similarity . with circles , determines how similar are the hierarchies at day one , with the hierarchies days after . with triangles , determines how similar are the hierarchies of consecutive time windows , separated by 100 days , as time evolves . in all cases ,the bars represent standard - deviations around the mean ., title="fig : " ]in this work , the hierarchical mutual information has been introduced , a tool that generalizes the standard mutual information for the comparison of hierarchies .more specifically , for the comparison of hierarchical partitions , which take the form of trees where parts are subsequently subdivided further into sub - parts and so on .the hierarchical mutual information can be used to compare the hierarchical community structure of complex networks , in analogous way as the standard mutual information can be used to compare standard community structures .we define here a normalized hierarchical mutual information .the traditional normalized mutual information satisfy certain properties ; it is a quantity lying in 12 & 12#1212_12%12[1][0] * * ,  ( ) * * ,  ( ) _ _ ( , ) * * ,  ( ) _ _( , ) _ _ ( , ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) http://dx.doi.org/10.1007/s10618-013-0335-9 [ * * , ( ) ] * * ,  ( ) * * ,  ( ) in  _ _ ( , ) pp .  * * ,  ( ) * * ,  ( ) * * ,  ( ) _ _ ( , , )  * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( )  * * ,  ( ) * * ,  ( ) ( ) http://www.wcoomd.org [ ] * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) _ _ ( )  in  _ _ ( )  * * ,  ( ) http://finance.yahoo.com [ ] * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ( ) ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( )

Abstract: the quest for a quantitative characterization of community and modular structure of complex networks produced a variety of methods and algorithms to classify different networks . however , it is not clear if such methods provide consistent , robust and meaningful results when considering hierarchies as a whole . part of the problem is the lack of a similarity measure for the comparison of hierarchical community structures . in this work we give a contribution by introducing the _ hierarchical mutual information _ , which is a generalization of the traditional mutual information , and allows to compare hierarchical partitions and hierarchical community structures . the _ normalized _ version of the hierarchical mutual information should behave analogously to the traditional normalized mutual information . here , the correct behavior of the hierarchical mutual information is corroborated on an extensive battery of numerical experiments . the experiments are performed on artificial hierarchies , and on the hierarchical community structure of artificial and empirical networks . furthermore , the experiments illustrate some of the practical applications of the hierarchical mutual information . namely , the comparison of different community detection methods , and the study of the consistency , robustness and temporal evolution of the hierarchical modular structure of networks .


problem solving is an integral part of most physics classes . there is ample research on the benefits of developing problem - solving skills ( e.g. ) , and even a whole line of research on problem - based learningunfortunately , for the most part physics classes in high school and introductory physics in college still today heavily rely on end - of - chapter exercises .research has shown that proficiency at solving end - of - chapter exercises does not promote proficiency at solving conceptual problems , although proficiency at conceptual problem solving promotes proficiency at exercise solving .scaffolding students to use prescribed problem solving strategies is helpful for both kinds of problems .students prefer to frame problems narrowly , eschewing real - world connections and connections to deeper principles in favor of merely getting through the problem at hand . why should nt they ?they have a large number of problems to solve every week , and if their instructor uses an online grading management system like webassign or masteringphysics , they are graded only on answers , not framing or principles .it s time - consuming and grade - producing .this relates to the concept of utility value : the degree to which a task is perceived as being relevant beyond the immediate situation . in this paper , we take up the question `` what is the utility of exercises ? '' from the lens of what solving exercises teaches students about learning physics .recent research has shown that how learning is framed can influence what strategies students use , how much they enjoy learning activities , and ultimately what they transfer .in addition to the technical content that students learn while they solve problems , students also learn cultural content about what physicists value .for example , physicists value reducing complicated scenarios into simpler ones ( e.g. the spherical cow ; the frictionless table ; the massless string ; the point particle). physicists value problem solving , particularly in mathematical ways .however , the ways in which physicists value problem solving and the kinds of problems we value as physicists depend on setting of the problem and the people doing the solving .  to articulate the differences between physicists - solving - problems and physics - students - solving - problems , we turn to accountable disciplinary knowledge ( adk ) . from this perspective, knowledge is facts , information , and skills acquired by a person through experience or education . in turn, disciplinary knowledge is the knowledge related to a specific field or discipline , such as physics .accountable disciplinary knowledge ( adk) is `` what counts '' as important to a community or discipline : the kinds of activities that participants engage in , the problems they value and solve , and methods for solving those problems . demonstrating one s adk is central to being accepted as a member of a community of practice . because a community s adk is continually being rendered and refined by its members , the norms and expectations of the community ( or `` what counts '' ) will vary across both communities and across time .  in communities of professional physicists ,adk includes posing , solving , and evaluating novel research questions .often the same person ( or research group ) who poses the question will be the one to answer it .in contrast , introductory and high school physics classroom communities adk often include solving short problems with answers in the back of the textbook . in these classroom settingsteachers are typically responsible for assigning problems , and students are responsible for performing them .problem solving in either community is an integral part of their adk , but the details of what counts as a problem , how it should be solved , and what is important are different .as the physics classroom is seen as the pathway to preparing and inducing new members to the professional physics community , examining the disconnect between these versions of problem solving may provide insight into the nature of the broader populations lack of scientific literacy .in this paper , we use the adk framework to investigate what counts " as solving problems well in a high school physics classroom from the perspective of the students .we present the case of `` stu '' , a high school student presenting an end - of - chapter - style problem to his peers , to argue that students adk around problem solving centers around use of correct procedure to produce correct answers .we use this example , drawn from a reform - based , epistemologically - aware classroom , to argue for the existence of a disconnect in adk between physicists and physics students , even in `` best case '' scenarios like stu s class .we frame this paper as an existence proof of what can be learned about students adk from their problem solving , not as an exhaustive study into all the things students do with screencasts .we use adk ( outlined above ) in concert with problem solving to study students behaviors and values in physics classrooms . in the pastthere have been several discussions about the importance of teaching students the proper methods to solve problems .there are classes that are based on problem - based learning , because studies have shown that solving problems develops students critical thinking skills , which in turn helps them develop a better understanding of the concepts behind the problems .cognitive theories describe the problem - solving process in four steps: ( 1 ) understand the problem ; ( 2 ) devise a plan ; ( 3 ) carry out the plan ; ( 4 ) verify or evaluate solution . of course , these steps may be recursive for large or complex problems .good solving - problem skills are an important part of adk for both physicists and students . however , the kinds of problems that students solve are simpler , requiring few sub - problems or recursions .physicists believe that the process of solving problems in physics classes is necessary and important practice for future professional situations .in contrast , most students seem to believe that the process of solving problems is a requirement for them to do well at school , just another step they need to get through .students epistemological beliefs affect how and what they learn , as well as their future professional aspirations . in this paperwe will discuss how discrepancies in students and teachers adk may also have an effect on the student learning process .this research is part of a larger project that was conducted in five high school physics classes ( four regular classes and one advanced class ) in an urban area of a u.s .western state. the school was in a lower socio - economic status community which was primarily composed of students from non - dominant backgrounds .table [ tab : schooldemo ] shows the school demographics .minority groups composed the majority of the population in the school .average student enrollment in physics was around 140 students , in classes with an average enrollment of 35 .most physics students were juniors .the remainder were sophomores .this study focuses on a classroom artifact from a single ap physics class that was composed of 32 juniors and seniors .hispanic & white & asian & afr .+ 56% & 32% & 8% & 3% + & esl / fep & iep + & 49% & 11% +  the class , like all physics classes in this school , was taught by a teacher with a background in biology , including a ph.d . in biochemistry . like the majority of us high school teachers of physics, the teacher did not have a degree in physics .the teacher was recruited for the research project through her role as a teacher leader in a teacher - driven professional development community called streamline to mastery. as a member of the program , the teacher aided in researching her own classroom practices in addition to performing her own action research projects. because of the class participation in this program , the classroom was outfitted with enough ipads to have one for every student , a technological affordance which dramatically changed classroom practice .   as part of the class requirements , students had a notebook where they solve problems selected by the teacher .in addition to solving traditional end - of - chapter problems in their notebooks and worksheets , students selected one of their assigned problems for screencasting and sharing with classmates .screencasting is a technology that captures a video of the ipad s screen while using the microphone to capture and merge audio to that file .this technology allows students to create various types of dynamic presentations . within the physics class ,the students assignment prompts were to use screencasts to create a tutorial teaching how to solve a problem of their choice from a specific worksheet or chapter on their book . how these tutorials were to be created and what they should look like was not specified by the teacher and was left for the students to determine .the students posted their screencasts on the class edmodosite ( a facebook - like social media site created specifically for education ) .after students posted their screencasts online , the teacher prompted students to view several peers work and to provide their peers with feedback on the quality and usefulness of their screencasts .through viewing their peers work and commenting on it both online and in class the students developed norms for what constituted `` good '' screencasts.  typical screencasts consisted of students talking through their thought process for solving a problem while writing out the solutions in real - time or while showing specific parts of a solution they had written down prior to recording the screencast . because the screencasts record the display of the ipad , they do not include the students hands or gestures ; to draw their classmates attention to specific parts of the screen , students often circle or underline with emphasis .each screencast is a presentation " that the students create for their peers .students sometimes use funny voices or altered inflection to indicate that they are reading the problem statement . in this analysis , we are not concerned with the production value of the screencast , but with what the student is trying to communicate .we draw on the screencast as a source of evidence about what the student values and perceives as `` doing well '' when solving a physics problem .  even though the practice of making and sharing screencasts is novel, we contend that the practice of solving this kind of problem in this kind of method is deeply typical of us physics education : it s very well - aligned with how the ap physics test assigns credit ( and subsequently the teacher gives credit ) .it s also well - aligned with prescriptive models of problem solving ( e.g. ) .the screencast affords us a glimpse of the adk evinced in problem solving .because the screencasts were created by the students in their own time , there is no record of students initial reactions to the problem and how they tackled it .this performative , reflective aspect of their practice amplifies students behaviors around `` good '' problem solving , because they are highlighting for their peers what they believe it is important in the process .for the purpose of this study we chose to use case - based research method to study students adk through screencast .the decision of choosing case - based research was due to the fact that we are producing an in - depth descriptions and interpretations of a student s screencast solution of a physics problem to establish possible causal links between students belief of what counts to do well in physics and their performance while explaining his peers how to solve a physics problem . whilst it will not answer a question completely, it will give some indications and allow for further elaboration on the topic of adk in classroom behaviors and the goals of problem - solving skills .  to evaluate students beliefs of doing well ; i.e. what student perceived as adk we coded the screencasts students created for the class to classify students behaviors .the type of behaviors we were interested in were confidence level , signs of reasoning and understanding , such as language used to explain the solution of the problem , the degree of detail on the steps to solve the problem and the reasoning and sense making while explaining the solutions .  because we are interested in how students adk may be visible in their classroom behaviors , for the purpose of this paper we selected a screencast where there were interesting disconnects between stu s professed values in problem solving ( e.g. getting the right answer , using the right method ) and his performance of the problem in the screencast . in resolving these disconnects, stu makes parts of his adk apparent .even though we have chosen this particular screencast for this paper , other screencast that were analyzed presented similar behaviors .the steps taken to do the analysis began with several group viewings of the screencast . in these viewingswe discussed our interpretation of the screencast , differences of opinions regarding tone of voice or student s attitude .for example , there was a debate on whether the student sounded surprise or not when his answer did not match the book s answer .after reviewing and discussing it several times , we reach an agreement . once all differences were ironed out , the screencast was transcribed and each researcher conducted individual viewings of the screencast . through the use of micro - genetic analysis , each researcher individually examined stu s problem solving discourse as he solved a specific physics problem about air pressure .stu s expectations for adk were investigated through examination of his language use , prosody , and tone .because of the nature of screencasts , we could nt follow his body language ; however , we also examined his `` screen language '' : the ways he connected his argument visually and drew emphasis to particular parts of the screen .the screencast was viewed many times by all researchers .the individual analysis was followed by group consensus building , which led to the creation of a generative coding scheme .generative coding is the process of creating codes according to salient features of the data ( as opposed to _ a priori _ codes ) .the application of these codes to the screencast created the empirical foundation for our interpretation and claims about the student s adk . through multiple viewings , assisted by the transcript, we developed emergent codes and claims about the events in the screencast. through discussion , we reached consensus about our interpretation , claims , and evidence .our claim is that an element of students adk in physics class is to follow a defined procedure , which ( if applied correctly ) will certainly result in a correct answer .the answer can be verified in the back of the textbook .following procedure is more important than conceptual understanding , critical thinking , or real - world reasonability of results , and the student is so focused on the procedure in his performance that he neglects other important details .this adk is in stark contrast to professional physicists adk about problem solving and also to the teachers intended adk for the students .we support our claim with observations from the screencast .stu selects a problem about air pressure for his presentation . in the problem ,the students are to calculate the force on a sheet of paper measuring by due to air pressure , given an air pressure of .unknown to stu , the published answer on the back of the book is wrong .it reports , where instead it should report .as it was mentioned above stu had previously solved the problem and is creating the screencast with the purpose of teaching " his classmates how to solve that problem .   ]   ]   as he presents his solution , he follows the classical steps of problem solving , writing first the screen in figure [ fig : step1 ] and then the one in figure [ fig : step2 ] :  stating what is known : :  he writes on the ipad the variables the problem provides define the unknowns : :  in this case it is explicitly given on the problem , because it asks  for the force .although once he starts performing calculations , he  identifies another unknown , which is the area he will need to  calculate , in this case the force .setting up equations : :  stu writes no explanation is given about  why he decided to use that particular equation or what the equation  and its parts mean , other than the fact that is necessary .they were  given an area and a pressure and learned that   .once he writes the equation for force he  points out the variable pressure that was given and he moves on to say  that they need the area and writes the equation for area as width x  length .again , no explanation as to why that quantifies the area . performing calculations : :  in the calculation part he is careful to write on the screencast the  operations .he points out that before calculating the area he needs to  transform from cm to m the variables of width and length .he does not  comment on why he needs to change from cm to m , but he goes on detail  about the operational way of doing it , as with the other calculations .  from what we observed , it was more important to him to teach and communicate to his peers the operational way of doing the steps to solve the problem , as opposed to the reasons why those steps were taken .that behavior agrees with our claims about the student s adk .even more evidence of it comes when he reaches the answer and there is a discrepancy between his answer and the book s answer .  throughout the problem, he emphasizes that the steps he is taking to solve the problem are the correct procedure , but does not explain the conceptual reasoning behind his choices , such as why it s necessary to convert to si units .the number that he produces at the end ( figure [ fig : step2 ] ) is not the same as the one reported in the back of the textbook .one difference is that the book s answer is in scientific notation and has fewer significant figures , but we do not know if this is one of the facts that he sees as a problem or that the numbers even in the correct notation do not match .   at the end of his screencast , stu says :  stu : :  i checked with ( mumbled ) to see what answer we were supposed to get  and it s , so i do not know about that .you  just saw what i did here , so i guess it is possible i just mess  something up butall i know is that all i know is that this are  the numbers i got .i followed the equations , and this is the answer i  came up with here 6078 n. but according to this is supposed to be   .he states that he can not understand the discrepancies between his answer and the book s answer .he emphasizes that he followed the procedure , therefore his answer must be correct .he never mentions the difference in notations or shows a reasoning process for making sense between his answer and the book s answer. he does not show any evidence of valuing real - world reasoning or critical thinking ; however , he does reassure his audience ( his peers ) that if they follow the procedure correctly , they will get the correct answer . from thiswe infer that following procedures correctly to obtain correct answers ( which can be found in a trusted authority source ) is an important part of adk , but real - world reasoning or sense - making are not .a reasonable counterargument might be that the student was solving a problem for the first time , in real time , for the screencast .if he did nt get a chance to prepare his answer , his solution might be choppy .however , we notice that the student s performance in the screencast could not have been his first time solving the problem . in presenting the problem , he mistakenly reports an area as instead of .however , in following steps , his force calcuations use . also , we noticed that he does not do a self - check , not even to verify his procedure , which he seems so confident of .this could be because he had already solved the problem and tried to revise it before doing the screencast and could not find an error .additionally , his voice does not sound surprised or altered when reporting the book s answer to be different than his .this also indicates that he is not reasoning in the moment , but just doing mechanical steps because he can not reconcile the idea of his answer being incorrect even though he followed the appropriate procedures.  the student values the correct procedure as the way to get the correct answer , and he advises his peers to use that procedure , even preferring it over his own answer .again it is clear that what he thinks is important to teach to his peers is the steps needed to take in order to get the answer , not the reason for taking those steps .stu : :  so , if you see a mistake in this process here , go on and change it to  try to get this answer here i guess i do nt see why this is  the answer , but try to get using this  method , correcting wherever i made my mistake . andif my answer is the  right answer that is how you get it , but it s not though . aim for  this , but hopefully this process will help everyone .it seems like the student has a very clear idea of the procedure : he seems confident on what he is doing and the steps he needs to take to solve the problem .this is what counts for him as doing well ; this is his adk for a physics class .furthermore , as was mentioned earlier , as part of the class students had to complete sets of problems in their notebooks and one of the problems as a screencast . in both assignments ,the teacher never formally graded the solutions . in a previous study screencasts and notebookswere scored for correctness and completeness . in thispaper results show that when solving problems in screencasts students solutions were 21% more complete then when solving problems in their notebooks . at the same timethe students answers in the screencasts were 33% more likely to be correct than when solving the notebook problems .this corroborates our claim , since it shows that for students it was more important to detail steps and procedures when teaching their peers than when just solving a problem for the teacher to evaluate .in this paper , we present an existence proof that students adk about problem solving is both visible in their problem solving discourse and substantially different from that of physicists . on the one hand , this is not a revolutionary result : we already `` know '' that solving end - of - chapter style exercises is a different kind of thing than conducting physics research .however , through this work we bring forth the idea of adk as a useful theory for analyzing what counts as doing well as a function of what kind of physics ( e.g. high school classrooms ) we study .  to make a more solid conclusions about problem - based learning and changes in adk we would need to test different types of problems and control for variables , such as students reactions when they first read the problem and think about the solution , because these screencasts are a finalized version of the solution that the student has already edited. this will provide with more information about the thinking process of the students before embarking in the problem - solving procedure and a less mechanical response .future work could include involve interviews with students to establish what students are thinking and how they approach the problems initially .however , the performative aspects of interviews are substantially different than for these ecologically valid screencasts , and we anticipate that aspects of students discourse will change in the new setting.  as a technology for education , screencasts ( and ipads more broadly ) are important new research tools because they can give insight into how students understand physics and communicate it to their peers . in traditional pen - and - paper problem solving , in which researchers have access only to students submitted inert solutions , screencasts present a new lens into the processes of students problem - solving and communicating . in this sense, they are the polar opposites of online homework systems .pedagogically , it is an open question as to whether screencasts improve student learning , or whether this kind of problem solving is one we d like to promote . in related work , we show that screencasts increase students sense of ownership by acting as boundary objects , which is a valuable learning goal in and of itself . in a related data set from upper - division physics , students who complete more homework problems in pencasts ( a close technological relative to screencasts ) are more successful in the remainder of their work for the class .the authors are indebted to insightful colleagues who offered criticism on this work ( in alphabetical order ) , paul irving and valerie otero .we are also grateful to the instructor who shared her class with us .this work was partially supported by the ksu department of physics and the national science foundation ( due-934921 ) .any opinions , findings , and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the national science foundation or other funders .33ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1'''' [ 0]secondoftwosanitize [ 0 ] + 12$12 & 12#1212_12%12[1][0] http://www.engr.washington.edu/caee/caee{_}briefs{_}pdfs/becominganengineering{_}stevens{_}jee08.pdf [ * * ,  ( ) ] in  __ , ( , , ) * * ,  ( ) link:\doibase 10.1119/1.17117 [ * * ,  ( ) ] * * ,  ( ) link:\doibase 10.1080/10508400903452876 [ * * ,  ( ) ] link:\doibase 10.1103/physrevstper.9.010111 [ * * ,  ( ) ] * * ,  ( ) link:\doibase 10.1016/s0732 - 3123(99)00031 - 0 [ * * ,  ( ) ] link:\doibase10.1037/0022 - 0663.100.2.398 [ * * ,  ( ) ] http://www2.physics.umd.edu/{~}davidham/studentresources.html [ * * ,  ( ) ] link:\doibase 10.1103/physrevstper.11.020121 [ * * ,  ( ) ] ,  link:\doibase 10.1103/physrevstper.10.010109 [ * * ,  ( ) ] _ _ ( , ) _ _( , ) http://www.whitehouse.gov/ostp/pcast[__ ] , ( , , ) _ _ ( , , ) link:\doibase 10.1002/sce.21043 [ * * ,  ( ) ] link:\doibase  10.1207/s15327809jls1201_3 [ * * , ( ) ] \doibase 10.1023/a:1022939512735 [ * * ,  ( ) ] in  _ _ , ( , ) _ _ ,  ph.d . thesis ,  ( ) in link:\doibase 10.1119/perc.2013.pr.053 [ _ _ ] , ( , ) pp .link:\doibase 10.1063/1.4789739 [ * * , ( ) ] link:\doibase 10.1063/1.3086101 [ * * ,  ( ) ] in link:\doibase 10.1063/1.3680073 [ _ _ ] , , , ( , , ) pp .  inlink:\doibase 10.1063/1.3680061 [ _ _ ] , , , ( , , ) pp.  link:\doibase 10.1103/physrevstper.10.020106 [ * * ,  ( ) ] link:\doibase 10.1080/10508400701413443 [ _ _ ] , vol .( ) pp .  * * ,  ( ) link:\doibase  10.1207/s15327809jls0401_2 [ * * , ( ) ] link:\doibase 10.1002/sce.21014 [ ( ) ]  ( )

Abstract: we define knowledge as facts , information , and skills acquired by a person through experience or education . in turn , disciplinary knowledge is the knowledge related to a specific field or discipline , such as physics . accountable disciplinary knowledge ( adk ) is what counts `` as important to a community or discipline : the kinds of activities that participants engage in , the problems they value and solve , and methods for solving those problems . because adk is determined on a community basis , the details of ' ' what counts `` will vary with the norms and expectations of the community . in this paper , we use the adk framework to investigate ' ' what counts " as solving problems well in a high school physics classroom from the perspective of the students .


nonlinear dynamics is a topic that not only cover all the disciplines , in both natural and social sciences , but also is now becoming part of introductory level undergraduate courses in sciences . searching for affordable , easy to setup , and reconfigurable classroom demonstrations that allow to investigate the physical and mathematical nature of nonlinear dynamical systems has been always a matter of interest for instructors .this paper describes a simple apparatus used to explore the behavior of one - dimensional chaotic maps , like the logistic map , when different initial conditions are chosen .the device consists of an inexpensive open - source microcontroller connected to an array of light emitting diodes ( leds ) , and programmed to iterate the logistic map in the one - dimensional interval  ] . the function is called an _ iteration function _ because we may find the population fraction in the following period cycle by repeating the mathematical operations expressed in . it is also one - dimensional since there is only a single variable , and the resulting curve is a line .we show a photograph of the apparatus setup in fig .[ fig01 ] , and its schematics in fig .[ fig02 ] .  setup of the logistic map device .an array of ten leds are connected to the arduino microcontroller using the same number of resistors . ]circuit diagram of the ten leds array connected to the arduino . ]the apparatus uses an open - source arduino prototyping platform made up of an atmel avr processor ( microcontroller ) .it has 14 digital input / output pins , 6 analog inputs , a 16 mhz crystal oscillator , a usb connection , a power jack , an icsp header , and a reset button . arduino is a registered trademark only the official boards are named `` arduino''so clones usually have names ending with `` duino . '' the arduino can be connected to a computer through the usb port and programmed using a language similar to c++ .the program is uploaded into the microcontroller using an integrated development environment ( ide ) . in the arduino world , programs are known as _we connected an array of ten leds to the digital pins of the microcontroller using resistors ( or for dimmer leds ) .we also used a solderless breadboard to hook up all the electric components to the microcontroller .listing [ list01 ] shows a simple program ( sketch ) used to operate the microcontroller , and _ visualize _ the logistic map by looking at the array of blinking leds .a glowing led represents an iteration of the one - dimensional map , and it is linked with a value in the interval  ] in ten identical segments and associate an led to each one of them .for example , the first led from the left represents the first interval segment , the second led represents the segment , and so on .the last led represents the final segment  ] forms a _ basin of attraction_.  another example of an attractor happens when and ( ) .we see that , after few iterations and blinking leds , there is only one led that remains on indefinitely .this led corresponds to the interval . in this case, we have a _ fixed point_ occurring at . as it is shown in fig .[ fig03](b ) , the orbit of the map follows a square spiral that converges into that fixed point .now , if and ( ) , we can observe a 2-_period cycle _ ; the value of oscillates between 0.558 and 0.765 , and the two corresponding leds turn on and off intermittently . here , the fixed point becomes a _repeller_. figure [ fig03](c ) shows the periodic behavior of the logistic map for this set of parameters .unfortunately , higher periodicity ( like 4-period , 8-period cycles , or higher ) can not be observed clearly with this device because of the way the leds are mapped into the  ] .figure [ fig03](d ) shows all points visited by the logistic map orbit after iterations .  ....// blinking logistic map   //choose the pin for each led  const int nbrleds  = 10 ;  const int ledpin [ ] = { 2,3,4,5,6,7,8,9,10,11 } ;   const int wait = 500 ;  //wait for 500 milliseconds   //logistic map parameters  const double a  = 3.7 ;  // logistic map constant   double x0 = 0.2 ;// initial position ( 0 < x0 < 1.0 )  double x  = x0 ;  // use x0 as your first calculated point   // setup ( ) initializes the led pins  void setup ( ) {    for ( int i = 0 ; i < nbrleds ; i++ ) {      pinmode(ledpin[i ] , output ) ;    }  }   // loop ( ) iterates the logistic map and turn on / off leds  void loop ( )  {    if ( x < 0.1 )      blinkled(ledpin[0 ] ) ;    else if ( ( x > = 0.1 ) & & ( x < 0.2 ) )      blinkled(ledpin[1 ] ) ;    else if ( ( x > = 0.2 ) & & ( x < 0.3 ) )      blinkled(ledpin[2 ] ) ;    else if ( ( x > = 0.3 ) & & ( x < 0.4 ) )      blinkled(ledpin[3 ] ) ;    else if ( ( x > = 0.4 ) & & ( x < 0.5 ) )      blinkled(ledpin[4 ] ) ;    else if ( ( x > = 0.5 ) & & ( x < 0.6 ) )      blinkled(ledpin[5 ] ) ;    else if ( ( x > = 0.6 ) & & ( x < 0.7 ) )      blinkled(ledpin[6 ] ) ;    else if ( ( x > = 0.7 ) & & ( x < 0.8 ) )      blinkled(ledpin[7 ] ) ;    else if ( ( x > = 0.8 ) & & ( x < 0.9 ) )      blinkled(ledpin[8 ] ) ;    else      blinkled(ledpin[9 ] ) ;     // iterates the logistic map function    x0 = x ;    x  = a * x0 * ( 1.0 - x0 ) ;  }   // blinkled function  // turn on / off leds  void blinkled(const int pin ) {    digitalwrite(pin , high ) ;  //turn led on    delay(wait ) ;        // wait 500 milliseconds    digitalwrite(pin , low ) ;  //turn led off  } ....with this apparatus , students may understand more easily the behavior of one - dimensional chaotic maps looking at an _ actual _ one - dimensional array of leds ; every point in the interval [ 0,1 ] produces another point within the same interval after the logistic map is iterated . by controlling the parameter and the initial value , the instructor and students can observe under what conditions periodic and chaotic behavior may occur , providing a better understanding of periodicity , bifurcations , and the route to chaos of a nonlinear dynamical system .the setup described here is inexpensive , easy , and fun to assemble , also enhances computer programming and electronics assembly skills .

Abstract: the logistic map is one of the simplest nonlinear dynamical systems that clearly exhibit the route to chaos . in this paper , we explored the evolution of the logistic map using an open - source microcontroller connected to an array of light emitting diodes ( leds ) . we divided the one - dimensional interval $ ] into ten equal parts , and associated and led to each segment . every time an iteration took place a corresponding led turned on indicating the value returned by the logistic map . by changing some initial conditions of the system , we observed the transition from order to chaos exhibited by the map .


the standard cosmological model successfully describes a wide range of observational phenomena using just six parameters in the context of a framework that requires only a handful of basic assumptions .the ` ' model is so successful in fact that attention has focussed on deviations from this simple picture . just like with particle physics, one can gauge the maturity of the field through the fact that activity switches from establishing the validity of the framework to searching for physics ` beyond the standard model ' .the most precise cosmological data come from careful measurements of the anisotropies in the cosmic microwave background ( cmb ) , as surveyed by the _ , _ wmap _ and _ planck _ satellites , as well as with a suite of ground - based and balloon - borne experiments ._ cobe _ helped establish as the standard cosmological model , and for many people the biggest news from both _wmap _ and _ planck _ was that there is no news , i.e.the standard model continues to be a good fit , even as the precision has improved dramatically . since the days of _ cobe _( e.g.ref . ) there has been intensive investigation into deviations from gaussianity or the breaking of statistical isotropy on large scales , motivated by the fact that they could be smoking guns for new physics in the early universe .this process has continued to the present day , with a huge number of studies searching for blemishes of various forms in the cmb sky , which might be evidence for chinks in the armour of the standard model .now it seems like this search has become the main industry in cosmology .hundreds of papers have been written on this topic .we can not possibly refer to all of them here , but many can be found in the reference lists of several overviews of the subject . in the cmb sky ,such deviations from perfection are often referred to as ` anomalies ' , and there are several distinct features of this sort that have been identified .sarkar et al . pointed out that there are at least two unrelated kinds of anomalies and hence one can take the product of the chance probabilities to determine how unlikely our observed universe is in the standard model .recently schwarz et al . have argued that there are at least _ three _ distinct kinds of anomaly , and since each of them separately has a probability of in the standard model , then our cmb sky is unlikely at the roughly 1-in - a - million level .similar conclusions were also made in another recent paper by melia .in fact considerably more than three kinds of anomaly have already been identified , and no doubt more remain to be uncovered .they include : the cold spot ; the low- deficit ; the dip ; hemispheric asymmetry ; low variance ; dipole modulation ; odd / even multipole asymmetry ; and other specific features , such as rings , stripes , wiggles , bites and even letters .when considered together , the combined probability is vanishingly small that our sky could be a realisation of a gaussian random process within .amazing though these cmb features are , we have discovered that a much more familiar data set , namely the digits of  , contains equally astonishing anomalies . through the use of advanced analysis methods , using pattern recognition software and bayesian search algorithms , we have discovered many distinct features in .in fact , for each cmb anomaly there appears to be a corresponding effect in the digits of  we call these analogous anisotropy anomalies ( or aaas ). these features in are _ at least _ as significant as the cosmological ones , and given that the cmb anomalies are patently evidence for new physics , this means that the patterns in must demonstrate the existence of _ mathematics _ beyond the standard model .the irrational number  occurs throughout physics , in anything related to rotations , waves , vibrations or phases .it occurs explicitly in coulomb s law , kepler s third law , einstein s field equations , the fourier transform , the normalization of a gaussian , the reduced planck constant , etc .and perhaps most astonishingly , if you divide the circumference of a circle by its diameter , you get exactly !  unlike other numbers , attracts enormous attention , with contests to memorise its digits , and it even has its own day . the number has also inspired a style of writing called ` pilish ' ( whose origin has been attributed to sir james jeans ) , in which one uses words whose lengths match sequentially to the digits of .furthermore it is known that the digits of eventually have to contain messages , and there is speculation that these may reveal basic truths about the nature of reality .  by developing novel computational techniques, we have performed extensive searches for patterns in the digits of , similar to those already carried out for the cmb sky . for definiteness , in most of the investigations that we describe below , we focus on the first 10,000 digits of  , since this is similar to the number of modes probed by today s cmb mapping experiments .we leave it to other researchers to extend our study to the trillions of additional digits that are now known . like the cmb sky , we also assume that the digits of are unchanging , although previous studies have already been carried out to investigate possible temporal or spatial variations of .let us begin with one of the most widely discussed features in the cmb sky , namely the ` cold spot ' .despite the fact that the power spectrum of the cmb anisotropies carries around worth of information , the apparently 23 significance of the cold spot is fantastically important . a quick glance at fig . [ fig : nocoldspot](a ) shows just how prominent this feature is on the sky . although not the large blue region near the centre , and not actually the coldest place on the cmb sky , the chances of finding a cold region of exactly this size and shape in gaussian random skies is quite small . andthe chances of finding such a spot in precisely this direction is almost vanishingly small .  in the digits of is an analogous feature , which might be called a ` hot spot ' , since it involves a cluster of the number ` 9 ' . specifically , ` 9 ' occurs six times concurrently after the 762nd digit of , as shown in fig .[ fig : nocoldspot](b ) .this is sometimes called the ` feynman point ' , and appears fantastically earlier in the digits of than one would expect for such a repeated run .one can estimate the probability of this aaa in the following way : the number of possible outcomes for 762 digits is ; the number of places the run ` 999999 ' could be located within the first 762 digits is and the number of possibilities for the other 756 digits is ; and hence the probability is .  it has also been pointed out that the digits of contain _ seven _ consecutive 9s at about the same position as the six 9s in .this is of course even less likely to occur by chance .there are many other examples of unlikely runs of digits within  .for example , if we focus on the number ` 4 ' , we find that the sequence ` 4444 ' occurs much _ later _ than expected ( in fact it is the last of the sets of four identical numbers ) , not until digit 54525 . and amazingly ` 44444 ' is also the last sequence of its type to occur ( i.e. after ` 00000 ' , ` 11111 ' , etc . ) , not until digit 808651 .this of course leads us to ask what is special about 4 and about 9 what previously hidden mathematical principle could explain this behaviour ?the large - scale cmb sky has lower variance than expected , and this can be examined in several different ways , e.g. by estimating the variance of the temperature field or some filtered power spectrum estimator .one popular statistical approach is to examine the correlation function of the cmb , which is shown in fig .[ fig : lowvariance](a ). it is very noticeable that above a precisely defined scale ( say about ) the correlation function is very nearly approximately zero . and the chances of this are surely tiny .this result is so dramatic that the universe is clearly trying to tell us something perhaps there is an anthropic connection between the lack of large - scale correlations and our existence ?this deficiency in the variance at low multipoles has an analogy in the behaviour of the early digits of . in the first 100 digits ,the numbers 0 to 9 all occur fairly close to the average number of times ( which is 10 ) , as shown in fig .[ fig : lowvariance](b ) .the biggest deviation is for the number 9 , which occurs 13 times , this being exactly from the mean , with no other number occurring with a frequency that is more than from the expectation .this therefore represents a very ` quiet ' distribution . using poisson statistics ,the chances of picking 100 random digits and finding one digit more than 13 times is 0.220 and the chances of finding it less than 8 times is 0.136 ; hence for each of the 10 digits we have and then the total probability of having such a quiet distribution is .it is clear that the early digits of are already telling us something through the fact that none of these integers wants to stand out .in fact there are indications that this low variance continues further , and we can determine that the expected number for digits is , with , and the variance is . in the first 1,000 digits only one number ( ` 1 ' in this case ) occurs more than from the expected frequency , and a similar pattern is even seen out to 10,000,000 digits , where ` 4 ' is the only number occurring more than from the expected fraction .   another often - discussed anomaly is the lack of power in the at low multipoles .this manifests itself as a general lowering of power for multipoles below about 40 and a specific ` dip ' that is noticed at .such features would be buried at the lowest multipoles if we were to plot versus linear , but can easily be seen when we carefully plot the lowest multipoles on a logarithmic scale , as shown in fig .[ fig : nodip](a ) .this low- dip has a corresponding aaa in the digits of , manifesting itself in the lack of the number ` 0 ' .[ fig : nodip](b ) shows the cumulative count of 0 in the digits of . for the lowest digitsthere is an obvious lack of 0s .in fact the first 0 does not occur until the 33rd digit , the probability of which can easily be estimated to be a few percent .hence this is interesting , but perhaps not very remarkable in itself .however , we find that consecutive strings of the number 0 are also under - represented .for example the pattern ` 000000 ' is the only string of six consecutive numbers that does _ not _ occur in the first million digits of ; in fact it does not show up until digit 1,699,927 .furthermore , if we consider the binary digits of we discover that there are _ many more _ instances of 0 than 1 for the lowest digits .in particular 61% of the first 164 bits are 0 , which is extreme enough to have a probability of just 0.3% ( using the binomial distribution for 164 trials ) .the fraction of occurrences of ` 0 ' stays well above 50% for several hundred more bits , showing that the binary digits also exhibit non - random features that require an explanation .many studies have pointed out that the cmb sky can be split into two hemispheres in which there is a dramatic difference in the power between each half of the sky .this can be studied equivalently by determining the amplitude of the dipole modulation of the sky . in fig .[ fig : hemisphere](a ) we reproduce a plot of the probability of obtaining the observed dipole amplitude in simulations of gaussian skies ( plotted here for each of the four distinct foreground - separated cmb maps produced by the _ planck _ team , but these differences are not relevant here ) .it is clear that the scale is quite special , and yields a low probability of about .we can examine an analogous statistic in the digits of .to do so we consider the digits up to some maximum digit , and split them into the first half and the second half . by removing the average ( i.e. the value 4.5 for the set of integers 0 to 9 ), we calculate the sum of the two separate halves of the digits , and plot this against the maximum digit in fig .[ fig : hemisphere](b ) .the two halves of the digits of behave in a dramatically different way , with an index around 2700 being the preferred scale here , but the difference continuing out to the highest digits examined .this aaa thus demonstrates that the digits of have the same sort of ` lop - sidedness ' as the cmb sky .obviously the lowest - order cmb multipoles are special , and so should be examined particularly closely .it has been noticed that there is an ` axis of evil ' , in that the preferred directions of these lowest multipoles are dramatically lined up with each other .the extraordinarily tight alignment of the axes of the dipole , quadrupole and octupole are demonstrated in fig .[ fig : alignment](a ) .this has been estimated to have a probability of order 0.1% .we can search for an aaa alignment in itself , but in order to have something to align it with we need to find another number .the obvious choice is ` e ' , which occurs in physics and mathematics almost as often as .indeed both numbers occur in the famous euler identity . this may be the most compact and meaningful mathematical expression found by humanity so far , since it includes the operations of subtraction , multiplication and exponentiation , as well as the numbers , 1 and 0 , in addition to and .another reason to combine with is the obvious significance of these two numbers for humans , since the number of pairs of chromosomes possessed by homo sapiens is 23 , and this is the integer bracketed by and  .  when we compare and digit by digit , we find that there are systematically fewer matches than expected . in the first 130 or sodigits there are quite a few matches ( e.g. in the 13th , 17th and 18th digits ) , occurring at a little more than the expect rate of 1 time in 10 . however , after the 130th digitthe fraction of matches is consistently low all the way out to the 10,000th digit , as shown in fig .[ fig : alignment](b ) .one can estimate the significance of this result , e.g. using a kolmogorov - smirnov test on the deviation , yielding a probability corresponding to approximately a effect .this is a remarkable result , showing that although and _ should _ be composed of random digits , and hence entirely uncorrelated , they are in fact _ anti_-correlated .there are two obvious possibilities : this relationship could come from a subtle connection through the euler identity ; or , more fundamentally , it could be pointing to these digits being far from random , but actually contain meaningful patterns .another effect seen in the cmb power spectrum is that odd multipoles are systematically higher than even multipoles , a property sometimes referred to as ` point - parity asymmetry ' .this can be seen in the _ planck _ data plotted in fig .[ fig : nosawtooth](a ) , where there is a very striking ` sawtooth ' pattern for odd versus even multipoles .there is an analogy when we compare the odd and even digits of . if we take the first digits of and split them into pairs , we find that the first digit is sometimes bigger than the second one in the pair , but sometimes the opposite is true . for example in , and , but . however , if we repeat this exercise , then more often than not we find that the odd digits are larger than the even digits .this is shown in fig .[ fig : nosawtooth](b ) , where we have plotted the sum of the difference between odd digits and even digits of .the probability of achieving such a systematic difference over so many pairs of digits is exceedingly small .one lesson here is that the odd digits of are generally bigger than the even digits , and hence if you wanted , as a party trick , to learn _ either _ the odd digits or even digits of , you would be better to pick the odd digits .but what could this ` parity ' aaa be telling us ?it was pointed out by the _ wmap _ team that one can find stephen hawking s initials in the cmb sky , as shown in fig . [ fig : initials](a ) .these same features , clearly being two roman characters in roughly the same size , font and orientation , are also present in the _ planck _ data .we are not aware of any systematic search for all combinations of letters , with arbitrary position , font , size and orientation .but presumably such a search would turn up many other interesting examples of writing on the ` cosmic billboard ' .words can also be searched for in the digits of . in order to dothat one needs to select a scheme for converting the digits into letters .we are aware of four separate methods to do this .firstly one could write the digits of in base 36 , so that after ` 0 , 1 , 9 ' we have ` a , b , c , ' , including the whole of the alphabet .secondly one could just take base 26 and map it on to the alphabet ( or base 27 , to include a space ) ; using this method we find that the word ` odd ' occurs considerably more often than the word ` even ' , which is just like the cmb point - parity asymmetry .thirdly one could determine in binary and then use groups of five binary digits to map to the alphabet ( plus a few punctuation marks ) ; in this scheme in fact the word ` anomaly ' appears at binary digit 1,769,000,467 .lastly one can use the more complicated transformation that we adopted to generate fig .[ fig : initials](b ).  one sees in this figure that it is possible to find much more noteworthy patterns than the mere pair of initials found in the cmb sky .for example , across the top of the plot we can just about discern the middle initial for ` s.h . ' and the right - hand half of the figure contains several aligned multipoles . for readers interested in further studying this` word search ' diagram , we have provided details of our own investigations in an appendix .it is obvious that for this aaa , it is relatively easy to find features in the digits of that are substantially more anomalous than what we see in the cmb .the cmb sky contains a combination of several independent anomalies , with a joint probability that makes what we observe extremely unlikely in the standard model , hence indicating evidence for several kinds of new physics .but perhaps even more amazingly , there are similar kinds of feature in the digits of and these are every bit as significant .the large - scale cmb sky presents a fixed pattern to us , and hence if one finds anomalies in this pattern , there is no more sky to use to confirm or improve the significance of what was initially found .the situation with the digits of is analogous , since any anomalies discovered can not be corroborated by performing experiments on these fixed digits .  some have argued that such features in the digits of need to be considered in the light of a posteriori statistics , or in other words that one should consider the multiplicity of other tests that could have been carried out , to marginalise over similar features that _ might _ have been found , using what is sometimes called the ` look - elsewhere effect ' .others have argued that we should look elsewhere for an explanation , that resorting to such statistical sleight - of - hand is the last bastion of scoundrels , because it can be used to torpedo _ any _ apparently effect .be that as it may , this issue about overestimating the statistical significance of the anomalies surely can not apply to the many dramatic features seen on the large - scale cmb sky .and hence we must also accept the importance of all the structures that also exist within the digits of .what could these aberrations be telling us ? andhow are they related to other apparent coincidences in nature ?if we are seeing anomalies in the decimal digits of , then presumably there is something special about base 10 .and hence we are indeed created in the image of the creator of not only the entire physical universe , but also the reality of mathematics .probably there is an anthropic argument that we could only exist in a reality in which mathematics and physics contain these signatures .no doubt further study of these joint anomalies will give us insight into other dimensions and universes that are otherwise unobservable .indeed the ` pi theorem ' of buckingham is a constraint on the number of dimensions within physics .can we conceive of being different in these other dimensions ? and does this teach us about the nature of the relationship between physics , mathematics and theology ?another connection is that the cmb temperature was once equal to , and that epoch corresponded to a redshift of , about 2billion years in the past  did something happen on earth around this time that connected the development of life with the background radiation and the digits of ?  as a final remark , we add this utilitarian view if the aaas mean that the cmb information is somehow _ already _ encoded in , then perhaps in future we can avoid all the fuss and bother of building real cmb experiments , minimising systematic effects while operating them , painstakingly analysing the data , and debating the statistical interpretation of the results and instead simply look more carefully at the digits of , or in any other random string of digits .= 1.6pt  9 scott d. , narimani a. , page d.n .canada , 2014 , 70 ( 4 ) , 258 [ arxiv:1309.2381 ] . for what this acronym might stand for ,see http://www.astro.ubc.ca/people/scott/cmbstandfor.html .smoot g.f . , et al . , 1992 , apj , 396 , l1 .bennett c.l ., et al . , 2003 , apjs , 148 , 1 .planck collaboration 2013 results i , 2014 , a&a , 571 , 1 ; planck collaboration 2015 results i , arxiv:1502.01582 .bromley b.c ., tegmark m. , 1999 , apj , 524 , l79 .it has been said , for example , that if the _ planck _results were an episode of sesame street , they would be brought to you by the words ` tension ' and ` anomaly ' .de oliveira - costa a. , et al . ,prd , 69 , 3516 ; copi c.j ., et al . , 2010 , adast , 2010 , 847541 .bennett c.l . ,et al . , 2011 , apjs , 192 , 17 .planck collaboration 2013 results xxiii , a&a , 571 , a23 ; planck collaboration 2015 results xvi , arxiv:1506.07135 .sarkar d. , et al ., 2011 , aph , 34 , 591 .schwarz d.j ., copi c.j . , huterer d. , starkman g.d . , arxiv:1510.07929 .melia f. , 2015 , aj , 149 , 6 .a selection of such claims are to be found in these papers : cayon l. , smoot g. , 1995 , apj , 452 , 487 ; pando j. , valls - gabaud d. , fang l. , 1998 , prl , 81 , 4568 ; magueijo j. , ferreira p.g ., grski k.m ., 1999 , aipc , 478 , 176 ; tegmark m. , de oliveira - costa a. , hamilton a.j . , 2003 , prd , 68 , 123523 ; mukherjee p. , wang y. , 2003 , apj , 598 , 779 ; eriksen h.k . , et al . , 2004 , apj , 605 , 14 ; roukema b.f . , et al ., 2004 , a&a , 423 , 821 ; hunt p. , sarkar s. , 2004 , prd , 70 , 103518 ; land k. , magueijo j. , 2005 , mnras , 357 , 994 ; hsu s. , zee a. , 2006 , mprla , 21 , 1495 ; covi l. , et al ., 2006 , prd , 74 , 083509 ; tocchini - valentini d. , hoffmann y. , silk j. , 2006 , mnras , 367 , 1095 ; bernui a. , et al ., 2007 , a&a , 464 , 479 ; robitaille p.m. , 2007 , prog . in phys . ,vol . 3 , iss . 1 , p. 3 ; cruz m. , et al . , 2007 , sci , 318 , 1612 ; aurich r. , et al . , 2008 ,cqgra , 25 , 125006 ; pietrobon d. , et al . , 2008 ,prd , 78 , 103504 ; rossmanith g. , et al ., 2009 , mnras , 399 , 1921 ; ichiki k. , nagata r. , 2009 , prd , 80 , 083002 ; gurzadyan v.g . , penrose r. , 2010 , arxiv:1011.3706 ; ayaita y. , weber m. , wetterich c. , 2010 , prd , 81 , 023507 ; vielva p. , 2010, advast , 592094 ; kovetz e.d . ,ben - david a. , itzhaki n. , 2010 , apj , 724 , 374 ; verschuur g.l . , 2010 ,apj , 711 , 1208 ; zuntz j. , zibin j.p ., zunckel c. , zwart j. , 2011 , arxiv:1103.6262 ; yoho a. , ferrer f. , starkman g.d ., 2011 , prd , 83 , 083525 ; santos l. , et al . , 2014 , a&a , 569 , a75 ; achcarro a. , et al . , 2014 ,prd , 89 , 3006 ; hu n. , torrafo j. , 2015 , prd , 91 , 4039 ; hunt p. , sarkar s. , 2015 , jcap , 12 , 052 ; chary r. , 2016 , apj , 817 , 33 .there are also the killjoys who have tried to refute some of the claims , or reassess their statistical significance , including : banday a.j ., zaroubi s. , grski k.m . , 2000 , apj , 533 , 575 ; mukherjee p. , hobson m.p ., lasenby a.n . , 2000 ,mnras , 318 , 1157 ; efstathiou g. , 2003 , mnras , 346 , l26 ; slosar a. , seljak u. , 2004 , prd , 70 , 083002 ; scott d. , zibin j.p . , 2005 , arxiv : physics.11135 ;zhang r. , huterer d. , 2010 , aph , 33 , 69 ; moss a. , scott d. , zibin j.p . , 2011 , jcap , 04 , 033 ; feeney s.m . , et al . , 2012 , prl , 108 , 241301 ; peiris h.v ., 2014 , iau symp . ,306 , 124 ; de abreau a. , contreras , d. , scott d. , 2015 , jcap , 12 , 031 .there are many stories about why is used as the symbol for this number , for example that the reflection of 3.14 looks like .tegmark m. , ` our mathematical universe , my quest for the ultimate nature of reality ' , 2014 , knopf .we all learn that 22/7 is an approximation for , but in fact the ratio 355/113 ( first obtained by the chinese mathematician and astronomer zu chongzhi 1500 years ago ) is considerably more accurate , so accurate ( correct to less than 0.00001% ) that the chances of finding such a ratio with two 3-digit numbers is extremely small , and one has to get to 52163/16604 in order to do better .similar amazing ratios for the cmb temperature are listed here : http://www.astro.ubc.ca/people/scott/whochosetemp.html .reciting the digits of rational numbers ( like ` 3 ' for example ) would be much more straightforward .although only really in north america , and not , for example , in europe , since there is no 31st of april . jeans is meant to have composed this pilish sentence corresponding to the first 15 digits of : ` how i need a drink , alcoholic in nature , after the heavy lectures involving quantum mechanics ! ' for the topic of this paper we offer this clumsy effort : ` and a mark i found entwining pi , belies skies !cmb alike isotropy anomalies dwindle ' .sagan c. , ` contact ' , 1985 , simon and schuster .images of the microwave sky of this sort are usually called ` cmb maps ' , while diagrams showing the digits of are referred to as ` charts ' .obtained from the internet , and so almost certainly correct .zibin j.p . , moss a. , scott d. , 2007 , prd , 76 , 13010 ; lange s. , page l. , 2007 , apj , 671 , 1075 ; moss a. , zibin j.p . , scott d. , 2008 , prd , 77 , 3505 .scherrer r.j . ,knox , l. , 2012 , arxiv:1204.0298 .scott d. , et al ., arxiv:1603.03550 .sometimes ( but not often ) called the ` smoot pit ' ; c. lineweaver , private communication .moreover , most of the rest of the sky is a dramatic hotter .after richard feynman , who is sometimes credited with the story that if you could learn to 762 digits , you could say ` nine , nine , nine , nine , nine , nine , and so on ' . however , there is no documented evidence that feynman ever said this .it is probably an example of stigler s law of eponymy .emphasized by michael hartl as part of a claim that is more than twice as good as .see tauday.com .one can find more examples using this search facility : www.angio.net/pi/piquery .das s. , et al ., 2014 , jcap , 04 , 014 .story , k.t ., et al . , 2013 , apj , 779 , 86 .land k. , magueijo j. , 2005 , prl , 95 , 1301 .although we do nt understand what is ` evil ' about it , and instead prefer the term ` shaft of light ' .perhaps even more important than some of the patterns we have found in the digits of .we leave as a future exercise the search for euler s identity encoded into the digits of and themselves .although it is a bit harder to see the zero than the other numbers .this number also being connected with humans through the well - known ` birthday problem ' , where the chances of two people sharing a birthday is above 50% if you have a group of 23 or more people .there is a related coincidence that 365 times is approximately the number of pairs of 23 items ( d. page , private communication ) .furthermore , everyone knows that if you combine pi and e you get pie , which is usually a good thing .this study can easily be extended to a much larger number of digits ( m. kopp , private communication ) . of course you could easily trick people by claiming to know all the odd digits of , and then simply reciting ` 1 , 3 , 5 , 7 and 9 ' .see the page http://www.cadaeic.net/picode.htm . this is the method used at david bailey s site pi.nersc.gov , where one can search for one s name in the first 4 billion binary digits of .the fact that the word ` hot ' occurs in these digits dramatically earlier than the word ` cold ' is further demonstration that prefers hot spots rather than cold spots . however , for the cmb , it may be possible to search for related anomalies in the polarization data , .see for example ` the garden of forking paths : why multiple comparisons can be a problem , even when there is no `` fishing expedition '' or `` p - hacking '' and the research hypothesis was posited ahead of time ' , gelman a. , loken e. , 2013 , columbia university and to mix metaphors where no one has gone before .mark twain would no doubt have quipped that this is a debate about ` anomalies , damned anomalies , and statistics ' .see some earlier papers with related ideas : scott d. , frolop a. , 2006 , astro - ph/0604011 , 2007 , astro - ph/0703783 , 2008 , arxiv:0803.4378 , 2014 , arxiv:1403.8145 ; ali frolop , ali and frolov , arxiv:1504.00108 .buckingham e. , 1914 , pr , 4 , 345 ; another example of stigler s law .barrow j.d ., 1993 , ` pi in the sky : counting , thinking , and being ' , back bay books ; we particularly like the first part of the title .speaking of which , the phrase ` pie in the sky ' was coined by joe hill in his satirical song ` long - haired preachers ' , published in 1911 ( a. melott , private communication ) ; hill anticipated much of what is in this paper `you will eat , bye and bye , in that glorious land above the sky , work and pray , live on hay , you ll get pie in the sky when you die . that s a lie . ' c. uhlenberg , private communication ; this may be consistent with the time of the first binary black hole merger detected by ligo .it may also be worth noting that at the last - scattering epoch the cmb temperature was .comments on this paper should be sent to the lead author , dr . frolop .we would like to thank caastro for their hospitality when the last parts of this paper were being completed one of us spent time at the university of queensland and the other at mount stromlo observatory , and both of us enjoyed our time in australia , where they really like pies .adams d. , 1979 , ` the hitchhiker s guide to the galaxy ' , pan .[ fig : initials](b ) presented some of the digits of transformed into the normal letters of the alphabet .precisely which digits these are , and how we performed the translation , are uninteresting details we guarantee that this set occurs _ somewhere _ in .we have performed extensive analysis of this set of letters , searching for meaningful words .we believe we can find all of the following : stephen william hawking ; peebles , penzias , wilson , mather , smoot ; euler , feynman , gauss ; pi ( several times ) , pie ( twice ) , digit ; cmb , cosmic , microwave , background , anisotropy ; multipole , monopole , dipole , quadrupole , octupole , hexadecapole , dotriacontapole ; act , bicep , cobe , planck , spt , wmap ; cold spot , hot spot , low - l dip , alignment , parity , anomaly , hemispheric , asymmetry , stripe , coldest , deviations ; a posteriori , look elsewhere , marginalize , fluke ; forty - two ; ali frolop ; april fool .since the _ lack _ of a deviation may carry the most information , when all these words have been found , the remaining letters may spell out the ultimate question .

Abstract: deviations of the observed cosmic microwave background ( cmb ) from the standard model , known as ` anomalies ' , are obviously highly significant and deserve to be pursued more aggressively in order to discover the physical phenomena underlying them . through intensive investigation we have discovered that there are equally surprising features in the digits of the number , and moreover there is a remarkable correspondence between each type of peculiarity in the digits of and the anomalies in the cmb . putting aside the unreasonable possibility that these are just the sort of flukes that appear when one looks hard enough , the only conceivable conclusion is that , however the cmb anomalies were created , a similar process imprinted patterns in the digits of .