text:
Talk:Language teaching/Teaching reading

Hilaryy 20:57, 26 February 2011 (UTC)
 * I made some minor edits to the actual chapter itself, mostly just typos and little spelling/grammar mistakes

Research
A good variety of academic and government-issued sources was used throughout the chapter, and I thought it was evident that the topic had been thoroughly investigated. There were a few occasions when pioneers in reading education were mentioned, and I thought this chapter could have benefitted from going back and citing their original work instead of a subsequent review of their studies (e.g. Huey, Webster). I think the reader would benefit from more detail on the reading programs mentioned, such as Kumon and Sylvan, and what they aim to do and the different ways they go about addressing reading problems.

Logic/Organization/Structure
The chapter flowed nicely, and I thought overall it progressed logically, answering questions as they seemed to naturally occur to the reader. The only suggestion I would make in terms of organization is to consider moving the ‘Roles of a Teacher’ section and making it section 5, just before ‘Assessing Reading Ability’. The teacher section could also be grouped with the section on teacher intervention, so that all of the discussion about teachers and reading occurs at the same point. I would also suggest adding a Conclusion or Summary in subsequent drafts to revisit the key points of the chapter and revisit the questions that were asked in the Introduction. Overall, I think the chapter was divided into appropriate sections that made the material more manageable for the reader. The pictures along the side were perfect, although I would suggest adding one or two more so they run the length of the paper. They really help keep the reader engaged, and I’m impressed by the html formatting they required!

Answers Questions/Makes Arguments
I thought the section `Which is the Best Methodology` may benefit from some restructuring. It almost seemed to get distracted half way through, and I don`t think it answered the question at hand. I think the second half of the section on teacher observation may fit better towards the end of the chapter with the teacher intervention section. I think that while the question of the best methodology may not have a definitive answer, it could still benefit from a little more development or exploration.

Writing Style
I thought the overall tone of the chapter was very professional and academically appropriate. I did struggle a little bit with the amount of rhetorical questions that popped up throughout the chapter. I think some of the weight of the academic material is lost because of the number of questions, and I think the paper would come across as more authoritative if some of them were reworked. In my opinion, the section on writing styles would benefit from a table with examples of how the different writing styles would approach a simple sentence or word differently. I found it hard to keep track of the differences between the different approaches. This was a first draft, and it was evident that there were some sections that will flow better once they have been further developed. I think that the inevitable revisions will only keep improving the flow of the paper.

Hilaryy 04:03, 28 February 2011 (UTC)

title: Talk:Language teaching/Teaching reading


text: File:Fe1.s11.team1.hw4.Basic Calculix Manual.pdf

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
User:EGM6341.s11.TEAM1.Sujin.HW4

Problem 4: Rolle's Theorem thrid derivative error demonstration
 Solved without assistance   Referenced [link here], but solved with s11 values

Find
Show Eq.4.1

Solution

 * {| style="width:100%" border="0" align="left"

$$\displaystyle e^{(3)}(t)=-\frac{t}{3}[-F^{(3)}(-t)+F^{(3)}(-t)] $$
 * style="width:35%; padding:10px; border:2px solid #8888aa" |
 * style="width:35%; padding:10px; border:2px solid #8888aa" |
 * style= |
 * }
 * }

Author

Problem 5: Show relationship in Simple Simpson's Error Theorem
 Solved without assistance   Referenced [link here], but solved with s11 values

Find
Find relation between $$ \xi\ \ $$ and $$ \zeta\ \ $$

Solution
Author

title: User:EGM6341.s11.TEAM1.Sujin.HW4


text:
Talk:Psycholinguistics/Neural Components of Speech Production

When sources were used they were most often relevant and usually recent and interesting research, but there were quite a few instances where citations are needed and were not used. For example when you are talking about the research that has shown that “patients with right hemisphere localized strokes or brain damage have a disrupted ability or inability to maintain or initiate normal speech prosody”, there is no source used, which also occurs on a number of other occasions. Just make sure that any information you state has been backed up fully by support which allows the reader to see where your information is coming from. I thought the sources you used in supporting both research on Broca’s area and speech production and the Wernicke-Geschwind model was intriguing and pertinent. As a suggestion I would perhaps include supplicant, current research for the disorders of speech production and maybe add to the PET imaging studies section. Also don’t forget that titles of Journals in your research section need to be italicized.

There is an apparent logical flow to your chapter but at times the writing can seem a bit “choppy” because the sentences don’t seem to move into each other. There are just a few points in which you will be talking about one subject and the next sentence will be very different from the previous topic. So just make sure to use transitional sentences between your information and between headings. However the organization of your information seems to be well done, especially with the physiological production of sound section considering the complexity of the material. One small recommendation would be to change the order a bit in this section by just switching the phonetics information with the evolution information. I think you should cover the basics first of physiological and then introduce further information. But in general the content of your topic seems to be in logical order as you move from first introducing the physiology of speech and then eventually move on to things such as disorders associated with speech production and recent models of speech production.

For the most part you have answered the question as to what the neural components of speech production are but I think that maybe you should briefly outline what topics you will cover in the introduction just so the reader knows what to expect from your chapter. You only really give an introduction of speech production in general but don’t really mention anything about the neural components of speech production in your introduction. I do appreciate the fact that you have also included other areas besides Broca’s and Wernicke’s in the involvement of speech production but it would be better if you expanded the information even more with recent research which you didn’t really do in your chapter. There is a lacking of supportive research to confirm the role of other areas of the brain in speech production in the chapter as well. However, your chapter does contribute to the reader’s understanding of the neural components of speech production but would again help to include a clear outline.

There are a few spelling errors and in some sentences there are minor words missing such an “a” or “the” and at times the sentences can be just slightly hard to understand. Sometimes your sentences can be really short or incomplete but really there are only minor spelling errors and your verb tense and tone were pretty good. Also watch your punctuation (vs. : etc.) but the APA formatting was well done. I have a few suggestions just for things like formatting such as making your heading font a bit bigger so it is easier to follow, adding pictures beside the physiological production of sound if you can find one because it would help in the understanding and maybe a picture also beside the context describing Broca’s and Wernicke’s areas for those who don’t know where it is located in the brain. You might want to also add a subheading for Wernicke-Geschwind Model.

Overall well done but make sure to watch the flow of your chapter, add your sources and also make sure to edit very critically! Megan J Mac Donald 23:34, 26 February 2011 (UTC)


text:
Talk:Game mechanics

More precise title
Are we developing a resource here about analog games (i.e. board an card games)? Leutha 23:42, 26 February 2011 (UTC)
 * Hi, I would say 'all games' for now. If the page gets very long, we can always reconsider. Cheers, Guido den Broeder 23:47, 26 February 2011 (UTC)

title: Talk:Game mechanics


text:
Talk:Psycholinguistics/Semantics in the Brain

 * I did edit the chapter as I reviewed it, If whoever is marking the edits would like to see the comments, it is in a word file and I have attached it to the assignment submission. I will send it to the author of the chapter, I simply did not feel comfortable changing someones hard work without their knowledge/permission.

Semantics in the Brain
As I was reading the introduction of semantics in the brain I found myself very intrigued in the topic due to the presentation and style of writing used. I began wondering how processing of language occurs, and where and what functions specific areas perform. I was also interested in understanding the connections between involved areas and how the synaptic connections work. Overall, the introduction did a wonderful job of making the reader interested and gave a good basis of what will be discussed in the chapter. I did make a few comments but it was more based on grammar than content, the content itself leaves you well informed about what you will be reading. The biggest suggestion I have to make is to not start a sentence with a referent such as “it” although in general people will know what you are referring to it is not proper grammar and requires longer processing by the reader.

Distributed Networks
The distributed networks section does a good job at demonstrating how processing generally occurs in the brain, with certain areas having higher activation for certain things compared to others. As a reader of a psycholinguistics chapter I was uncertain as to why object form and motion was used to describe distributed networks as opposed to a seemingly more relevant topic. However, if there has been a fair bit of research on these topics I can understand why it was used. If this is the case, I would suggest saying why you selected those topics. I was interested by the difference in activation for unique vs. generic items. I find it fascinating to think that our brain, and neural circuits can link past knowledge and memories to a presented item such as a person’s dog vs. a generic dog. The ability of the brain to do this is essential yet how it does so still eludes researchers around the world. The opening sentence did a wonderful job at getting the reader’s attention and really other than a few word choice disagreements the only comment I would make is to fully explain everything because one has to assume that a person reading the chapter may have no previous experience with the topic or even research for that matter. Overall, the information was presented in a clean, well-written manner that was stimulating to me as a reader.

Neural Network “Hub”
The neural network “Hub” section is a good introduction to the concept of a neural network hub and the controversy surrounding it. I was unaware of the contradictory results from neuroimaging such as those of PET vs fMRI and found it very interesting to learn about how much technology can influence opinions and results. However, a few suggestions is I would remove the direct quote in the introduction, I believe it was suggested that we not use direct quotes and paraphrase instead. Also the sentence “This means that due to a lack of ROI ATL activation may have been overlooked or unnoticed due to insensitive neuroimaging studies (Visser et al., 2010)” does not make sense, I would reword or check syntax

Semantic Priming
The semantic priming section was really well explained. Not only was semantic priming explained but also one of the main tasks that use semantic priming (the lexical decision task) as well. It was an overall goo account of the function and use of semantic priming.

Formal thought disorder (FTD)
I was unaware of FTD in general and its prevalence in schizophrenic patients; therefore, I required a good introduction to the disorder. The section really explained what the disorder is, its connections and where it is thought to have an impact in the brain. My only critique is that in the conclusion the far-spreading hypothesis was introduced with no previous mention of it, therefore, I believe it should be mentioned prior to the conclusion with a brief explanation of what it is.

Overall, the semantics in the brain chapter was really well written and informative. There was evidence of ample research and it was organized in a very orderly manner with the presentation of the basics preceding the more advanced topics such as disorders. I really found the information quite interesting and look forward to reading the final product.

Answers to Part I

 * 1) Neurological
 * 2) Schizophrenia
 * 3) Anterior
 * 4) False
 * 5) Movement
 * 6) Motor
 * 7) Greater
 * 8) Lexical
 * 9) Thought
 * 10) Priming


text:
Michel Foucault/Reading group

"Here we seek to describe how power seeps into the very grain of individuals, reaches right inside their bodies, permeates their gestures, their posture, what they say, how they learn to live and work with other people” (Foucault: 1979: 28)"

A reading group for discussion of Michel Foucault's Governmentality.

If you are interested in these events, please add your name to the participants list at the event you'd like to attend.

Meetings
Unpacking neo-liberal technologies of government in Australian higher education social work departments Journal of Social Work April 2011 11: 222-236, doi:10.1177/1468017310386696 Article can be found at -http://jsw.sagepub.com/content/11/2.author-index “ I can't help but dream about a kind of criticism that would try not to judge but to bring a work, a book, a sentence, and idea to life; it would light fires, watch the grass grow, listen to the wind, and catch the sea foam in the breeze and scatter it. It would multiply not judgments but signs of existence; it would summon them, drag them from their sleep. Perhaps it would invent them sometimes ‐ all the better. Criticism that hands down sentences send[s]me to sleep; I'd like a criticism of scintillating leaps of the imagination. It would not be sovereign or dressed in red. It would bear the lightning of possible storms”. (Foucault, 1997:323)
 * /27 May 2011/ - Professions and experts- University of Canberra Teaching Commons 10.00 am - 12.00 noon
 * /13 April 2011/ - The Birth of Bio Politics
 * /23 February 2011/ - Starting the reading group
 * 27th of June 2011 in Teaching Commons at 10.00 am/ - Reading by Bay, Uschi
 * 25th of July 2011 in Teaching Commons at 10.00 am/ Reading by Bent Flyvbjerg, "Habermas and Foucault: Thinkers for Civil Society?" British Journal of Sociology, vol. 49, no. 2, June 1998, 208–233. http://en.wikipedia.org/wiki/The_Foucault/Habermas_debate
 * August date TBA - I was thinking we might have a discussion around this quote

Current reading list

 * 1) Johnson, T. 1993 "Expertise and the State" in Gane, M and Johnson T (eds) Foucault's New Domains. London:Routledge. I am trying to find out how to download the pdf file!!

Related readings

 * 1) Ivan Illich, 1978. Disabling Professions
 * 2) Raymond Geuss on Real Politics - Philosophy Bites Sun, 19 October 2008. Raymond Geuss wants political philosophers to focus on real politics rather than abstract notions. In this interview with Nigel Warburton for Philosophy Bites he explains why he believes philosophers such as Robert Nozick and John Rawls were fundamentally misguided in the way they approached political philosophy.

Introductory resources

 * Foucault: 25 years on - A conference hosted by the Centre for Post-colonial and Globalisation Studies, Adelaide, 25 June 2009. Proceedings Edited by Ian Goodwin-Smith. Published online by the Hawke Research Institute, University of South Australia, Underdale, SA, 2009. These conference proceedings have been peer reviewed. ISBN 978-0-86803-827-8
 * Michel Foucault - The Culture of the Self, First Lecture, Part 1 of 7 A series of audio recording published to Youtube. This is the first in a series of three lectures in which French philosopher Michel Foucault examines Western culture's conceptual development of individual subjectivity. He gave these lectures, in English, at UC Berkeley, beginning on April 12, 1983, roughly a year before he died. There are some negligable distortions in the tape. http://plato.stanford.edu/entries/foucault/

title: Michel Foucault/Reading group


text:
Talk:Michel Foucault/Reading group

(A reading group for discussion of Michel Foucault Archeology, Geneology and Ethics work. If you are interested in this text, please add your name to the list of participants and put the chapter notes pages on your watchlist.)

Interesting Resource from UNISA
Foucault: 25 years on A conference hosted by the Centre for Post-colonial and Globalisation Studies, Adelaide, 25 June 2009 Edited by Ian Goodwin-Smith Published online by the Hawke Research Institute, University of South Australia, Underdale, SA, 2009.

These conference proceedings have been peer reviewed. ISBN 978-0-86803-827-8

title: Talk:Michel Foucault/Reading group


text:
Michel Foucault/Reading group/23 February 2011

Participants

 * 1) Diane Phillips
 * 2) Leigh Blackall
 * 3) Robyn Faulkner


text:
Digital Age/Knowledge Management and Wikis

The question of what is knowledge becomes even more interesting when we try to apply it. Knowledge Management, as a discipline, lacks a single definition of knowledge. This raises an interesting question - if it isn't possible to form a common definition of knowledge, how is it possible to evaluate a knowledge management system?

To examine this, let's look at three different knowledge management systems that have quite a bit in common, yet, when we apply different understandings of what knowledge is, all reach different results when we try to evaluate them against it. For this we'll use three online encyclopedias: Nupedia, Wikipedia and Citizendium.

A pocket history
Nupedia was developed by Larry Sanger and Jimmy Wales as a free-content encyclopedia, in much the same sense as Wikipedia and Citizendium which were to follow. Wales formulated the concept in 1999, and hired Sanger (who was nearing the completion of his PhD in Epistemology, the philosophical study of knowledge) to be the Editor-in-chief. Seemingly informed by traditional peer review models, Sanger, working with a Nupedia board, developed a seven step model for approving material in the encyclopedia. The editor in charge of a topic area, who was required to be highly qualified, was to work with an author whom he or she appointed to write a particular article and a team of reviewers and copywriters in order to develop, evaluate and prepare material for publication. The result was a fairly tortuous approach, especially given that all involved were volunteers – and, indeed, Sanger recommended simplifying the model later in the life of Nupedia.

In January, 2001, with very few articles having progressed through the complete process, the idea emerged to use a Wiki in order to develop material which could then be fed into Nupedia's review process. Thus Wikipedia was born, not as encyclopedia in and of itself, but as a feeder for the Nupedia project. However, the design of Wikipedia was such that its growth soon outstripped that of Nupedia, to the point where Nupedia lost funding and was allowed to quietly pass away, while Wikipedia continued to develop until it reached the stage where it is today.

In spite of its success by any measures, problems emerged with Wikipedia. Wikipedia suffers from a number of issues, not the least of which is the lack of qualified editorial review. Wikipedia relies on the idea of crowdsourcing - huge numbers of people combining their knowledge to create material, rather than a single, knowledgeable person generating each individual article. You also encounter a second problem with Wikipedia – as all articles are in a state of flux, constantly being modified over time, it is difficult to know if any particular version of an article is accurate.

To counter this, after leaving the Wikipedia project, Larry Sanger founded Citizendium. Citizendium, like Wikipedia, uses a wiki to allow for collaborative development. But unlike Wikipedia, not everyone can edit - contributors need to apply, presenting their qualifications as part of the process. In addition, those with acknowledged expertise in a topic area can "approve" versions of articles, and these approved versions are recognised as such. Thus, in theory readers can be confident that an approved version has been checked by an expert and therefore is unlikely to contain errors.

Evaluating with the standard definition of knowledge
We've been discussing a definition of knowledge based on Justified True Belief. So, let's try evaluating these three projects against that definition. To start with, how do they handle belief?

Belief
Belief is a tricky concept when evaluating knowledge management systems. Clearly, encyclopedias don't have beliefs of their own - that is something we reserve for thinking beings such as ourselves. In one sense, this means that these projects can't contain knowledge, because they don't have beliefs. However, there is another sense in which we can interpret belief - the ability to create beliefs in readers. It is viable to argue that knowledge management systems don't contain knowledge as such, but that they are able to manage knowledge by storing information (or data) that becomes knowledge in users of the system. In this sense, do these three systems have the ability to create beliefs?

I'm inclined to answer that they all do, to varying degrees. However, the extent to which people believe what they read in the three projects is likely to vary based on the answer to the next two questions, so I'm putting that one aside while we move on.

Justification
There are a few means by which "justification" can be interpreted. But for this discussion, the focus is on a simple idea: if these projects are going to provide or create knowledge, does the end user (the reader) have justification for believing the information contained therein?

There are a few means by which justification can be provided. One of them is justification by authority. If someone who should know something tells it to you, you can generally feel justified in believing that it is true. Thus if I ask a good friend for his or her phone number, and they tell it to me, I have justification for believing that I now know their number. I might not - it might be the wrong number, either because they made an error or because they deliberately wanted to give me the wrong one - but that comes down to the truth condition. I am still justified in believing that it is true, even if I might be wrong. Justification by authority is a major part of Citizendium and Nupedia. Both systems employed experts to write the material: in Nupedia's case as the author, reviewer and editor, and in Citizendium as some of the authors and, most importantly, as the approvers. Thus the reader can feel that the material is justified because an expert has written and/or approved what is being presented.

Wikipedia has a much harder time. While some authors on Wikipedia are experts, (for example, the UNU-MERIT survey found that a majority of editors on Wikipedia self-identified as experts in the field that they were working), the pseudo-anonymous status of contributors on Wikipedia, and the motto "the encyclopaedia that anyone can edit", provide little reason to be confident that this is the case. For even if it is true that the majority of editors are experts, there are still questions that can't be answered. After all, you can't tell if someone is an expert, even if they claim to be one; their area of expertise may not extend to the specific matters that they are editing; and there is no way to you tell that the particular person who added the material you are reading right now was an expert, even if the majority of the material was written by one.


text:
Michel Foucault/Reading group/13 April 2011

Readings to be discussed

 * 1) The Birth of Bio-Politics – Michel Foucault's Lecture at the Collège de France on Neo-Liberal Governmentality
 * 2) Full-text: Birth of Biopolitics Chapter 1
 * 3) Michel Foucault - The Culture of the Self, First Lecture, Part 1 of 7

Participants

 * 1) Robyn Faulkner
 * 2) Leighblackall
 * 3) Diane Phillips
 * 4) Phil Hutchinson


title: File:Fe1.s11.team5.HW4.fig4.5.1.jpg


title: File:Fe1.s11.team5.HW4.fig4.5.2.jpg


title: File:Fe1.s11.team5.HW4.fig4.5.3.jpg


title: File:Fe1.s11.team5.HW4.fig4.5.4.jpg


title: File:Fe1.s11.team5.HW4.fig4.5.5.jpg


text:
User:EGM6341.s11.TEAM1.HW5

Problem 1: Integrate using Composite Trapezoidal Rule and Composite Simpson's
 Referenced the open-source code gaussleg on the MathWorks File Exchange

Find: Plots and error associated with the different discretizations
A.) For Eq. 1.1, plot $$ f_n^T(x) \ $$ (piecewise linear) and $$ f_n^S(x) \ $$ (piecewise quadratic) on $$ f(x) \ $$ B.) For Eq. 1.2, apply Trapezoidal and Simpson's Rule using uniform discretization C.) For Eq. 1.2, apply Trapezoidal and Simpson's Rule using non-uniform discretization D.) For Eq. 1.2, apply Trapezoidal and Simpson's Rule using Gauss-Legendre quadrature E.) For Eq. 1.3, apply Trapezoidal and Simpson's Rule using uniform discretization

Plot Trapezoidal Rule and Simpson's Rule
The following MATLAB (v2009b) code was developed to display area differences of integrand between Composite Trapezoidal rule and Composite Simpson's rule. Note that Langrange polynomials with n=2 were used to fit parabolas to $$f(x) \ $$.

Uniform discretization
The exact value of the integral in Eq. 1.2 is:

The following MATLAB code then developed to numerically integrate Eq 1.2 using uniform discretization and produce the following table.

Non-uniform discretization
In order to make h small close to zero, the following method of choosing x values was used:

The following MATLAB code then developed to numerically integrate Eq 1.2 using non-uniform discretization and produce the following table.

Gauss-Legendre Quadrature
The following MATLAB code was developed to evaluate the integral in Eq. 1.2.

NOTE: The user defined function legendrepoly was adapted from the work done in the open-source function gaussleg, available on the MathWorks File Exchange.

Periodic functions
Wolfram Alpha was used to calculate the exact value of the integral in Eq. 1.3.

The following MATLAB code was developed to numerically integrate Eq. 1.3.

Problem 2: Linear State Space Model
Referenced the open-source code cauchyrnd on the MathWorks File Exchange

Given: Linear State Space Model
Given the following:

Find: Model plot with various random noise
A.) Run model and plot $$\underline{x} _0 \ $$ B.) Find the equilibrium point as $$ \lim_{k \to \infty} \underline{x} _{k+1} = \lim_{k \to \infty} \underline{F} ^{k+1} \underline{x} _0 =: \underline{\hat{a} } $$ C.) Use MATLAB randn to generate $$ \begin{Bmatrix} w_j \end{Bmatrix} $$, for $$ j=0,1,2,... \ $$ and plot $$ \begin{Bmatrix} \underline{x} _j \end{Bmatrix} $$, for $$ j=0,1,2,... \ $$. Use a big blue dot for $$ j=0 \ $$, use small dots for the other points. All for $$ \alpha\ = 0.5 $$, $$ \alpha\ = 1 $$  and  $$ \alpha\ = 2 $$ D.) Run model with Cauchy random noise and plot $$\underline{x} _0 \ $$

Run linear state space model
The following MATLAB (v2009b) code was developed to evaluate the state space model until an equilibrium point was reached. The results are shown in Figure 5.2.1.

 * Nm1.s11.team1.HW4.fig5.2.1.png

Find equilibrium point
In order to determine the equilibrium point, the Euclidean distance between the next $$x_{n+1} \ $$ and $$ x_{n} \ $$ was calculated for each step in the model. In this case, the Euclidean distance is defined as:

When this distance became less than $$ 10^{-6} \ $$, the equilibrium point was taken as "found".

The MATLAB code presented above displays the equilibrium point, which is also plotted in Figure 5.2.1 above.

Gaussian Random Noise
The following MATLAB (v2009b) code was developed to evaluate the state space model for 1000 iterations, since an equilibrium point was not reached. The results are shown in Figure 5.2.2.

 * Nm1.s11.team1.HW4.fig5.2.2.png

Cauchy Random Noise
A set of MATLAB functions freely available on the MathWorks File Exchange called cauchy was used to generate random values from the Cauchy distribution.

The following MATLAB (v2009b) code was developed to evaluate the state space model for 1000 iterations, since an equilibrium point was not reached. The results are shown in Figure 5.2.2.

 * Nm1.s11.team1.HW4.fig5.2.3.png

Problem 3: Cauchy Heavy Tails
 Solved without assistance

Given Quartile points: $$ Q_1 \ $$, $$ Q_2 \ $$ and $$ Q_3 \ $$, where

Find the first and third Quartile points for Cauchy distribution with parameter ( x0, gamma )
Note: Culmulative distribution function(CDF) of Cauchy distribution:

Plugging Q1 into CDF function and value of 1/4 yields,

Plugging Q3 into CDF function and value of 3/4 yields,

 * Nm1 s11 team1 HW5 p3 fig1.png

Find the first and third Quartile points for Normal distribution with parameter ( u, sigma )
Note: Cumulative distribution function (CDF):

Plugging x=Q1 into the CDF function,

Multiplying 2 on both side of the eqn,

Subtracting 1 from the both side,

Tanking inverse erf function on both side of the eqn,

Multiplying sqrt(2) and sigma from the both side,

Adding mu on the both side yields,

Similarlly,

Multiplying 2 on both side of the eqn,

Subtracting 1 from the both side,

Tanking inverse erf function on both side of the eqn,

Multiplying sqrt(2) and sigma from the both side,

Adding mu on the both side yields,

 * Nm1 s11 team1 HW5 p3 fig1.png

Find appropriate sigma^1 such that gamma^G=1 and plot C(0,1) and N(0, sigma^1)
Given:

Where $$ \gamma\ ^c = $$ half width of $$ C \left ( x_0, \gamma\ ^c \right ) $$ and $$ \gamma\ ^G = $$ half width of $$ N \left ( \mu\, \sigma\ \right ) $$

Result Matlab

 * Nm1 s11 team1 HW5 p3 fig2.png

==== Find $$ \left \{ Q_1^c, Q_3^c \right \}  $$ for $$ C \left ( x_0, \gamma\ \right ) $$ and $$ C \left ( 0, 1 \right ) $$ also find $$ \left \{ Q_1^G,Q_3^G  \right \}  $$ for $$ N \left ( \mu\, \sigma\ \right ) $$ and $$ N \left ( 0, \sigma\ ^1 \right ) $$ ====

Note:  General form of $$\displaystyle \{\ Q_1^c, Q_3^c \}\ $$ for $$\displaystyle C \left( x_0, \gamma \right) $$ is already derived in the Part.A as follows,

Simply, plugging the given parameters(0,1) into the general form is the best way to find $$\{\ Q_1^c, Q_3^c \}\ $$ for $$\displaystyle C \left( 0, 1 \right) $$

Note:  General form of $$\displaystyle \{\ Q_1^G, Q_3^G \}\ $$ for $$\displaystyle N \left( x_0, \gamma \right) $$ is already derived in the Part.B as follows,

Plugging the parameters N(0,sigma) we got in part.3 into the general form,

Plot $$ \left \{ Q_1^c, Q_3^c \right \}  $$ for  $$ C \left ( 0, 1 \right ) $$ also plot $$ \left \{ Q_1^G,Q_3^G  \right \}  $$ for  $$ N \left ( 0, \sigma\ ^1 \right ) $$ and comment on results

Matlab

 * Nm1 s11 team1 HW5 p3 fig2.png
 * Nm1 s11 team1 HW5 p3 fig cdf.png

Problem 4: Spring and Dampener system
 Solved without assistance

Find: Equation of the motion and control scheme
A.) Equation of motion B.) State space model of system C.) $$c_{cr}$$ D.) Plot $$x_k \ $$ for $$c=\frac{1}{2}c_{cr},c_{cr},\frac{3}{2}c_{cr} $$, and with Gaussian and Cauchy noise.

Equation of motion
The system(Spring-mass-damper) equation can be defined as,

Each of the elements are

Plugging each of the elements in to the eq4.1 becomes EOM(Equation of the motion) as

Define state space model
State space

Applying the given expression on p29-7, Eq(4):

Rewriting to discrete system equation, as

where I is the identity matrix, and A, B, and h are defined as above

Critical damping
Frequency and damping ratio are given as

Thus,

State Space Models with Random Noise
When external force(Control force) u=0, plot $$\displaystyle \mathbf$$ $$\displaystyle c=0.5C_{cr}$$, $$\displaystyle c=C_{cr}$$, $$\displaystyle c=1.5C_{cr}$$

 * Nm1 s11 team1 HW5 p4 fig a.png

Part.2 with Gaussian noise

 * Nm1 s11 team1 HW5 p4 fig b.png

Part3. With Cauchy noise

 * Nm1 s11 team1 HW5 p4 fig c.png

Problem 5: Modify Trapezoidal Rule MATLAB code and create Romberg Table
 Solved without assistance

Find: Improved Trapezoidal Rule and Romberg Table
A.) Modify your Trapezoidal Rule MATLAB code to make it more efficient B.) Create Romberg Table and compare to previous results in HW2.4

More Efficient Trapezoidal Rule Code
Refer to Homework 2.4 for original code. Improved code is shown below, along with the resulting table similar to the one generated from the original code. The original code took 0.016680 seconds to run; the modified code below took 0.004149 seconds to run, meaning the new code runs in approximately 1/4th of the time.

Create Romberg Table
Using equation 3 on [[media:Nm1.s11.mtg29.djvu|Lecture page 7-5]] for $$T_k \ $$ and the MATLAB code written below, the following Romberg Integration Table was generated.

Note that $$T_4(4) \ $$ results in an error of 1.3323e-015. Compared to $$T_0(64) \ $$, which has an error of 3.742e-006, $$T_4(4) \ $$ is approximately $$10^9$$ times better than the uncorrected trapezoidal rule at n=64.

Problem 6: Compute Integral until error order of magnitude reaches 10^-6
 Solved without assistance

Reference [[media:Nm1.s11.mtg7.djvu|HW2.4 from Lecture 7-3]] and [[media:Nm1.s11.mtg30.djvu|HW5.4 from Lecture 29-6]] Compute $$ I_n \ $$ using $$ CT_k(n) \ $$, for $$ k=1,2,3 \ $$ for $$ n=2,4,8,16... \ $$ until error $$ I-I_n= O(10^{-6}) \ $$

Solution

From [[media:Nm1.s11.mtg30.djvu|Lecture 30-1]], the corrected trapezoidal rule states that

where

The following MATLAB code was developed to evaluate the corrected trapezoidal rule using $$ k=1,2,3 \ $$ until the error $$ I - I_n^T \leq O(10^{-6}) \ $$

Problem 7: Discuss the pros and cons of the following integration methods
 Solved with references listed here:  http://en.wikipedia.org/wiki/Taylor_series

http://en.wikipedia.org/wiki/Maclaurin%27s_series

http://en.wikipedia.org/wiki/Taylor_approximation

http://www.answers.com/topic/trapezium-rule

http://en.wikipedia.org/wiki/Trapezium_rule

http://en.wikiversity.org/w/index.php?title=User:Egm6341.s2010.Team1/HW4&oldid=542621

http://en.wikiversity.org/w/index.php?title=Egm6341.s10.Team2/HW4&oldid=541906

http://en.wikiversity.org/w/index.php?title=User:Egm6341.s10.team3.sa/HW4&oldid=541673

http://en.wikiversity.org/w/index.php?title=User:Egm6341.s10.Team4/HW4&oldid=541867

Taylor Series
 Positives:  1) The partial sums of the series can be used as approximations of the entire function.

2) Differentiation and integration of power series can be performed term by term and is hence particularly easy.

3) The solution is highly accurate for a small order compared to Trapezoidal and Simpson's rule.

4) It is one step and explicit.

5) Accuracy and speediness.

6) Powerful tool that helps to approximate any type of simple or complex functions into a polynomial form $$ P_n(x) \ $$ with an error in the form of remainder $$ R_{n+1}(x) \ $$

 Negatives:  1) The partial sums of the series are good if sufficiently many terms are included.

2) Problems arise with natural logarithm functions, such as log(1 + x), and some of its Taylor polynomials around a = 0. These approximations converge to the function only in the region −1 < x ≤ 1; outside of this region the higher-degree Taylor polynomials are worse approximations for the function. This is similar to Runge's phenomenon.

3) It needs the explicit form of derivatives of the function.

4) Number of operations can be large.

5) Need to know about smoothness of solution.It is senseless to use high order method when the solution has unbounded high order derivatives.

6) For oscillatory functions, a high order Taylor seems a good choice, but can lead to problems of loss of significance in the computation if not programmed carefully.

7) The function $$ F(x) \ $$ should be differentiable in the given limits 'a' and 'b' in which it is approximated as a polynomial.

8) The product of two sub-functions in a given function $$ F(x) \ $$ like for example $$ e^xcos(x) \ $$ would make it more difficult and more costly (in terms of computation) to evaluate successive derivatives.

9) It is only accurate in the area of the point that the Taylor series is calculated for (for example, $$ x=0 \ $$)

Composite Trapezoidal
 Positives:  1) The error in using the trapezium rule is approximately proportional to $$ \frac{1}{n^2} $$, so that if the number of sub-intervals is doubled, the error is reduced by a factor of 4.

2) Often converges very quickly for periodic functions

3) The method is quite simple in implementing, compared to other methods.

4) It takes a piecewise linear approximation of the function and hence execution of the solution is faster.

5) The solution is very rapidly convergent.

6) It consists of the fact that weighting coefficients are nearly equal to each other.

7) Approximate derivative by constant

8) Average (not discriminate) endpoints.

9) Very intuitive and easy to use

10) It yields better accuracy than simple trapezoidal rule in calculating the numerical integral of a function.

 Negatives:  1) Error is difficult to identify, can be eaily over or under estimated

2) The error is much higher compared to the other methods.

3) This method is less accurate for non linear functions since it has straight line approximation.

4) For various classes of functions that are not twice-differentiable,the trapezoidal rule has sharper bounds than Simpson's rule.

5) Needs many subintervals to converge and the convergence rate is low for non-periodic functions.

6) Runge Phenomena.

Composite Simpson
 Positives:  1) If the function being integrated is relatively smooth over the interval [a,b], a smooth quadratic interpolant like the one used in Simpson's rule will give good results. This is achieved with composite Simpson's by breaking up larger interval into smaller intervals.

2) It assumes piecewise quadratic interpolation and hence the accuracy is much higher compared to trapezoidal rule.

3) Exact for cubic polynomials

4) The error is less compared to that of trapezoidal rule.

5) Weighting coefficients are simple and do not fluctuate in terms of magnitude.

6) It is still intuitive and easy to use

 Negatives:  1) A large number of ordinates are needed in between the interval.

2) for various classes of rougher functions (ones with weaker smoothness conditions), the trapezoidal rule has faster convergence in general than Simpson's rule.

3) This rule can be applied to only even number of finite intervals between the limits- 'a' and 'b'. So the value of 'n' in Simpson's rule is always an even number. i.e. n=2i; where (i=1,2,3...)

4) Runge Phenomena

Romberg (Richardson)
 Positives:  1) Evaluates the integrand at equally-spaced points. The integrand must have continuous derivatives, though fairly good results may be obtained if only a few derivatives exist.

2) Newton Cotes methods fail to converge for many integrals, while Romberg integration is more stable.

3) Error is reduced subsequently as we go from $$ T_K(n) \ $$ to $$ T_{K+1}(n) \ $$.

4) It is efficient in the sense that $$ T_{K+1}(n) $$ is calculated from the already computed values of $$ T_K(n) \ $$ and $$ T_K(2n) \ $$

5) The above condition reduces the time and space requirement.

6) By using Richardson extrapolation repeatedly on the trap. rule is more efficient.

7) Converge to high accuracy.

8) The successive computation of $$ I(2n) \ $$ is cheaper and faster when $$ I(2n) \ $$ is already computed.

9) The accuracy of the numerical integration results obtained are better than other methods. The higher the power of 'h' term in the trapezoidal error, the better the accuracy.

 Negatives:  1) In Richardson extrapolation, We are neglecting the higher orders of $$ a_ih^{2i} \ $$.

2) The success of Romberg integration is only justified if the integrand-'f' satisfies the hypotheses of the Euler–Maclaurin Theorem.For example, as illustrated in the An Introduction to Numerical Analysis by Suli and Mayers Text Pg.218 (http://www.netlibrary.com/Search/SearchResults.aspx?t1=Mayers%2c+D.+F.&tt1=Author&ql=ENG%7C) The function $$ x \ $$ going to $$ x^{\frac{1}{3}}\ $$ is not differentiable at $$ x=0 \ $$, so the required conditions are not satisfied for any extrapolation.

Corrected Trapezoidal Rules: $$ cT_k(n) \ $$
 Positives:  1) When integrals of periodic functions are approximated numerically, Corrected Trapezoidal functions are the best choice.

2) More accurate than composite trapezoidal rule and Romberg table.

3) Converge more quickly than trapezoidal rule.

 Negatives:  1) This method demands derivatives of the function at the end points of the interval.

2) Bernoulli's number should be calculated for each order of derivatives.

Problem 8: The Theorem of Higher Order Trapezoidal Rule Error(HOTRE)
 Solved without assistance

Given
,where $$\begin{align} x_k=a+kh, \; \ h=(b-a)/n, \; \ x(t)= t*\frac{h}{2} + \frac{x_i+x_{i+1}}{2}, \; \ t \epsilon [-1,1] \end{align}$$ ,where $$\begin{align} g_k(t):=f(x(t)) \; \ x\in[x_k,x_{k+1}] \end{align}$$

Show eqn.8.1 is equal to eqn.8.2
from eqn.8.3 we have $$\begin{align} x_k=-1, \; x_{k+1}=1 \end{align}$$ $$\begin{align} \Rightarrow f(x_k)=g(-1)\end{align}$$ $$\begin{align}, \; f(x_{k+1})=g(1)\end{align}$$ $$\begin{align}, \; \frac{dx(t)}{dt}=\frac{h}{2} \end{align}$$

Plugging above terms and eqn7.3 into eqn.7.1

Show HOTRE for Trapezoidal rule Step 1 is true
From lecture note [[media:Nm1.s11.mtg30.djvu|30-3]], First step of HOTRE for Trap. rule is as following; Using Integration by parts, we can rewrite LHS of eqn.8.5; $$\displaystyle\begin{align} \int u {v}^{'}= uv - \int v {u}^{'} \end{align}$$ Let $$\displaystyle\begin{align} u=t \ \ {v}^{'}=g(t) \end{align}$$

$$\displaystyle \begin{align} = \int_{-1}^{1}g(t)dt-(g(-1)+g(1)) \\ \end{align}$$

So, HOTRE for Trap. rule Step 1(eqn.8.5) is true

Show HOTRE for Trap. rule Step 2a is true
From lecture note [[media:Nm1.s11.mtg30.djvu|30-3]], Step 2a of HOTRE for Trap. rule is as following; where $$\displaystyle \begin{align} \; g_k^{(i)}(t)=\frac{d^i}{dt^i} g_k^{(t)}, \;x \epsilon[x_k, x_{k+1}] \end{align}$$ From eqn.8.1 and eqn.8.2,

=Contributing Members & Referenced Lecture=

title: User:EGM6341.s11.TEAM1.HW5


text:
The resolution of the problem of the Prime Sequence

Link to any subpages this page might have

The Resolution of the Problem of the Prime Sequence.[research] [THIS CONTRIBUTION HAS BEEN REPLACED BY THE CONTRIBUTION ENTITLED: " The Prime Sequence Problem " IN WHICH THE QUESTION IMPLIED BY THE ABOVE TITLES IS TREATED as an exercise in the linear algebra of a four dimensional Euclidean space and a finite discrete version of the Legendre transformation. The principle is that all the information about the prime sequence can be transferred from a 4D space to a 2D space wherein regularity in finite discrete functions might be found.

The reader who is interested in the prime sequence is respectfully redirected to the above contribution: " The Prime Sequence Problem ".

There is of course much other literature on this, as yet unsolved mystery of the erratic prime sequence, but nowhere else as far as can be seen are primes treated as co-ordinates in a hyperspace of even dimension: [2, 4, 6, 8, ... ].SHAWWPG19410425 (talk) 23:03, 1 December 2012 (UTC)


text:
User:Egm6341.s11.team2/hwk5

=Problem 5.1: Error By Different Discretizations=

Ref: Lecture 23-1,Lecture 26-4

''' This problem was solved without referring to S10. '''

Given: The Following Functions
Consider the following three functions:

$$\displaystyle f(x) = \frac{1} $$

$$\displaystyle g(x) = {x^{1/3}} $$

$$\displaystyle h(x) = \exp (\sin (x)) $$

Find: The Error Based on the Following Discretizations
 A)  Use uniform discretization and find $$ {I_1} = \int\limits_0^6 {f(x)} .dx $$ and Find the error and the error ratio as the number of the intervals n are doubled

 B)  Use uniform, non-uniform discretization and Gauss-Legendre quadrature and find $$ {I_2} = \int\limits_0^1 {g(x)} .dx $$ and find the error and the error ratio as the number of the intervals n are doubled

 C)  Use uniform discretization and find $$ {I_3} = \int\limits_0^{2\pi } {h(x)} .dx $$ and find the error and the error ratio as the number of the intervals n are doubled

The Error of Numerical Integration Using Uniform Discretization

 * HW5 P1 A Trap.png

 * HW5_P1_A_TrapError.png

 * HW5_P1_A_SimpsonError.png

Comment: As the result shows, the critical error ratio of 4 and 16 are achieved using Trapezoid and Simpson rules, respectively.

The Error of Numerical Integration Using Uniform Discretization, Non-uniform Discretization, and Gauss-Legendre Quadrature

 * HW5 P1 B TrapError uniform.png

 * HW5 P1 B SimpsonError uniform.png

Comment: As the result shows that in case of using the uniform discretization, the error ratio never reaches to the critical ratio of 4 and 16 for Trapezoid and Simpson rules, respectively.

 * HW5 P1 B TrapError Non-uniform.png

 * HW5 P1 B SimpsonError Non-uniform.png

Comment It seems even using Chebychev non-uniform discretization does not help the error ratio to reach the critical ration of 4 and 16 for Trapezoid and Simpson rules, respectively.

 * HW5 P1 B Gauss.png

The Error of Numerical Integration Using Uniform Discretization for a Periodic Function

 * HW5 P1 C TrapError uniform.png

 * HW5 P1 C SimpsonError uniform.png

Comment: The result shows that the Trapezoid rule shows a much better error decrease ratio at the the same n value in comparison with Simpson rule.

=Problem 5.2: Linear State Space Model =

Given: The Linear State Space Model (LSSM)
Where LSSM has the general form
 * {| style="width:100%" border="0"

$$ \displaystyle \mathbf{x_{k+1}}_{\color{red}(nx1)} = \mathbf{F}_{\color{red}(nxn)} \mathbf{x_{k}}_{\color{red}(nx1)} $$     (2.1)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

And $$ \displaystyle \mathbf{F}_{\color{red}(nxn)} $$ is defined as
 * {| style="width:100%" border="0"

$$ \displaystyle \mathbf{F}_{\color{red}(nxn)} = \mathbf{I}_{\color{red}(nxn)} + \Delta \mathbf{A}_{\color{red}(nxn)} $$     (2.2)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

If we choose $$ \displaystyle {\color{red} n=2} $$ and define the matrices in (2.2) as
 * {| style="width:100%" border="0"

$$ \displaystyle \mathbf{I}_{\color{red}(2x2)} = \left[ \begin{matrix} 1 & 0 \\   0 & 1  \\ \end{matrix} \right], \quad \mathbf{\Delta} = 0.02, \quad \mathbf{A}_{\color{red}(2x2)} = \left[ \begin{matrix} -0.2 & 1    \\   -1   & -0.2  \\ \end{matrix} \right] $$     (2.3)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

In addition we will consider the k-th step of the system $$ \displaystyle \mathbf{x_k} $$ and initial point $$ \displaystyle \mathbf{x_0} $$, in (2.1) as
 * {| style="width:100%" border="0"

$$ \displaystyle \mathbf{x_k}_{\color{red}(2x1)} = \left\{ \begin{matrix} x_k^ \\ x_k^ \\ \end{matrix} \right\} \quad and \quad
 * style="width:95%" |
 * style="width:95%" |

\mathbf{x_0}_{\color{red}(2x1)} = \left\{ \begin{matrix} x_k^ \\ x_k^ \\ \end{matrix} \right\} = \left\{ \begin{matrix} 3 \\   -2  \\ \end{matrix} \right\} $$     (2.3)
 *
 * }

1. Run LSSM
And plot $$ \displaystyle \{ \mathbf{x_j}, j=0,1,...\} $$ in the state space $$ \displaystyle \left( x^1,x^2\right) =\left( x,y\right) $$

2. Find the Equilibrium Point
As $$ \displaystyle \underset{x \to \infty}{\mathop{\lim }} \mathbf{x_{k+1}} = \underset{x \to \infty}{\mathop{\lim }} \mathbf{F^{k+1}\cdot x_0} =: \mathbf{ \hat{x}} $$

a) Plot $$ \mathbf{ \hat{x}} $$
Using a BIG RED DOT

b) Plot $$ \displaystyle \mathbf{x_} $$
Using a BIG BLUE DOT in the same plane as $$ \displaystyle \{ \mathbf{x_j}, j=0,1,...\} $$ using small dots

3. Gaussian Random Noise:
====a) Let $$ \displaystyle \mathbf{G} = \alpha \cdot \left\{ \begin{matrix}  1  \\   1  \\ \end{matrix} \right\}_{\color{red}2x1}  $$.====

4. Cauchy Random Noise:
====a) Let $$ \displaystyle \mathbf{G} = \alpha \cdot \left\{ \begin{matrix}  1  \\   1  \\ \end{matrix} \right\}_{\color{red}2x1}  $$.====

Hint:
Find a Matlab command to generate $$ \displaystyle \{ \mathbf{\theta_{j}}, j=0,1,2,... \} $$ in single-slit diffraction experiment.

1. Run LSSM
And plot $$ \displaystyle \{ \mathbf{x_j}, j=0,1,...\} $$ in the state space $$ \displaystyle \left( x^1,x^2\right) =\left( x,y\right) $$

Matlab Code: Matlab Plot of the LSSM:
 * HW5-P2-1.tif

Using the above Matlab code we were able to track the time evolution over time to the equilibrium point. We considered $$ \displaystyle n = 1001 $$ points to obtain the equilibrium position. It is reasonable to infer that this behavior continues as $$ \displaystyle n \rightarrow \infty $$ since the stability of the phase space by eigenvalue analysis holds. Namely, for a given $$ \displaystyle 2x2 $$ matrix $$ \displaystyle \mathbf{A} $$ of the general form,

 * {| style="width:100%" border="0"

$$ \displaystyle \mathbf{A}_{\color{red}(2x2)} = \left[ \begin{matrix} a & b \\ c & d \\ \end{matrix} \right] $$     (2.4)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Remarks:
Must satisfy the following two conditions to be classified as stable: i) The trace, $$ \displaystyle T = a+d < 0 $$ and ii) The determinant, $$ \displaystyle D = ad-bc > 0 $$

For our matrix $$ \displaystyle \mathbf{A} $$ we get: i) The trace, $$ \displaystyle T = -0.2+(-0.2) = -0.4 < 0 $$ and ii) The determinant, $$ \displaystyle D = (-0.2)(-0.2)-(-1)(1)=1.04 > 0 $$ Therefore, we observe a stable near-equilibrium at $$ \displaystyle n=1001 $$ which sufficiently demonstrates the behavior of this system. Since it was easy enough to carry out the numerical verification at larger $$ \displaystyle $$ a calculation for $$ \displaystyle n=10001 $$ which provided points around the order of $$ \displaystyle 10^{-17} $$.

2. Find the Equilibrium Point
As $$ \displaystyle k $$ goes to infinity,

a) Plot $$ \mathbf{ \hat{x}} $$
Using a BIG RED DOT

b) Plot $$ \displaystyle \mathbf{x_} $$
Using a BIG BLUE DOT in the same plane as $$ \displaystyle \{ \mathbf{x_j}, j=0,1,...\} $$ using small dots

Matlab Code:

Matlab Plot of the LSSM:
 * HW5-P2-1.tif

3. Gaussian Random Noise:
====a) Let $$ \displaystyle \mathbf{G} = \alpha \cdot \left\{ \begin{matrix}  1  \\   1  \\ \end{matrix} \right\}_{\color{red}2x1}  $$.====

c) Using the Matlab command randn to generate $$ \displaystyle \{ \mathbf{w_{j}}, j=0,1,2,... \} $$.
Matlab Code:

d) Plot $$ \displaystyle \{ \mathbf{x_{j}}, j=0,1,2,... \} $$ for $$ \displaystyle \alpha=0.5,1,2 $$.
Shown Below is the Linear State Space Model With Random Gaussian Noise and $$ \displaystyle {\color{red}\alpha = 0.5} $$:
 * HW5-P2-3-guassianNoiseAlpha_05.tif

Shown Below is the Linear State Space Model With Random Gaussian Noise and $$ \displaystyle {\color{red}\alpha = 1} $$:
 * HW5-P2-3-guassianNoiseAlpha_2.tif

Shown Below is the Linear State Space Model With Random Gaussian Noise and $$ \displaystyle {\color{red}\alpha = 2} $$:
 * HW5-P2-3-guassianNoiseAlpha_2.tif

4. Cauchy Random Noise:
====a) Let $$ \displaystyle \mathbf{G} = \alpha \cdot \left\{ \begin{matrix}  1  \\   1  \\ \end{matrix} \right\}_{\color{red}2x1}  $$.====

c) Using the Matlab command randn to generate $$ \displaystyle \{ \mathbf{w_{j}}, j=0,1,2,... \} $$.
Matlab Code:

d) Plot $$ \displaystyle \{ \mathbf{x_{j}}, j=0,1,2,... \} $$ for $$ \displaystyle \alpha=0.5,1,2 $$.
Shown Below is the Linear State Space Model With Cauchy Noise and $$ \displaystyle {\color{red}\alpha = 0.5} $$:
 * HW5-P2-4cauchyNoiseAlpha_05.tif

Shown Below is the Linear State Space Model With Cauchy Noise and $$ \displaystyle {\color{red}\alpha = 1} $$:
 * HW5-P2-4-cauchyNoiseAlpha_1.tif

Shown Below is the Linear State Space Model With Cauchy Noise and $$ \displaystyle {\color{red}\alpha = 2} $$:
 * HW5-P2-4cauchyNoiseAlpha_2.tif

Hint:
Find a Matlab command to generate $$ \displaystyle \{ \mathbf{\theta_{j}}, j=0,1,2,... \} $$ in single-slit diffraction experiment.

=Problem 5.3 Cauchy Distribution and Normal Distribution=

From the lecture slide Mtg26

Given:
The Cauchy distribution:

The Normal or Gauss distribution:

and the definition of quartiles $$\displaystyle {{Q}_{1}}$$, $$\displaystyle {{Q}_{2}}$$ and $$\displaystyle {{Q}_{3}}$$:

Find:
a) Three quartiles $$\displaystyle \left\{ Q_{1}^{C},Q_{3}^{C} \right\}$$ for Cauchy distribution

b) Three quartiles $$\displaystyle \left\{ Q_{1}^{G},Q_{3}^{G} \right\}$$ for Normal distribution

c) Let $$\displaystyle {{x}_{0}}=\mu =0$$ and $$\displaystyle {{\gamma }^{C}}=1$$ where $$\displaystyle {{\gamma }^{C}}$$ is the half width of $$\displaystyle C({{x}_{0}},\gamma )$$. Find $$\displaystyle {{\sigma }^{1}}$$ such that the half width of normal distribution $$\displaystyle {{\gamma }^{G}}=1$$. Then plot $$\displaystyle C(0,1)$$ and $$\displaystyle N(0,{{\sigma }^{1}})$$

d) Find $$\displaystyle \left\{ Q_{1}^{C},Q_{3}^{C} \right\}$$ for $$\displaystyle C(0,1)$$ and $$\displaystyle \left\{ Q_{1}^{G},Q_{3}^{G} \right\}$$ for $$\displaystyle N(0,{{\sigma }^{1}})$$. Then plot them with comments on results.

a) Three quartiles for Cauchy distribution
From the WolframAlpha we have got the general expression for the quartiles $$\displaystyle Q_{1}^{C}$$ and $$\displaystyle Q_{3}^{C}$$:

We can prove this by finding the cumulative distribution function of Cauchy distribution,

Then,

Similarly,

b) Three quartiles for Normal distribution
From the WolframAlpha we can find the quartiles for the normal distribution,

And also we can find the cumulative of normal distribution,

Then we can check the validity of the quartiles given.

c) Compare Cauchy and Gauss distribution with same half width.
With the information given we have our Cauchy distribution with half width equals 1,

To find a normal distribution we need to know the maximum value of

Hence,

Then we must find the x coordinates in the half width nodes.

The difference between the 2 roots should be twice the half width which is 2, hence,

which means the normal distribution is,

The plots of them can be achieved by WolframAlpha

d) Find quartiles for particular Cauchy and Normal distribution. Then plot them with comments on results.
By the formula from part a) and part b) we have,

The following figure is plotted by Adam Franklin,

Corrsponding Matlab Code

Comments

From the results above we can see that the range from $$\displaystyle Q_{1}^{C}$$ to $$\displaystyle Q_{3}^{C}$$ is significantly larger than that from $$\displaystyle Q_{1}^{G}$$ to $$\displaystyle Q_{3}^{G}$$. The probability density outside (-0.5729, 0.5729) decrease to very small in normal distribution while in Cauchy distribution this is until (-1, 1). And as we can see form the figures, the Cauchy pdf is generallu "under" normal distribution pdf around the peak, allowing more probability for the "tail".This means the Cauchy distribution allow more probability on the large outcome to occur.

=Problem 5.4 Mass-Spring-Damper with Gaussian noise and Cauchy noise=

Given: The Following Spring System
An ideal mass-spring-damper system with mass m, spring constant k and viscous damper of damping coefficient c is subject to an oscillatory force u. This can be illustrated in the following image.

Find:
1) Derive equations of motion in terms of d, c, k, m, u.

2) Let $$ {\mathbf{x}} = \left\{ {\begin{array}{ccccccccccccccc} d \\  {\dot d} \end{array}} \right\} = \left\{ {\begin{array}{ccccccccccccccc}   \\ \end{array}} \right\}$$. Find $$\left( {{\mathbf{F}},{\mathbf{G}}} \right) $$

3) Find $${c_{cr}}$$ in terms of k, m st. this system is critically damped.

4) Let k=1, m=1/2, $$ {{\mathbf{x}}_0} = {\left[ {0.8, - 0.4} \right]^T} $$

a) For u=0, plot $${{\mathbf{x}}_k}$$ for $$c = \frac{1}{2}{c_{cr}},{c_{cr}},\frac{3}{2}{c_{cr}}$$.

b) For u=0.5 Gaussian noise and $$c = \frac{3}{2}{c_{cr}}$$, plot $${{\mathbf{x}}_k}$$.

c) For u=0.5 Cauchy noise and $$c = \frac{3}{2}{c_{cr}}$$, plot $${{\mathbf{x}}_k}$$.

Solution:
1) Since the system is subject to an oscillatory force generated from the spring, we have this oscillatory force

and also a damping force from the damper:

Since we can use Newton's Second Law

Therefore we can derive equations of motion as following:

2) From Eq. 5.4.4, we can solve for $${\ddot d}$$ as following:

Therefore we can rewrite $${\dot x}$$ in matrix form

If we make a discretization as

and use forward Euler method

We can have

3) Since we can rewrite Eq. 5.4.4 as

where $$\omega_0 = \sqrt{ k \over m } $$, $$\zeta = { c \over 2 \sqrt{m k} }$$

When $$\zeta = 1$$, the system is said to be critically damped. A critically damped system converges to zero faster than any other and without oscillating. Therefore we have $${c_cr}$$ indicated in Eq. 5.4.12 and can look for damping for more explanations.

4) a) The following matlab codes are used to solve the problem without any noise, and the damping coefficients can be chose to create different cases with under-damping, critical damping or over-damping.

Comment:

Different c values cause different damping cases. From the figure above, we can find out a critically damped system converges to zero faster than any other and without oscillating.

b) The following matlab codes are used to solve the problem with the Gaussian Random Noise and over-damping case.

Comment:

When the system has Gaussian noise, the convergence can be archived, however the final result is not exactly zero since there is still noise all the time and the final result oscillated around zero.

c) The following matlab codes are used to solve the problem with the Cauchy Noise and over-damping case.

Comment:

When the system has Cauchy noise, the same result was generated as Gaussian noise. However the oscillation around the zero is much larger than the case with Gaussian noise.

=Problem 5.5: Richardson Extrapolation and Romberg Integration applied to HW* 2.4=

''' This problem was solved without referring to S10 homework. '''

Statement:
1) Modify the Matlab code from HW*2.4 to make it more efficient, i.e. use Richardson extrapolation to compute higher order integral estimates starting from the trapezoidal rule. 2) Construct a Romberg table for results in 1). Compare to results from HW* 2.4.

Solution:
1) From problem 2.4, the composite trapezoidal rule was implemented for 2,4,8,16,32, and 64 node points. Equation 5.5.1 was then implemented to find the higher order integral estimates. The code is easily modified to include greater numbers of nodes or even higher order integral estimates.

The order of the error on each integral estimate can be found using Equation 5.5.2:

2) The following table presents the results of using Romberg integration on problem 2.4. The subscript on T corresponds to the iteration. Since the method starts with the composite trapezoidal rule, T0 corresponds to integration by the trapezoidal rule for various numbers of nodes.

Using Equation 5.5.2, the error of T3(4) is on the order of ~O(-7). Thus, using this method the third order estimate using only 4 nodes has O(-7) accuracy. However, this requires knowing the integral estimate using the composite trapezoidal rule for 2,4,8,16 and 32 nodes. For comparison, the composite trapezoidal rule alone only reaches O(-6) accuracy using 512 nodes, and the composite Simpson rule reaches O(-6) accuracy with 16 nodes. Legendre-Gauss quadrature reaches O(-7) accuracy with only 4 nodes. If the quadrature weights are readily available, Legendre-Gauss quadrature converges to an accurate solution with many fewer nodes required than the other methods. The Romberg integration scheme provides another useful method for numerically calculating the value of an integral, especially because it uses previous results to improve the integral estimate.

=Problem 5.6: Computing CTk(n) =

''' This problem was solved without referring to S10. '''

From the lecture slide Mtg 30-2

Given:
From [[media:Nm1.s11.mtg30.djvu|Lecture 30-1]], the corrected Trap. Rule is

Find:
1) Compute In using CTk(n), for k = 1 and n = 2, 4, 8, ... until error is of order 10-6

2) Compute In using CTk(n), for k = 2 and n = 2, 4, 8, ... until error is of order 10-6

3) Compute In using CTk(n), for k = 3 and n = 2, 4, 8, ... until error is of order 10-6

Solution:
The corrected Trap. Rule is

where

and

$$B_{2i}$$ are Bernoulli numbers. The values for $$ d_{i}, i=1,2,3 $$ are

We can develop expressions for $$CT_1(n)$$,$$CT_2(n)$$ and  $$CT_3(n)$$ using Eq 6.1.

$$CT_1(n)$$

where $$T_0(n) $$ is the comp. trap. rule below.

 * {| style="width:100%" border="0" align="center"

T_0(n) = \frac{b-a}{n} \left[ \frac{1}{2}f_0 + f_1 + \ldots + f_{n-1} + \frac{1}{2}f_n \right]$$ (6.7)
 * $$\displaystyle
 * $$\displaystyle
 *
 * }
 * }

$$CT_2(n)$$

$$CT_3(n)$$

The matlab code below was used to generate the above table.

=Problem 5.7: Discuss Pros and Cons of Different Numeric Integration Methods =

''' This problem was solved by referencing Team 2, S10 homework. '''

From the lecture slide Mtg 30-1

Problem Statement:
Discuss the pros and cons of the following quadrature methodslecture 30-1:

 * 1) Taylor's Series
 * 2) Composite Trapezoidal Rule
 * 3) Composite Simpson's Rule
 * 4) Romberg Table (Including Richardson's Extrapolation)
 * 5) Corrected Trapezoidal Rule.

Pros:

 * 1) Local behavior about the point of expansion is very accurate, fast and computationally tractable.
 * 2) The number of operations are typically small for higher order approximation of simple functions, such as polynomials.
 * 3) With the aid of symbolic packages complex function behavior is easily found and accurate.
 * 4) The solution is highly accurate for a small order compared to Trapezoidal and Simpson's rule.
 * 5) It is one step and explicit.

Cons:

 * 1) One needs to know the "smoothness" and global behavior of the function since computing derivative of an unbounded solution would be pointless.
 * 2) The number of operations may be large for a complex functions. For example, functions containing non-distributable products, quotients or composite trigonometric and exponential, to name a few.
 * 3) In addition, oscillatory functions, especially those having a small period(high frequency) are more cumbersome to approximate using a Taylor's Series.
 * 4) It needs the explicit form of derivatives of the function.

Pros:

 * 1) The method is quite simple in implementing, compared to other methods.
 * 2) It takes a piecewise linear approximation of the function and hence execution of the solution is faster.
 * 3) The solution is very rapidly convergent.
 * 4) It consists of the fact that weighting coefficients are nearly equal to each other.
 * 5) Convergence for periodic functions is very quick when choosing to integrate over integer multiple of the period.
 * 6) The function (in the integrand) need only be twice continuously differentiable on the domain.

Cons:

 * 1) The error is much higher compared to the other methods.
 * 2) This method is less accurate for non linear functions since it has straight line approximation.
 * 3) Steep concavity (or convexity) of the integrand can result in under or overestimates due to the second order continuously differentiable criteria. This means for a concave up integrand, and thus a positive second derivative, one will observe negative error and therefore an overestimate. Similarly, for a concave down (convex) integrand, and thus a negative second derivative, one will observe positive error and therefor an underestimate.

Pros:

 * 1) It assumes piecewise quadratic interpolation and hence the accuracy is much higher compared to trapezoidal rule.
 * 2) It can integrate polynomials up to third order accurately.
 * 3) The error is less compared to that of trapezoidal rule.
 * 4) Weighting coefficients are simple and do not fluctuate in terms of magnitude.

Cons:

 * 1) A large number of ordinates are needed in between the interval.

Pros:

 * 1) Error is reduced subsequently as we go from TK(n) to TK+1(n).
 * 2) It is efficient in the sense that TK+1(n) is calculated from the already computed values of TK(n) and TK(2n).
 * 3) The above condition reduces the time and space requirement. Also, the table format is good for pursuing the history of calculations and making some comparisons.

Cons:

 * 1) In Richardson extrapolation, We are neglecting the higher orders of aih2i.

Pros:

 * 1) When integrals of periodic functions are approximated numerically, Corrected Trapezoidal functions are the best choice. Even we can reach to the error equal to zero in this condition.

Cons:

 * 1) This method demands derivatives of the function at the end points of the interval.
 * 2) Bernoulli's number should be calculated for each order of derivatives.

=Problem 5.8: Theorem of Higher Order Error for Trap. Rule =

''' This problem was solved without referring to S10. '''

From the lecture slide Mtg 30-2

Given
Higher Order Error for Trap. rule Eq (5) from lecture 30-2, which is:

Solution
Part 1: Change of Variables

First, transform the variables of the error function for higher order trapezoidal rule,

 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\begin{align} E_{n}^{T} = I-{{I}_{n}} &=\int\limits_{a}^{b}{f(x)dx}-{{T}_{0}}(n) \\ &=\sum\limits_{k=0}^{n-1}{[\int\limits_^{f(x)dx}-\frac{h}{2}\{f({{x}_{k}})+f({{x}_{k+1}})\}]} \\ \end{align}

$$

(8.4)
 *
 * }

Where $${{x}_{k}}:=a+kh,h=\frac{(b-a)}{n}$$.

Now, we transfer $$\int\limits_^{f(x)dx}$$ to $$\int\limits_{-1}^{+1}{f(x(t))dt}$$.
 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\begin{align} x(t)&=\frac{{{x}_{k+1}}-{{x}_{k}}}{2}t+\frac{{{x}_{k+1}}+{{x}_{k}}}{2} \\ & =\frac{h}{2}t+\frac{{{x}_{k+1}}+{{x}_{k}}}{2} \\ \end{align}

$$

(8.5)
 *
 * }

From the transformation, we see the following:

 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\begin{align} x(-1)&={{x}_{k}} \\ x(0)&=\frac{{{x}_{k+1}}+{{x}_{k}}}{2} \\ x(+1)&={{x}_{k+1}} \\ dx&=\frac{h}{2}dt \\ \end{align}

$$

(8.6)
 *
 * }.

By using Eq 8.6, we can rewrite that trapezoidal error as follows:

 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\begin{align} E_{n}^{T} &=\sum\limits_{k=0}^{n-1}{[\int\limits_{-1}^{+1}{f(x(t))\frac{h}{2}dt}-\frac{h}{2}\{f(x(-1))+f(x(+1))\}]} \\ & =\frac{h}{2}\sum\limits_{k=0}^{n-1}{[\int\limits_{-1}^{+1}{f(x(t))dt}-\{f(x(-1))+f(x(+1))\}]} \\ \end{align}

$$

(8.7)
 *
 * }

By defining $${{g}_{k}}(t)=f(x(t))_ – ^ – such_ – ^ – that_ – ^ – x\in [{{x}_{k}},{{x}_{k+1}}]$$, we have

 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

E_{n}^{T}=\frac{h}{2}\sum\limits_{k=0}^{n-1}{[\int\limits_{-1}^{+1}{{{g}_{k}}(t)dt}-\{{{g}_{k}}(-1)+{{g}_{k}}(+1)\}]}

$$

(8.8)
 *
 * }

Part 2: Show the Equality

Next, integrate by parts.

 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\int\limits_{-1}^{+1}{(-t)g_{k}^{(-1)}(t)dt} =[(-t){{g}_{k}}(t)]_{x=-1}^{x=1}-\int\limits_{-1}^{+1}{{{g}_{k}}(t)d(-t)}

$$

(8.9)
 *
 * }

Part 3: Higher Order Terms

Lastly, since we have already defined the function $${{g}_{k}}(t)=f(x(t)), x\in [{{x}_{k}},{{x}_{k+1}}]$$ and from the transformation above, we have $$dx=\frac{h}{2}dt$$.

Successively differentiating the function $$\displaystyle {{g}_{k}}(t)$$, wrt $$\displaystyle t$$, we get

=References=

=Contributing Members & Referenced Lecture=

title: User:Egm6341.s11.team2/hwk5


text:
User:Eml5526.s11.team6.joglekar/hwk4

=Problem 9=

Given
Solving a differential equation using Weight Residual Form. The partial differential equation is:

 * {| style="width:100%" border="0"

$$ \frac{\partial}{\partial x}\left(2\frac{\partial u}{\partial x}\right) + 3 = 0 $$
 * $$\displaystyle (Eq.9.1) $$
 * }
 * }

The boundary conditions are:

$$ u\left(1\right) = 0$$

$$\frac{du}{dx}\left(0\right) = -4 $$

Weighting funtion to be used:

 * {| style="width:100%" border="0"

$$ b_{i} = \left \{ 1, \sin\left ( x \right ),\sin\left (  2x \right )... \right \} $$

 * $$\displaystyle (Eq.9.2) $$
 * }
 * }

Which is correction function when $$\phi = \frac{\pi}{2}$$

Find

 * 1 Find an approximate solution of the form $$U^{h}=\sum_{i=0}^{n}d_{i}b_{i}$$ for n=2
 * 2 Find two equations that enforce the boundary conditions
 * 3 Project the weight residues
 * 4 Display the equations in matrix form
 * 5 Solve for $$\displaystyle d $$
 * 6 Construct $$U^{h}$$ and plot $$U^{h}$$ and $$u $$
 * 7 Repeat 9.1 to 9.6 for n = 4 and n = 6

9.1 Find an approximate solution with n=2
For n = 2:
 * {| style="width:100%" border="0"

$$ d_{i}=\begin{bmatrix} d0 & d1 & d2 \end{bmatrix} $$
 * $$\displaystyle (Eq.9.3) $$
 * }
 * }

and

 * {| style="width:100%" border="0"

$$ b_{i}=\begin{bmatrix} 1 & \sin  \left ( x\right ) & \sin  \left ( 2x\right ) \end{bmatrix} $$
 * $$\displaystyle (Eq.9.4) $$
 * }
 * }

Therefore:

9.2 Find two equations that enforce the boundary conditions
The boundary conditions are:

And

9.3 Project the weight residues
In this case changing the function u(x) by the approximated function $$U^{h}$$ in the PDE (Eq.9.1), we found that P(u) is:

 * {| style="width:100%" border="0"

$$ P\left ( U^{h} \right )=2\frac{\partial^2 U^{h}}{\partial x^2}+3=-2\left ( d1\sin \left (x \right ) +4d2\sin \left (2x \right )\right )+3 $$
 * $$\displaystyle (Eq.9.8) $$
 * }
 * }

The equation 9.8 can be written as:

 * {| style="width:100%" border="0"

$$ P\left ( U^{h} \right )= -2\begin{bmatrix}0 & \sin \left (x\right ) & 4\sin \left (2x \right ) \end{bmatrix}\begin{bmatrix} d0\\ d1\\

d2\end{bmatrix}+3 $$
 * $$\displaystyle (Eq.9.9) $$
 * }
 * }

Projecting the residue we have:

 * {| style="width:100%" border="0"

$$ \int_{a}^{b}b(x)P\left ( U^{h} \right )= 0 $$ After substitution of (Eq.9.4) and (Eq.9.9) in (Eq.9.10) the following equation is obtained
 * $$\displaystyle (Eq. 9.10) $$
 * }
 * }

9.4 Display the equations in matrix form
Performing the product and then the integration indicated in (Eq.9.11) we have:

The latter equation is clearly of the form $$\displaystyle Kd=F$$. We can observe that K is not symmetric.

9.5 Solve for $$ d $$
In (Eq.9.12) we have three equations; but, as we need to enforce the boundary conditions we take the only one of them and solve it together with (Eq.9.6) and (Eq.9.7). From these equations we now have the system:

 * {| style="width:100%" border="0"

$$ \begin{bmatrix} 1 &\sin\left (1 \right )   & \sin\left (2  \right ) \\ 0 &\cos \left (0 \right ) &2\cos \left (0  \right ) \\ 0 & 2\frac{\sin^3\left(1\right)}{3} & 2-\frac{\sin\left(4\right)}{2} \end{bmatrix}\begin{bmatrix} d0\\ d1\\

d2\end{bmatrix}=\begin{bmatrix} 0\\ 4\\

2.12\end{bmatrix}$$
 * $$\displaystyle (Eq.9.13) $$
 * }
 * }

After solving the system in (Eq.9.13) we obtain d:

9.6 Construct $$U^{h}$$ and plot $$U^{h}$$ and $$u$$
Replacing d in (Eq.9.5) we obtain the approximated solution:

 * {| style="width:100%" border="0"

$$ U^{h}=-3.11+3.33\sin \left (x \right )+0.33\sin \left ( 2x \right ) $$
 * $$\displaystyle (Eq.9.14) $$
 * }
 * }

On the other hand, the exact solution of the PDE (Eq.9.1) with the given boundary condition is:

 * {| style="width:100%" border="0"

$$ u\left ( x \right )=-\frac{3}{4}x^{2}-4x+\frac{19}{4} $$
 * $$\displaystyle (Eq.9.15) $$
 * }
 * }

In Fig.9.1 we show the exact and the approximated solution.

9.7 Plot the graph for $$ u\left(x\right)$$ and $$u^h\left(x\right)$$
Using following generalized code for any value of n, we get plots for n=4 and n=6

The error at x=0.5 is plotted for different values of n

title: User:Eml5526.s11.team6.joglekar/hwk4


title: User:WillWell777


text:
Talk:Psycholinguistics/Acquisition of More than One Language

Suggestions for classmate's chapter
The information was presented in a very easily understandable manner and in a complexity that is easily understandable by readers of all strengths. However in the first half of the chapter there were almost no in text citations and this could lead a reader to believe that the interpretations were your own. I understand that this section was an introduction and the information is fairly straightforward but just to avoid any conflict I would suggest adding citations wherever possible. There were a few sentences that were a bit long and vague but after reading them over a few times I was able to understand the message.

For the “Is there an Innate Component for learning a second Language?” section I am not sure if you intend on adding in information later or if you just forgot to add in the information, but that section is completely missing. If you have the information already then adding it should not be too much of a problem and will make your chapter a little more complete. I do believe that this section is quite an interesting component to your chapter so I do look forward to the information as it ties in the concept of whether humans are innately capable of multiple languages.

I enjoyed reading the section on emotional responses within Bilingualism but i felt that it did not really fit in the overall section of multiple language acquisition, but related more to another topic such as behavioural differences in bilinguals. I felt that the discussion/conclusion section was a bit weak as it did not wrap up or summarise the preceding information. It did state some interesting points but in order to end on a stronger note I believe a little summary would go a long way to help the reader remember the key points. Another small suggestion would be to add some graphics such as graphs or pictures. Graphics help strengthen the information already presented and gives the reader a bit of relief from the overwhelming text.

Overall the layout if great, with a logical flow but just a small criticism would be the length of each section. It seems that there are a great deal of sections which is wonderful but certain sections seem to have only one or two sentences in them and the information from certain sections seem to overlap in my opinion, so I would suggest amalgamating certain section to make a broader and more convincing paragraph (eg instead of having a small paragraph between bilingualism and SLA, I would suggest adding that paragraph into the language differentiation subheading. Also within the “socioculturists approach” section I believe you should just make one heading such as social factors of bilingualism rather than different subheading for additive, subtractive and codeswitching).

In summary I think you did a great job and I enjoyed reading your chapter as well as learning some interesting facts. A slight revision and few adjustments will help make this chapter an even better resource.

Joel Sardinha 17:03, 27 February 2011 (UTC)


text:
Portal:Swahili

Welcome to the Department of Swahili! Karibu kwenye Shule ya Kiswahili!

{| cellpadding="10" cellspacing="5" style="width: 99%; background-color:inherit ; margin-left: auto; margin-right: auto" Swahili Division Welcome to the Swahili Department at Wikiversity, part of the Center for Foreign Language Learning and the School of Language and Literature.
 * style="background-color: #ffffe0; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px; height: 60px;" colspan="2" |

Streams

 * List of Swahili Learners and Teachers

Courses Currently Offered

 * Swahili I

Courses Planned

 * Swahili II
 * Swahili III
 * Swahili IV

Department News
{| cellpadding="10" cellspacing="5" style="width: 99%; background-color:inherit ; margin-left: auto; margin-right: auto"
 * Department is founded: Sunday, February 27th 2011
 * Swahili I is in development: Sunday, February 27th 2011
 * }
 * style="background-color: #ffffe0; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px; height: 60px;" colspan="2" |

Language Learning Tips
If you are an English speaker, the chances are that you don't have many opportunities to speak Swahili. This is unfortunate, as one of the most important ingredients into learning a language is conversing in that language. It produces fluidity in speech and builds vocabulary and grammatical knowledge. It also develops the skill of understanding the language, which is a very different skill altogether. One way to overcome this obstacle is to talk to yourself in Swahili, but very slowly, to make sure that you have the correct grammar and pronunciation. As you progress in the courses, you should speed up the pace. To achieve understanding skills, you could listen to tapes or you could listen to movies with English subtitles to make sure you are getting the correct meaning of the sentence.
 * }

title: Portal:Swahili


text:
Category:Swahili Teacher

This category belongs to the Swahili Language Division, and its purpose is the coordination of teachers.

 * The main article of this category is the Swahili Language Division.

title: Category:Swahili Teacher


text:
Swahili I

{| cellpadding="10" cellspacing="5" style="width: 99%; background-color:inherit ; margin-left: auto; margin-right: auto"
 * style="background-color: #ffffe0; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px; height: 60px;" colspan="2" |

Swahili I Kiswahili I

Karibu!
 * style="width: 40%; background-color: #fffff0; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" |
 * style="width: 40%; background-color: #fffff0; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" |

Swahili I
"Swahili I" is a course in the Swahili Department. This course aims to give students a sense of elementary grammar and to build a substantial vocabulary through repetition, exercises, reading, and writing. It gives students an understanding of the noun class system and of the various affices that are so important to meanings of sentences. It also aims to start the study of culture by including chapters about things like sports played by the Waswahili (people who speak Swahili).
 * rowspan="2" style="width: 60%; background-color: #fffff0; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" |

Lessons

 * 1) /Grammar/
 * 2) /Greetings/
 * 3) /Numbers and Time/
 * 4) /Dates/
 * 5) /More Grammar/
 * 6) /Restaurant Vocabulary/
 * 7) /Family Vocabulary/
 * 8) /Shopping and Clothes Vocabulary/
 * 9) /African Geography/
 * 10) /House and Daily Activities Vocabulary/
 * 11) /Sports and Athletics/
 * 12) /Airport Vocabulary/
 * }

title: Swahili I


text:
Talk:Language teaching/Cochlear implants and language instruction

Chapter Review – Cochlear Implants and Language Instruction – Anna Richter
Reviewer: Emily McGuire Hi Anna, I enjoyed reading your chapter! I picked it because I have some background on the subject of cochlear implants, and was interested to see what you could dig up about language instruction! The following is a summary of the main points I wanted to make about your chapter, and I will also send you a Word document of your chapter with my comments in the side margin. The Word document contains the minor edits as well as some notes about the stuff I will cover on this Discussion page. The first thing I noticed about your page was that it was very well laid out. I appreciate your use of headings and subheadings and might follow your lead in formatting my own chapter! The layout gives the impression of a chapter that was well planned out and organized in a logical fashion. The picture you used for showing a cochlear implant was simple and clear, which is great. The worst thing I have ever come across in a textbook was a photo that was supposed to be clarifying something in the text, but was so complicated and full of detail that it only confused the reader. Your picture does not do that – awesome. I know it’s a bit tough to get pictures into Wikiversity if they aren’t already there (my chapter has no pictures yet…shame) but if you could find any more to add, they really make things look so much more interesting. First of all, let’s discuss citations. There are some areas where I feel citations are lacking, for example the sections on oral and total communication. Where is that information from? If you are simply summarizing the textbook, then Dr. Newman said citing isn’t necessary, but you will also get a better mark if you go out and find a different source for this information (and then cite it). This happens in a couple of other spots as well, and I have highlighted them in the Word document I will send you. Also, in some places you have cited using the Wikiversity format and in others you use APA and then sometimes you use both! I am guessing that this is a work in progress, but just in case, picking one or the other would be ideal. If you haven’t seen it already, there is a page explaining how to cite to an active link at the bottom of your chapter page, a la Wikipedia style (Dr. Newman included a link to this instructional page in his Peer Reviews PDF). I haven’t done this yet either but I am guessing it is required for our final versions. One of the areas I think you could improve on is the cochlear implant section. You are the only person who gets to write about them, and covering them in good detail would make your chapter more interesting. Like I said, I have some background in this area and would like to make some suggestions for further information you can include in this section. Firstly, I think it is crucial that you mention that there are two different kinds of deafness. In your chapter you make a global statement about deaf individuals having damage to their cochlea, and this is not always true. Deafness resulting from damage to the cochlea is called conductive deafness, while deafness resulting from impaired functioning of the auditory nerve is called nerve deafness (you can also become deaf through brain damage but this is rarer). Cochlear implants are helpful only to those with conductive deafness, because they take over auditory signalling at the level of the cochlea. You can definitely find good articles on this out there. Secondly, you fail to mention the most interesting and most important part of the cochlear implant, which is the microelectrode array that is inserted onto the bottom of the basilar membrane. The microelectrode array is the part of the implant that actually takes over signalling from the damaged cochlea, more specifically performs the signalling that would normally come from hair cells (which are the parts of the cochlea most likely to be damaged). I think it is important to give some detail on this. There are also some very cool pictures of it on Google… The only other section that I think needs a bit of work is the one on reading. I think you have some good information in there but it is not organized in a smooth and logical fashion. You end up repeating yourself a bit and bringing up topics two or three times instead of explaining it thoroughly the first time you mention it. It’s a simple matter of reading it through a few times though, and you should easily be able to rearrange it and simplify it so the reader gets all the facts in the right order. The rest of your sections are written very well (with a couple of small exceptions which I have highlighted in your Word document). You have done a great job of presenting multiple points of view when it comes to controversial research and conclusions that have been put out there over the years. Sometimes the direction of your arguments gets a little vague, but overall well done. Be careful to modify your statements with words like “possible”, “suggested”, and “proposed”, and to keep citing as much as you can, so that your writing doesn’t come across as a personal opinion rather than a research review. That’s all from me! I think this is a great first version and with a few small edits and bit of elaboration, it will be a phenomenal final chapter. Feel free to email me with any questions/concerns, at epmcguire@gmail.com or emily.mcguire@dal.ca :)


text:
Swahili I/Grammar

Swahili or Kiswahili is a language in which affixes are very prevalent. Oftentimes, these affixes, though usually prefixes, give the sentence a whole new meaning. This is one thing to be careful about when learning Swahili, as you must learn to hear eacher important thing to try to learn, which is especially difficult for native English speakers, is to learn the agreement between noun, adjective, pronoun, and even verb.

One helpful aspect of Kiswahili grammar is the fact that there are no articles. This can be confusing at first, but then proves to be useful, as they would have to agree with the nouns and that would get very confusing. One other very important feature of Swahili grammar is "-a," meaning "of." Each noun class has a version of this word, so it must agree with the word which it is modifying. For instance, "Chuo Kikuu cha Nairobi" means "Main School (University) of Nairobi." The "cha" is a form of "-a" that agrees with "chuo," which is in the 7th noun class.

Pronunciation
Reading Material: Swahili Wikibook

Word Order
Swahili generally has an SVO word order, though you can actually express a subject, verb, and object into one word. Even if you do not combine them into one word, though, you must put the proper affixes into the verb as though you hadn't. Unlike many other languages, however, words do not change when put into object form. This is why you can look "mimi" up in the dictionary and find a result of both "me" and "I." There are also no datives or genitives or ablatives. There are, however, locatives, such as "chuoni," meaning "in school/at school." This comes from chuo, meaning "school," only adding the locative suffix, "-ni." Other examples of this are nyumbani, or "in home/at home," and duniani, or "in the world," and kanisani, or "in church/at church."

Noun Classes
Like in most Bantu languages, Swahili has a system of noun classes, or ngeli. Originally, there were 22 noun classes in Swahili, but over time it evolved into less. There are now 16 noun classes in Swahili: As you can see, Swahili nouns are not pluralized in the same way as many other languages. Instead, the prefix of the word changes, transferring the word to a different noun class.

Verbs
Verbs in Swahili don't conjugate, but instead take subject prefixes. The pronouns I, you, he/she/it, we, y'all, they all have different subject prefixes, as well as separate words for each. These pronouns, however, change for negation of the verb. In addition, to mark the tense of a verb, there are affixes for each tense or mood. In addition, gerunds use the same words as infinitives. So "Ninapenda kula" means "I love eating," because "ninapenda" means "I love" and "kula" means "to eat." So "kula" can mean either "to eat" or its gerund, "eating." Also, after you have marked the subject and the tense, an object can be added in using an object affix. These are usually the same as the subject affixes, but this is not always the case. You can also add applicatives and causatives and passive markers into verbs, changing the meaning once again. For instance, "ninaua" means "I kill," whereas "ninauawa" means "I am killed." Lastly, you can add place or time markers to show when something happened or where something is. Subject Pronouns Object Pronouns Here is a table showing the basic tenses of Swahili. You place these affixes between the subject and object prefix and suffix to show the tense. You will learn others, like the conditional and the storytelling tense later in lesson 5.

Adjectives
Adjectives, like in other languages, must agree with the noun which they are modifying. This is often, but not always, done by adding the noun's prefix to the beginning of the adjective. So for instance, "ndoto nzuri" would mean "good dream", but "mtoto mzuri" would mean "good child." Notice how both "mzuri" and "nzuri" mean good, but their prefixes are different. This is because they have to agree with the noun which they are modifying. In this case, ndoto is in the 9th noun class, giving its adjective the n- prefix, whereas mtoto is in the 1st noun class, giving its adjective the m- prefix.

Concord
Concord, or agreement, is a very important part of Swahili grammar. As mentioned earlier, verbs and adjectives must agree with nouns, altering themselves to adapt to the noun class of the noun which they must agree with.

Exercises
Exercises

Vocab of Lesson 1
Here is all of the vocabulary mentioned in this lesson. Now, as many people advise, it is not good to translate. To achieve native fluency, you must think like a native. So when you hear the word "child," you see some image in your head, usually of a child. You should make a flashcard with "mtoto" written on it with that image (in color if you want) under the word "mtoto." Make a separate card for "watoto." Make a flashcard for each of the listed items below.
 * Mtoto = Child
 * Watoto = Children
 * Mti = Tree
 * Miti = Trees
 * Jicho = Eye
 * Macho = Eyes
 * Kisu = Knife
 * Visu = Knives
 * Ndoto = Dream
 * Ndoto = Dreams
 * Ua = Fence
 * Nyua = Fences
 * Utoto = Childhood
 * Kuwa = To Be
 * Chuo = School
 * Vyuo = Schools
 * -Zuri = Good
 * Nyumba = Home
 * Kanisa = Church
 * Dunia = World
 * Ngeli = Noun Class
 * Ngeli = Noun Classes
 * Mimi = I
 * Mimi = Me
 * -A = Of

title: Swahili I/Grammar


text:
User:Eml5526.s11.team2.brenner/hwk4

=Problem 4.2: Heat Conduction Fish & Belytschko Page 73 Problem 3.5 =

Given: The Following Boundary Conditions
Where (4.2.1) is the essential boundary condition defined at $$ \displaystyle x = 0 $$ and (4.2.2) is the natural boundary condition defined at $$ \displaystyle x = 10 $$ on the domain $$ \Omega = ]0,10[ \quad $$

Find:The Weak Form for 1-D Heat Conduction From the Strong Form
Obtain the weak form for the equation of the heat conduction with the boundary conditions stated above. The condition on the natural boundary condition is a convection condition.

Solution
The strong form has the following equation for a 1-D heat conduction body.

Assuming a steady-state from for the heat conduction and a constant area ( 4.2.3) reduces to

Multiplying by an arbitrary weighting function $$ \displaystyle w(x) $$ and integrating with respect to x across the domain leads to the following form

Making use of Integration by Parts on the first term of (4.2.5) leads to the following equation.

Using the requirement of the essential boundary condition on the weighting function satisfying the essential boundary location implies that $$ \displaystyle w(0) = 0 $$ and also using the natural boundary condition (4.2.2) $$ k \frac{\partial T}{\partial x}_{x=10} = hT $$ leads to the following

Therefore the weak form can be represented finding a $$ \displaystyle T(x) $$ such that for all smooth $$ \displaystyle T(x) $$ with $$ \displaystyle T(0) = 100 $$ (4.2.8) is satisfied.

=Problem 4.4: Torsion of a Bar Fish & Belytschko Page 73 Problem 3.7 =

Given
Given the strong form for the circular bar in torsion (Fig. 1):

Where the natural boundary condition is

Where the essential boundary condition is

Where the essential boundary condition $$ \displaystyle m(x) $$ is a distributed moment per unit length, M is the torsion moment, $$ \displaystyle \phi $$ is the angle of rotation, $$ \displaystyle G $$ is the shear modulus and $$ \displaystyle J $$ is the polar moment of intertia given by $$ \displaystyle J = \frac{\pi C^4}{2} $$, where $$ \displaystyle C $$ is the radius of the circular shaft.

Find
a. Construct the weak form for the circular bar in torsion.

b. Assume that $$ \displaystyle m(x) = 0 $$ and integrate the differential equation given above. Find the integration constants using the boundary conditions.

(a)
Multiplying (4.4.1) by an arbitrary weighting function $$ \displaystyle w(x) $$ and integrating with respect to x across the domain leads to the following form

Making use of Integration by Parts on the first term of (4.4.4) leads to the following equation.

Using the requirement of the essential boundary condition on the weighting function satisfying the essential boundary location implies that $$ \displaystyle w(0) = 0 $$ and also using the natural boundary condition (4.4.2) $$ JG \frac{\partial \phi}{\partial x}_{x=l} = M $$ leads to the following

Therefore the weak form can be represented finding a $$ \displaystyle \phi(x) $$ such that for all smooth $$ \displaystyle \phi(x) $$ with $$ \displaystyle \phi(0) = 0 $$ (4.4.7) is satisfied.

(b)
Applying equation (4.4.1) and with $$ \displaystyle m(x) = 0 $$ leads to the following form.

Integrating (4.4.8) with respect to x generates a constant of integration and provides the following form.

Applying the natural boundary condition from (4.4.2) leads to the following

Solving (4.4.9) for $$ \displaystyle \frac{\partial \phi}{\partial x} $$ and substituting in $$ \displaystyle C_1 = -M $$ leads to the following

Integrating (4.4.11) with respect to x generates a constant of integration and provides the following form, which is the well known equation for angle of twist of a bar.

Applying the essential boundary condition from (4.4.3) to (4.4.12) leads to the following

Therefore the constants and the angle of twist are as follows.

=Problem 4.5: =

title: User:Eml5526.s11.team2.brenner/hwk4


text:
Talk:Psycholinguistics/Aphasia Rehabilitation

Peer Review

Research The research in your chapter seems to be very well done. You have a wide variety of research that you have properly cited throughout your. Further your “aphasia sites of interest” section is a good addition to your chapter and allows the reader to pursue their interest.

Logic/Organization Your chapter is outlined in a logic fashion. By providing the origins and history of the topic first and then moving into the different methods used today is a great way to organise this topic. However a conclusion section could be useful to sum up the information in the chapter. This could include any possible research occurring in the field (if this can be found), it could incorporate your new trends section that you have already made, or it could explain which method of rehabilitation is most widely used and why this is so.

Analysis/Integration Overall your sections are good but could use further discussion of the topics. This is mostly seen in the “Different Approaches” section. For most of the methods you provide a good definition of the approach but further information could be given. This could include the effectiveness of each method and percentage of success, how long rehabilitation typically takes for each method and why a certain approach would be used rather than another. A great example of this is your section on “Cognitive Neuropsychological Approach.” If it is a problem that there is little research and work done in the other methods of rehabilitation than perhaps just explain that when you are defining about them so the reader can focus on the more important and more widely used methods.

Another thing to be careful about is bring up things and then not explaining them. In the “Focus Today” section you brought up “Western Aphasia Battery,” stated its use but never explained how it works, which could be helpful to the reader.

Answer Questions/Makes Argument Your chapter answer the question of what is Aphasia Rehabilitation. However it does not answer the questions of how it was developed or which method is best used today. By providing an outline of the steps taken in history to get aphasia rehabilitation to where it is today and further explanation of approaches that I mentioned above could allow for these things to be answered. For the history sections, including information such as; who are the major contributors that made aphasia rehabilitation what it is today? What research did they do? And so on would be of great use to a reader.

Writing Style As with all first drafts, it would be a good idea for you to read over your chapter and make sure all of your sentences make sense and are easily read. For the most part they are but at time I found myself reading a sentence a couple of times to understand what you were trying to say. I especially found this in your last section “Book Clubs”, I knew what you wanted to say but I found your wording a bit confusing and it seemed that you were missing some words.

Overall your chapter is a good introduction to the aphasia rehabilitation. However by adding the historical development and further explanation of the rehabilitation approaches, you could make greatly improve on this chapter.