To my MEC2010 group members,
The test function you should implement in your expression is as follows:
$$\theta(i,j)=\varepsilon \cdot \sin \bigg( \frac{ij\pi}{n_{x}} \bigg)$$
The test case was the diffusion (heat) equation:
$$\frac{\partial U}{\partial t}=\kappa\frac{\partial^{2}U}{\partial x^2}$$
In your response, you need to discuss the conditions that ensure numerical stability and produce a plot using meshgrid() to show the stable and unstable cases respectively.
Email me if further clarification is required.
Wednesday, June 10, 2015
Thursday, April 23, 2015
Hi All,
I wanted to make this post to address some of the concerns expressed in your emails regarding 'distributions' and 'functions'. This is covered extensively by online resources. I've had a brief look at the Wikipedia page and it seems fit for consumption. Please don't ask me anything about compact support. I don't know anything about it. Anyway, there are many different flavours of explanations and the preference of one over others differs between authors. I'm going to offer one of those flavours.
A distribution has some 'function-like' properties but not enough of those to make it a function. One of those properties is that it houses a class or family of functions AND, for each of those functions, points to a set of numbers (a field) that relates to that function. The class of functions that lie inside the distribution MUST be well-behaving. I think distributions are only accepted very reluctantly by mathematicians and recognition comes on the basis that (1) it is valuable as an analysis tool and (2) they have an explicit, analytic form.
Linearity is preferred but not necessary; see examples from stochastic calculus and operational calculus for additional explanation. This is not an issue for us because we only look at linear problems that do not involved coupled dependent variables and their derivatives. The only distributions we will EVER look at (maybe) are the PDF and the Dirac Delta. And yes, it is possible to multiply PDFs but there are constraints that limit those possibilities. I will talk more about these in class. There are a number of transforms that can change distributions to 'nicer' forms to allow for that operation.
I'll go through a few concrete examples during our next extension class on Tuesday 28 April. Email me if you have any special requests.
I wanted to make this post to address some of the concerns expressed in your emails regarding 'distributions' and 'functions'. This is covered extensively by online resources. I've had a brief look at the Wikipedia page and it seems fit for consumption. Please don't ask me anything about compact support. I don't know anything about it. Anyway, there are many different flavours of explanations and the preference of one over others differs between authors. I'm going to offer one of those flavours.
A distribution has some 'function-like' properties but not enough of those to make it a function. One of those properties is that it houses a class or family of functions AND, for each of those functions, points to a set of numbers (a field) that relates to that function. The class of functions that lie inside the distribution MUST be well-behaving. I think distributions are only accepted very reluctantly by mathematicians and recognition comes on the basis that (1) it is valuable as an analysis tool and (2) they have an explicit, analytic form.
Linearity is preferred but not necessary; see examples from stochastic calculus and operational calculus for additional explanation. This is not an issue for us because we only look at linear problems that do not involved coupled dependent variables and their derivatives. The only distributions we will EVER look at (maybe) are the PDF and the Dirac Delta. And yes, it is possible to multiply PDFs but there are constraints that limit those possibilities. I will talk more about these in class. There are a number of transforms that can change distributions to 'nicer' forms to allow for that operation.
I'll go through a few concrete examples during our next extension class on Tuesday 28 April. Email me if you have any special requests.
Saturday, September 20, 2014
Congratulations to all my UMAT students - all fourteen of you. The range of results was bounded at the high end which is indicative of the level of competitiveness demonstrated by this year's cohort. The following is a summary of your combined achievements.
Range: 77 - 100 percentile
Mean: 89.1 percentile
Median: 87.7 percentile
Stdev: 4.3 percentile
Students who sat the exam: 14
Range: 77 - 100 percentile
Mean: 89.1 percentile
Median: 87.7 percentile
Stdev: 4.3 percentile
Students who sat the exam: 14
Tuesday, May 27, 2014
Thursday, December 19, 2013
VCE
Humanities: English / ESL, English Language, History
Social Sciences: Accounting, Economics, Legal Studies, Psychology, Business Management
Physical Sciences: Specialist Maths, Maths Methods, Chemistry, Physics, Biology
IB (SL & HL)
Humanities: English, History
Social Sciences: Economics, Information Technology, Psychology
Physical Sciences: Mathematics, Chemistry, Physics, Computer Science
Tertiary
Finance, Accounting, Business Law, Economics, Econometrics/Statistics
Special Examinations
UMAT & GAMSAT
CFA (All Levels)
Classes (Structured Accelerated Learning Program)
English
English Language
Specialist Maths
Maths Methods
Physics
Chemistry
Humanities: English / ESL, English Language, History
Social Sciences: Accounting, Economics, Legal Studies, Psychology, Business Management
Physical Sciences: Specialist Maths, Maths Methods, Chemistry, Physics, Biology
IB (SL & HL)
Humanities: English, History
Social Sciences: Economics, Information Technology, Psychology
Physical Sciences: Mathematics, Chemistry, Physics, Computer Science
Tertiary
Finance, Accounting, Business Law, Economics, Econometrics/Statistics
Special Examinations
UMAT & GAMSAT
CFA (All Levels)
Classes (Structured Accelerated Learning Program)
English
English Language
Specialist Maths
Maths Methods
Physics
Chemistry
Thursday, October 3, 2013
English Exam and Context Writing: Expository Essay vs. Expository Article
Hi Guys,
It's about 6 weeks until D-Day and I'm sure you can't wait to have me out of your lives. On that note, I've spoken to a selection of VCE English teachers from a cross section of schools over the break and there has been a number of developments regarding the approach students should take when formulating their response to Task 2: Context on the exam. The information below is especially relevant to those students who wish to write expository for that particular section.
As we're all aware, the final exam comes in three flavours: text response, context writing and language analysis. Task 1: Text Response is generally formulaic. It involves your stock standard 1-3-1 introduction, body and conclusion. Students make their case in their introduction, provide evidence and supporting arguments in the body and then a closing statement to wrap things up. The more creative pieces may try to take on a contrarian perspective and challenge the status quo but generally the writer's creative options, in particular their style and form, are limited to the rigmarole of text response writing as outlined in the study guide.
Task 3: Language Analysis is even more formulaic. Students are provided with a fictitious piece of stimulus to which they must pick apart and analyse the various elements of suasion. There's generally a regime in place where students are taught to identify and comment on the effectiveness of structural and emotive mechanisms that help the piece and its author connect with the target demographic. The stimulus is generic and contrived as its purpose is to provide enough talk points for students to pick up and expound.
The crux of the exam lies with Task 2: Context. Let's first discuss the merits of an Expository Essay. This is a piece that takes an opportunity to discuss and dissect the essence of a particular text with corroborative evidence from external references. The split between internal vs external evidence gathering is around 30:70 give or take. Students may wish to adopt the golden triangle approach (text-author-audience) whereby they formulate a piece that draws upon the dominant themes, symbols and motifs present within the text and its genre with respect to the author and target audience. The Expository Essay is structurally similar to that of a Text Response but the content is augmented with exogenous references to provide additional insight and broaden the reader's worldview. it is, for the most part, formulaic and procedural.
Conversely, an Expository Article is a persuasive piece that also draws heavily on imaginative and creative resources. The form is flexible: a letter to a Special Interest Group, correspondence between the author and a related party, an exchange of dialogue between characters from the text, a personal reflective journal etc. The choices are endless. The article format is flexible and highly responsive as it enables each student to customise and tailor their piece to their sentimentality and to address a particular section of subtext. Students are encouraged to extract a segment of the plot and take a tangential approach by proposing an alternative plausible reality whereby they create and introduce an extension to the pre-existing body of work. It is important to realise that the expository article can manifest into different things for different writers but it is not an essay in its own right.
The free form feature article or letter aspires to the spirit and essence of exposition. Expository writing was never about conformance. In an otherwise regimented exam, it presents students with a unique opportunity to deviate from the norm and address the prompt from a vantage point of their choice. The decisions students make with respect to language, narrative, tone, form and delivery help shape the nature of the beast and the impact it has on the reader. Hence, and most importantly, it arms students with the ability to surprise the assessor, perhaps even shocking them, and to challenge their priors. Expository writing using free form style such as feature article is a pioneering endeavour that is naturally innovative but also necessarily risky.
In general, the uncertainty in marking associated with Task 1 and 3 are low-moderate and low respectively on a risk matrix. For Task 2, it is moderate-high. As we're all aware, there's a positive correlation between subjectivity and marking uncertainty. It is entirely possible, and not altogether uncommon, for a perfect piece to be given an 8.0 or 8.5; perhaps even a 7.0 if the assessor is a real [REDACTED]. On average, each piece is allocated 4 minutes of assessment time; 2 minutes from each assessor. The average of their combined mark is the final mark you get (exceptions are rare).
As you can imagine, the margin for error in judgment and subjective interpretation is significant. If you write a mediocre piece and your message is unclear to the examiner, you will not get the mark. If you write a brilliant piece full of nuanced and eloquent ideas but the message is unclear to the examiner, you will not get the mark. And unlike your SACs which allow students to guide markers with a Statement of Intent, such a device is not available for consideration on the exam. So be clear, be concise and be succinct. Take the examiners for the rider of their lives; just don't expect them to pick themselves back up if they fall off your bandwagon.
It's about 6 weeks until D-Day and I'm sure you can't wait to have me out of your lives. On that note, I've spoken to a selection of VCE English teachers from a cross section of schools over the break and there has been a number of developments regarding the approach students should take when formulating their response to Task 2: Context on the exam. The information below is especially relevant to those students who wish to write expository for that particular section.
As we're all aware, the final exam comes in three flavours: text response, context writing and language analysis. Task 1: Text Response is generally formulaic. It involves your stock standard 1-3-1 introduction, body and conclusion. Students make their case in their introduction, provide evidence and supporting arguments in the body and then a closing statement to wrap things up. The more creative pieces may try to take on a contrarian perspective and challenge the status quo but generally the writer's creative options, in particular their style and form, are limited to the rigmarole of text response writing as outlined in the study guide.
Task 3: Language Analysis is even more formulaic. Students are provided with a fictitious piece of stimulus to which they must pick apart and analyse the various elements of suasion. There's generally a regime in place where students are taught to identify and comment on the effectiveness of structural and emotive mechanisms that help the piece and its author connect with the target demographic. The stimulus is generic and contrived as its purpose is to provide enough talk points for students to pick up and expound.
The crux of the exam lies with Task 2: Context. Let's first discuss the merits of an Expository Essay. This is a piece that takes an opportunity to discuss and dissect the essence of a particular text with corroborative evidence from external references. The split between internal vs external evidence gathering is around 30:70 give or take. Students may wish to adopt the golden triangle approach (text-author-audience) whereby they formulate a piece that draws upon the dominant themes, symbols and motifs present within the text and its genre with respect to the author and target audience. The Expository Essay is structurally similar to that of a Text Response but the content is augmented with exogenous references to provide additional insight and broaden the reader's worldview. it is, for the most part, formulaic and procedural.
Conversely, an Expository Article is a persuasive piece that also draws heavily on imaginative and creative resources. The form is flexible: a letter to a Special Interest Group, correspondence between the author and a related party, an exchange of dialogue between characters from the text, a personal reflective journal etc. The choices are endless. The article format is flexible and highly responsive as it enables each student to customise and tailor their piece to their sentimentality and to address a particular section of subtext. Students are encouraged to extract a segment of the plot and take a tangential approach by proposing an alternative plausible reality whereby they create and introduce an extension to the pre-existing body of work. It is important to realise that the expository article can manifest into different things for different writers but it is not an essay in its own right.
The free form feature article or letter aspires to the spirit and essence of exposition. Expository writing was never about conformance. In an otherwise regimented exam, it presents students with a unique opportunity to deviate from the norm and address the prompt from a vantage point of their choice. The decisions students make with respect to language, narrative, tone, form and delivery help shape the nature of the beast and the impact it has on the reader. Hence, and most importantly, it arms students with the ability to surprise the assessor, perhaps even shocking them, and to challenge their priors. Expository writing using free form style such as feature article is a pioneering endeavour that is naturally innovative but also necessarily risky.
In general, the uncertainty in marking associated with Task 1 and 3 are low-moderate and low respectively on a risk matrix. For Task 2, it is moderate-high. As we're all aware, there's a positive correlation between subjectivity and marking uncertainty. It is entirely possible, and not altogether uncommon, for a perfect piece to be given an 8.0 or 8.5; perhaps even a 7.0 if the assessor is a real [REDACTED]. On average, each piece is allocated 4 minutes of assessment time; 2 minutes from each assessor. The average of their combined mark is the final mark you get (exceptions are rare).
As you can imagine, the margin for error in judgment and subjective interpretation is significant. If you write a mediocre piece and your message is unclear to the examiner, you will not get the mark. If you write a brilliant piece full of nuanced and eloquent ideas but the message is unclear to the examiner, you will not get the mark. And unlike your SACs which allow students to guide markers with a Statement of Intent, such a device is not available for consideration on the exam. So be clear, be concise and be succinct. Take the examiners for the rider of their lives; just don't expect them to pick themselves back up if they fall off your bandwagon.
Saturday, August 17, 2013
Evans et al 'Essential: Specialist Maths'
Hi All,
To all the guys and gals studying Specialist Maths this year, I have managed to dig up a pdf of all the solutions to the chapter problems and review sections. Note that these are actual worked solutions with step by step guidance as to how the end result is derived. I think most of you should have received a copy by now, if not from me than certainly from your school teacher.
Anyhow, it has been brought to my attention that the Solution Manual does not contain solutions to the extended response questions. I've looked around but it appears that solutions to these questions don't actually exist, at least not in any official form. I'm currently in the process of compiling a set of handwritten notes which provides worked steps to all these problems. Please be patient as some of the proofs take time and can take up more than two A4 pages including graphics.
Since there are no formal solutions to compare against, I have used computational methods in many instances to derive an approximation to check my answers. However, this is not possible for every single type of question as some are theoretical proofs. If anyone spots a mistake, please bring it to my attention immediately. Also, I'm completely open to others having a go and submitting their own attempts at some of these problems. I'm about halfway through the long form questions and the plan is to get the first draft out around mid Sept, have you guys error check and then use the feedback to finalise the document hopefully 4-5 weeks before final exams.
Watch this space and stay tuned!
To all the guys and gals studying Specialist Maths this year, I have managed to dig up a pdf of all the solutions to the chapter problems and review sections. Note that these are actual worked solutions with step by step guidance as to how the end result is derived. I think most of you should have received a copy by now, if not from me than certainly from your school teacher.
Anyhow, it has been brought to my attention that the Solution Manual does not contain solutions to the extended response questions. I've looked around but it appears that solutions to these questions don't actually exist, at least not in any official form. I'm currently in the process of compiling a set of handwritten notes which provides worked steps to all these problems. Please be patient as some of the proofs take time and can take up more than two A4 pages including graphics.
Since there are no formal solutions to compare against, I have used computational methods in many instances to derive an approximation to check my answers. However, this is not possible for every single type of question as some are theoretical proofs. If anyone spots a mistake, please bring it to my attention immediately. Also, I'm completely open to others having a go and submitting their own attempts at some of these problems. I'm about halfway through the long form questions and the plan is to get the first draft out around mid Sept, have you guys error check and then use the feedback to finalise the document hopefully 4-5 weeks before final exams.
Watch this space and stay tuned!
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