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Home Assignment 2 / Re: Problem 3
« on: September 26, 2012, 05:21:59 AM »Part (a) and (b), part (c) and (d) the same questions? Thanks
There was misprint in (c) fixed almost immediately. Compare fourth lines in (a) vs (b) , (c) vs (d)
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There was misprint in (c) fixed almost immediately. Compare fourth lines in (a) vs (b) , (c) vs (d)
2267
Home Assignment 1 / Re: Problem 4
« on: September 26, 2012, 05:19:38 AM »
Peishan, the question now boils down to: why solution with r.h.e. $x^2+y^2$ does not exist and one really needs to use polar coordinates (or to use equivalent geometric motivation)
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Misc Math / Re: Heat Wave Equation Solution Integral
« on: September 26, 2012, 02:28:48 AM »After we determined the solution to the heat wave equation, we took the integral from negative infinity to some x. The solution was in the form of a Normal Distribution function where in statistics the integral indicates the probability that we randomly choose a value within that interval. I was wondering what are the physical implications of this same integral in the wave equation? Does the area under the solution indicate the energy in the system up to that point or does it have some other meaning?
Heat wave equation? WTH are you talking about? There is a wave equation (and its ilk) and the heat equation (and its ilk) describing either very different processes or the same process under very different assumption. The properties of these equations are really different.
PS Sure more complex phenomena can be described by systems containing both equations but for the purpose of this class ...
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Home Assignment 1 / Re: Problem 6
« on: September 26, 2012, 02:20:39 AM »
Intuitively we know that driver's speed has negative correlation with traffic density. A more realistic choice for $c$ is to let it be a monotone decreasing function of $\rho$. In this case however, conservation of cars equation is not linear anymore. This is discussed in detail by Prof. Ivrii in last year's forum.
The really interesting thing is a distinction between the speed of the individual cars $c(\rho)$ and the group speed $v(\rho)= (c(\rho)\rho)'= c'(\rho)\rho + c(\rho)$ of the group of cars. In fact the group does not have a constant "crew": if there is a place with higher density of cars than an average, it moves but the leading cars in the group have a lesser density in front, accelerate and leave the group while cars behind the group catch with it, and join it.
Such distinction is common in wave motion
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Home Assignment 1 / Re: Problem 4
« on: September 26, 2012, 02:11:08 AM »
c) In the latter case, if $(x,y)=(0,0)$ is in domain of $u$, general solution does not exist as $\arcsin{\frac{x}{\sqrt{x^2+y^2}}}$ is not well-defined.
Not persuasive: may be we just were not smart enough? You need to demonstrate that solution really does not exist and explain what is an obstacle.
Hint Use polar coordinates
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Home Assignment 1 / Re: Problem 2
« on: September 25, 2012, 06:53:47 PM »
The difference between two cases is that in one of them all trajectories have $(0,0)$ as the limit points and in another only those with $x=0$ or $y=0$ (node vs saddle).
Actually since in the saddle case $x^4y=C$ for $C\ne 0$ consists of two disjoint parts (with $x>0$ and with $x<0$), the values of $u$ on these parts are not necessarily equal and $u=f(x^4y)+x|x|^{-1}g(x^4y)$ with $g(0)=0$.
Actually since in the saddle case $x^4y=C$ for $C\ne 0$ consists of two disjoint parts (with $x>0$ and with $x<0$), the values of $u$ on these parts are not necessarily equal and $u=f(x^4y)+x|x|^{-1}g(x^4y)$ with $g(0)=0$.
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Home Assignment 1 / Re: Problem 5
« on: September 25, 2012, 06:42:11 PM »
The previous post is definitely an improvement over the preceding one
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Home Assignment 1 / Re: Problem 5
« on: September 25, 2012, 03:12:39 PM »
Scanning is barely passable
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Home Assignment 1 / Re: Problem 1
« on: September 25, 2012, 03:08:21 PM »
So, what are the answers for (c), (d)? Especially (d). Formulate them explicitly
Scanning is passable but not spectacular
Scanning is passable but not spectacular
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Home Assignment 1 / Re: WTH?
« on: September 25, 2012, 10:14:20 AM »Dear professor Ivrii,
Thank you for remembering meBut I spent too much time to resize them and post them; Some of them is readable, can you please consider them?
Next time I will use scanner.
Thank you,
Aida
Barely readable. You don't need to resize (may be just rotate). However correct initial settings of the scanner/camera are crucial.
I definitely do not want to encourage this type of submission. If it was from some clueless newbie I could consider, but from veteran-poster -- no way
BTW, has anyone any idea how much time it takes to prepare 1 hour of lecture notes or 1 home assignment? 2-3 hours :-)
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Home Assignment 1 / WTH?
« on: September 25, 2012, 02:08:17 AM »
Aida, WTH? In MAT244 your scan was a golden standard of scanning
http://weyl.math.toronto.edu:8888/MAT244-2012S-Forum/index.php?topic=38.msg111#msg111 (see, I remember, so good it was), and now this? Such posts defy the whole purpose of this forum, it is not to submit your papers for grading (you submit it to TA) but even this is difficult as quality of scan is poor (actually you used cellphone without taking care of settings), but to share the solution with your classmates who can comment, find errors or correct them.
From this point the typed solution using MathJax is far the best as I can edit it, just marking the place where I see an error, and anyone can copy-paste code from it. But apart of this your former clean black-white scan with perfect position of the paper was the very best thing.
Here you use colour (not even grayscale) scan and some papers are horizontal and some diagonal ... everyone has a really hard time to read them. I admit, some of your submissions a better than othersbut from the point of view of this forum I must consider all of them non-existent
. It looks like you just decided to capture the space preventing anyone else from posting the solutions.
So, everyone should feel to post solutions.
------
Also:
some people used paper clips despite our request to staple and some even tried to use "poor man paper clips" just folding several times the corner of the paper and adding a bit of saliva. Sorry, no of these constructions is robust enough to prevent separation of pages. I hope I caught every such attempt and stapled, but I am not sure. I am also not sure if Prof Colliander managed with this. So, if you have not stapled and your pages are separated and lost, you know whom to blame.
Not everyone indicated the section where to bring your papers back (so get such papers from TA who graded it).
Please use the standard paper (letter size). Someone used A4 (a bit more narrow and a longer overseas size). It looks like a little thing but also causes problems when dealing with tens of papers.
And finally, again: please, do not come to submit papers during the class -- only before lecture, during the break or after. Everyone in the night class feels tired (instructor in the least degree because of adrenalin rush) and such distractions are rather disruptive.
http://weyl.math.toronto.edu:8888/MAT244-2012S-Forum/index.php?topic=38.msg111#msg111 (see, I remember, so good it was), and now this? Such posts defy the whole purpose of this forum, it is not to submit your papers for grading (you submit it to TA) but even this is difficult as quality of scan is poor (actually you used cellphone without taking care of settings), but to share the solution with your classmates who can comment, find errors or correct them.
From this point the typed solution using MathJax is far the best as I can edit it, just marking the place where I see an error, and anyone can copy-paste code from it. But apart of this your former clean black-white scan with perfect position of the paper was the very best thing.
Here you use colour (not even grayscale) scan and some papers are horizontal and some diagonal ... everyone has a really hard time to read them. I admit, some of your submissions a better than othersbut from the point of view of this forum I must consider all of them non-existent
. It looks like you just decided to capture the space preventing anyone else from posting the solutions.So, everyone should feel to post solutions.
------
Also:
some people used paper clips despite our request to staple and some even tried to use "poor man paper clips" just folding several times the corner of the paper and adding a bit of saliva. Sorry, no of these constructions is robust enough to prevent separation of pages. I hope I caught every such attempt and stapled, but I am not sure. I am also not sure if Prof Colliander managed with this. So, if you have not stapled and your pages are separated and lost, you know whom to blame.
Not everyone indicated the section where to bring your papers back (so get such papers from TA who graded it).
Please use the standard paper (letter size). Someone used A4 (a bit more narrow and a longer overseas size). It looks like a little thing but also causes problems when dealing with tens of papers.
And finally, again: please, do not come to submit papers during the class -- only before lecture, during the break or after. Everyone in the night class feels tired (instructor in the least degree because of adrenalin rush) and such distractions are rather disruptive.
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Home Assignment 1 / Re: HA1-pdf
« on: September 24, 2012, 10:35:46 PM »
Is it an ancient manuscript? Can anyone read this?
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Home Assignment 1 / Re: Problem 6
« on: September 24, 2012, 11:46:56 AM »
One-way (no way to reverse the direction)
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Home Assignment 1 / Re: Problem 5
« on: September 23, 2012, 11:28:50 PM »
The easiest way is to write the general solution and then trying to satisfy boundary conditions. This works for correct settings (and for those with the typo as well).