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### Messages - Victor Ivrii

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1
##### MAT244--Misc / FE Marks
« on: April 20, 2018, 05:22:07 PM »
I removed this topic since it does not carry any useful information but also some students exposed some personal info. Not much, but better to be on the safe side.

2
##### Final Exam / FE-P5 Comments
« on: April 19, 2018, 01:16:12 PM »
a. Some students missed some stationary points and/or reported wrong points. All further analysis in the wrong points was ignored as irrelevant. For all three correct points found I gave 10pts, with a reduction of 3 pts for each missed points, and 1pts for extra points (only for those who found 3 correct points).

b. Linearization was easy, but some borked it. Finding eigenvalues was supposed to be a breeze but ...
One does not need to solve any equations to find eigenvalues of the diagonal or triangular matrices (some students wrote wrong equations and found wrong eigenvalues). Also eigenvectors of the diagonal matrices are obvious, and of the triangular are easy.

Not everyone found correctly eigenvalues of the matrix at $(2,3)$. T recall, that they are $1\pm i\sqrt{5}$. One does not need to look for eigenvectors, however one should look at the sign of bottom left element of the matrix and conclude what is the direction of rotation (I have not subtracted points for missing justification that it is counter-clockwise)

c. Drawing of the local pictures. At (0,0) mant=y draw incoming and outgoing lines like X instead of +, others indicated the wrong directions. Point $(-1,0)$ was more difficult. And in $(2,3)$ some draw "hairy monsters"

d. Even when all local pictures were drawn correctly, some students draw intersecting lines (trajectories do not intersect!) and not everyone observed that $x=0$ and $y=0$ consist of trajectories (see "skeleton" in my post above)

In some papers with no calculations or with calculations, leading to wrong conclusions) there are "miraculously" correct pictures. Those were discarded because "only solutions (not just answers) are evaluated".

3
##### Final Exam / FE-P6 Comments
« on: April 19, 2018, 11:45:12 AM »
Observe that Hessian of $H(x,y)$ is
$$\begin{pmatrix} 2x &2y\\ 2y &2x \end{pmatrix};$$
compare with the  Jacobi matrix (Jacobian is its determinant). In this particular case (of exact system) sometimes it is called skew-Hessian.

I attach the Contour plot of $H(x,y)$; note that $(-1,0)$ is the local maximum and $(1,0)$ is the local minimum, while $(0,\pm 1)$ are two saddle points

4
##### Final Exam / Re: FE-P5--solution
« on: April 18, 2018, 06:48:47 AM »
a.  Solving $x(x-y+1)=0$, $y(x-2)=0$ we get cases
\begin{align*}
&x=y=0  &&\implies A_1=(0,0),\\
&x=x-2=0  &&\implies \text{impossible}\\
&y=x-y+1=0 &&\implies A_2=(-1,0),\\
&x-y+1=x-2=0 &&\implies A_3=(2,3).
\end{align*}
b. Linearizations at these points have matrices
\begin{align*}
&
\begin{pmatrix}
1 &\ \ 0\\
0 &-2
\end{pmatrix}
&&
\begin{pmatrix}
-1 &\ \ 1\\
0 &-3
\end{pmatrix}
&&
\begin{pmatrix}
2 &-2\\
3 &0
\end{pmatrix}
\\[5pt]
\text{with eigenvalues    }&\{1,-2\} &&\{-1,-3\} && \{1-\sqrt{5}i,1+\sqrt{5}i \}
\end{align*}
and therefore
* $A_1$ is a saddle,
* $A_2$ is a stable node, and
* $A_3$ is unstable focal point and since left bottom number is $3>0$ it is counterclockwise oriented.

c. Axis are:
in $A_1$:  $\mathbf{e}_1=(1,0)^T$ unstable ($\lambda_1=1$), $\mathbf{e}_2=(0,1)^T$ stable ($\lambda_2=-2$).

in $A_2$: $\mathbf{f}_1=(1,0)^T$ ($\lambda_1=-1$), $\mathbf{f}_2=(1,-2)^T$ ($\lambda_1=-3$). Since $\lambda_1 >\lambda_2$, all trajectories have an entry directions $\pm \mathbf{f}_1$ (except two, which have entry directions $\pm \mathbf{f}_2$). Then we draw trajectories near critical points (See attachment  P5-loc.png).

d. One should observe that either $x=0$ in every point of the trajectory, or in no point; and that $y=0$ in every point of the trajectory, or in no point. It allows us to make a "skeleton'' of the phase portrait (see attachment), impose local pictures on it and finally draw a global portrait

5
##### Final Exam / Re: FE-P4
« on: April 15, 2018, 03:26:19 AM »
Solution is complete, but the answer must be written.

Also a simpler form $\ln(\tan(t))$ is preferable.

Finally \sin, \cos, and so on must be escaped by \ to provide upright letters and a proper spacing

6
##### Final Exam / Re: FE-P3
« on: April 15, 2018, 03:15:29 AM »
Since the solution is incomplete after Y(x),
I am attaching a copy of my solution
The only thing which was missing in the solution, is the final answer, but it warrants neither such claim, nor uploading your solution.

General remark:
It would be better to denote "parameters" by uppercase letters $C_1(x)$, $C_2(x)$,... and constants by lowercase letters $c_1$, $c_2$,...

7
##### Final Exam / Re: FE-P6
« on: April 14, 2018, 02:25:56 PM »
There is nothing missing.

8
##### Final Exam / Re: FE-P6
« on: April 14, 2018, 01:19:45 PM »
.

9
##### APM346--Announcements / Re: Grading FE
« on: April 14, 2018, 01:13:44 PM »
E-marks: submitted. As soon as they are approved (Monday?) I will make detailed exam marks available

10
« on: April 14, 2018, 10:40:19 AM »
I finished grading FE. Need to enter marks, calculate etc. Bonus (karma) will be done about 5pm. After this: no changes

11
##### Final Exam / Re: FE-P7
« on: April 14, 2018, 09:48:43 AM »
Indeed, George and Tristan lost "$-$" (but in Tristan's exam paper there was one).

How to calculate F.T. of $h(y)$? We observe that $h(y)'=-\partial_y \phi(y)$ $\phi(y)=2(1+y^2)^{-1}$ and therefore $\hat{h}(\eta)= -i\eta \hat{\phi}(\eta)= -i \eta e^{-|\eta|}$.

More detailed final calculations:
\begin{align*}
u(x,y)=&\int_{-\infty}^\infty \hat{u}(x,\eta)e^{i\eta y}\,d\eta =
i\int_{-\infty}^\infty \sigma(y) e^{-|\eta|x+i\eta y}\,d\eta\\[3pt]
=&-i\int_{-\infty}^0 e^{\eta (1+x+yi)}\,d\eta + i\int_0^{-\infty} e^{-\eta (1+x-yi)}\,d\eta \\[3pt]
=&-\frac{i}{1+x+yi}+\frac{i}{1+x-yi}= -\frac{2y}{(1+x)^2+y^2}.
\end{align*}

12
##### Final Exam / Re: FE-P6
« on: April 14, 2018, 05:23:18 AM »
Unfortunately, I wrote "by separation of variables" and almost everybody got confused. If it was more advanced class, I would shrug "So what? You should think rather than follow wrong advices, even from Professor, Supervisor, ..." but it would be too cruel and unfair here.

So, I decided for those who got completely confused, to give mark "-" (effectively 0). So I am taking the sum of all problems except P5, and multiply by 7/6. I also take the sum of all problems. And finally I take the maximum of these two numbers.

Still, correct solution is pending.

13
##### Web Bonus Problems / Re: Exam Week
« on: April 14, 2018, 05:07:22 AM »
Sheng
correct, but one needs to solve the problem for $w$ by the standard separation of variables.

14
##### Final Exam / Re: FE-P5
« on: April 13, 2018, 09:13:38 PM »
OK

15
##### Final Exam / Re: FE-P5
« on: April 13, 2018, 01:18:03 PM »
Consolation prize

Solve this problem but with (\ref{5-2}) replaced by (\ref{5-X})

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