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Messages - Victor Ivrii

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APM346––Home Assignments / Re: An ODE
« on: Today at 10:55:58 AM »
We are talking about different things, $(x\pm yi)^{|m|}$ is a homogeneous harmonic polynomial of two variables.

Let me give you example. Consider harmonic polynomial of order $3$, containing $z^3$ (there is only one of them, the rest differ by polynomials, which do not contain $z^3$). First of all find it: We need only $z^1$ (as we separate odd and even with respect to $z$; $zxy$ is harmonic by its own, so we take $z^3- az(x^2+y^2)$ for symmetry. Obviously, this is harmonic if $a=\frac{3}{4}$. So, consider
Plugging $z=\rho\cos(\phi)$, $x=\rho\sin(\phi)\cos(\theta)$, $y=\rho\sin(\phi)\sin(\theta)$ we get
\rho^3\Bigl(\cos^3(\phi) -\frac{3}{4}\cos(\phi)\sin^2(\phi)\Bigr)=\rho^3\Bigl(\frac{7}{4}\cos^3(\phi)-\frac{3}{4}\Bigr).
Here $m=0$, obviously.

Consider harmonic polynomial of order $3$, containing $z^2$. Well it must contain $z^2x $ or $z^2y$, or,  better $z^2(x\pm iy)$. This is not harmonic polynomial, to make it correct to $z^2(x\pm yi) + a(x^2+y^2)(x\pm yi)$. Obviously, this is harmonic if $a=\frac{1}{2}$.
\bigl(z^2-\frac{1}{2}(x^2+y^2)\bigr)(x\pm yi) = \rho^3 \bigl(\cos^2(\phi)-\frac{1}[2}\sin^2(\phi)\bigl)\sin(\phi) e^{\pm i\theta}.
Here $m=\pm 1$.

To get to $m=\pm 2$ consider this way $z(x\pm yi)^2$. $m=\pm 3$ consider this way $(x\pm yi)^3$. All four are harmonic without corrections.

Web Bonus Problems / Re: Phaseportrait
« on: Today at 01:55:44 AM »
See my comment above

Term Test 2 / Do not Post until Wednesday, 3--4 pm
« on: March 20, 2018, 08:18:45 PM »
At Wed, 3--4 pm I will post all problems, for all sittings.

Until the problem is posted, you solution is considered unauthorized and will be deleted. There is a student, who writes it today

APM346--Lectures / Re: Fourier Transform and IFT table
« on: March 20, 2018, 01:35:56 PM »
I provide either FT or hint

APM346––Home Assignments / Re: An ODE
« on: March 20, 2018, 01:34:59 PM »
In case of 2 variables $(x\pm iy)^{|m|}$ are solutions (harmonic functions)

APM346––Home Assignments / Re: An ODE
« on: March 20, 2018, 08:49:19 AM »
$|m|\le l$ and both are integers

Web Bonus Problems / Re: Phaseportrait
« on: March 20, 2018, 06:21:29 AM »
there is the big difference between describe (what you see) and explain (why it is so).

APM346--Lectures / Re: Unbounded Terms
« on: March 19, 2018, 07:48:49 PM »
We are looking for a bounded solution. You may consider this as "boundary condition at infinity"

APM346--Announcements / Re: Term Test 2 arrangement
« on: March 19, 2018, 04:14:30 PM »
Late sitting:

18:00--22:00 in Huron 215, room 1008
(come as early as you can)

If the door to the building our floor is locked––email me or call me (cellphone number will be provided during the lecture)

It happens when we have several eigenvectors with the same eigenvalue; then we get different fundamental systems, but we get different descriptions of the same solutions

Web Bonus Problems / Re: Phaseportrait
« on: March 19, 2018, 01:21:44 PM »
Now explain it!!!

Web Bonus Problems / Phaseportrait
« on: March 18, 2018, 12:43:58 PM »
Sketch the phaseportrait for the system below. Anyone can post a different solution, but there is a catch: it should be drawn with different s/w than already used and this s/w must be reported
&x'= \sin(x)\cos(y)\\
&y'=-\cos(x)\sin(y) && -4\le x\le 4, \ -4\le y \le 4

MAT244--Misc / What s/w have you used for plots?
« on: March 18, 2018, 11:46:11 AM »
I see that some used s/w to plot which I am not familiar. Please post here!

Quiz-6 / Re: Q6--T0101
« on: March 18, 2018, 11:44:41 AM »
\det  should be also "escaped" (\ is an escape character)

Technical Questions / Re: Scans and snapshots
« on: March 18, 2018, 04:39:19 AM »
How to make a snapshot:

1) Put you paper on the even dark surface properly

2) Provide a good light

3) Use dedicated "scanning" s/w, not general photo s/w (f.e. I use free Scannable for iOS). It will understand what kind of the document you have, capture it without its surroundings.

4) If needed, manually post-process snapshot on smartphone, or on computer. With Scannable you can even crop and  rotate your picture

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