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### Topics - Jingxuan Zhang

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1
##### APM346--Misc / perturbed NLS
« on: February 19, 2019, 06:57:24 AM »
Consider the perturbed NLS
$$\partial_t q=-\partial_x^2 q-2|q|^2q+\epsilon R\qquad R=R(q,-q^*).\label{1}$$
It is suggested that we consider also the conjugated equation at the same time
$$\partial_t r=\partial_x^2 r+2|r|^2r-\epsilon R^*\qquad r=-q^*.\label{2}$$
It seems to me that whenever $q$ solves \eqref{1}, $r$ solves \eqref{2}.Then how does \eqref{2} help? What puzzles me more is that when we do the 1st order perturbation theory and collect coefficients according to $\epsilon^1$, it is suggested to write a coupled system of PDE involving $q,r$. Why is this better than just considering one equation, say \eqref{1} alone?

Afterwards we also consider the spectrum of that coupled linear operator, but is that why?

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##### MAT334--Lectures & Home Assignments / Bounding gamma function
« on: January 27, 2019, 08:50:05 PM »
I have almost forgotten how to due with

Any hint?

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##### APM346--Misc / analytic extension
« on: January 26, 2019, 10:29:57 AM »
Suppose I have a absolutely convergent series solution of an ODE on the real line and I want to extend this to the whole plane. Naïvely I will just write the same formula, replaced with a complex variable. But to what extent is this justifiable?

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##### MAT334--Lectures & Home Assignments / What can I say about f'(0)?
« on: November 21, 2018, 04:45:22 PM »
Suppose $f:D\to\mathbb{C}$ is analytic near 0, such that $\|x\|=1\implies |f(x)|=1$. Does it follow that $f'(0)$ is purely imaginary?

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##### APM346––Home Assignments / problem in problem
« on: March 30, 2018, 06:02:18 PM »
I assume problem 2.1 of http://www.math.toronto.edu/ivrii/PDE-textbook/Chapter10/S10.P.html intends to ask "Minimizing T" subject to no constrain?

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##### APM346--Lectures / consequence of EL
« on: March 30, 2018, 09:37:25 AM »
From (what I consider to be Euler-Lagrange)
$$\label{1}L_u=(L_{u'})_t$$
how can I derive
$$\label{2}L=u'L_{u'}+C?$$

Or is \eqref{2} even right? are they derived independently? if I integrate both sides of \eqref{1}, what exactly will be on the left?

7
##### Quiz-7 / Thursday's quiz
« on: March 29, 2018, 03:21:42 PM »
It was question 3.3 as of
http://www.math.toronto.edu/courses/apm346h1/20181/PDE-textbook/Chapter8/S8.P.html

Since $g$ is already harmonic, it's harmonic extension in unit ball has the same formula! Moreover $g(kx)=k^3g(x)$ so this formula is in fact a sum of homogeneous harmonic polynomial consisting of one term! So
$$u(x,y,z)=xyz,x^2+y^2+z^2\leq 1$$ or $$\tilde{u}(\rho,\theta,\phi)=\rho^3\cos^2\phi\sin\phi\cos\theta\sin\theta,0\leq\rho\leq1,0\leq\theta\leq 2\pi, 0\leq\phi\leq\pi.$$

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##### Quiz-7 / Wednesday's quiz
« on: March 29, 2018, 09:20:21 AM »
I was not there but I heard from my friend that they were asked to find harmonic extension in $B_1(0)$ of $g(x,y,z)=x^4+y^4+z^4$ given on $C_1(0)$.

Solution $u$ is sought in the form
$$\label{1}u=g-P(x,y,z)(\rho^2-1),\,\rho=\sqrt{x^2+y^2+z^2}.$$
Where $P$ is a polynomial even and symmetric in $x,y,z$, as does $g$, and $\deg(P)=\deg(g)-2=2$ . Therefore $P=P(\rho)=a\rho^2+b$ for some constant $a,b$, and so \eqref{1} becomes
$$\label{2}u=g-(a\rho^4+(b-a)\rho^2-b).$$
Observe $$\Delta\rho^2=6,\,\Delta\rho^4=20\rho^2,\,\Delta g=12\rho^2.$$ Now set $\Delta u=0$ and \eqref{2} gives
$$\label{3}0=12\rho^2-(20a\rho^2+6(b-a)).$$
Equating both sides of \eqref{3} term by term we find $a=b=\frac{3}{5}$ and so \eqref{2} becomes
$$u=g-\frac{3}{5}(\rho^4-1)=\frac{2}{5}(x^4+y^4+z^4)-\frac{6}{5}(x^2y^2+x^2z^2+y^2z^2)+\frac{3}{5}.$$

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##### APM346––Home Assignments / An ODE
« on: March 20, 2018, 08:24:05 AM »
I am referring to Q2.3 on
http://www.math.toronto.edu/courses/apm346h1/20181/PDE-textbook/Chapter8/S8.P.html

So how do I actually solve
$$\sin^2\phi \Phi''+\sin\phi\cos\phi \Phi' -(l(l+1)\sin^2\phi-m^2)\Phi=0$$
which, suppose it's correctly derived, has cost me an entire afternoon? I remember I certain remark
in lecture that there should be constrain on $m$ and some thing like $(x+iy)^m$, but that part of my note
is very much blurred.

I heuristically plugged in $\sin,\cos,\sin^2,\cos^2,\sin\cos$ but there does not seem to be a good cancellation.

10
##### APM346--Lectures / excersice 1, chap 7.3
« on: March 17, 2018, 05:34:56 PM »
http://www.math.toronto.edu/courses/apm346h1/20181/PDE-textbook/Chapter7/S7.3.html#sect-7.3.1

Exercise 1 appears quite strange. if $f=0$ on $\Omega=\{\|x\|\geq R\}$ and $u(y)=\int_\Omega G(x,y)f(x)\,dx$, then shouldn't this integral vanish instead of giving that curious form?

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##### APM346--Lectures / technical typos
« on: March 15, 2018, 04:40:02 PM »
1. in chap 7.3 between eq $(12),(13)$ the codes are there.
2. in chap 8.1 between eq $(15),(16)$ methinks the last term in the enumerator of the first selected fraction should have a factor $\rho^2$.

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##### APM346––Home Assignments / To drag
« on: March 13, 2018, 04:39:00 PM »
I am referring to the mean value theorem proof in
http://www.math.toronto.edu/courses/apm346h1/20181/PDE-textbook/Chapter7/S7.2.html

So twice we dragged the kernel $G$ out of the integral. How can we actually do that? if $\Delta u \gtrless$ can we still drag? can we in general drag?

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##### APM346--Lectures / potential
« on: March 11, 2018, 05:46:54 PM »
http://www.math.toronto.edu/courses/apm346h1/20181/PDE-textbook/Chapter7/S7.2.html

My question is concerning equations $(9),(10)$. First what is $u(y)$ in fact? is that some sort of charge? is "charge" and "potential" the same? are they sort of energy? (You must know the asker is barely exposed to Physics.) Secondly, if $u=u(y)$ then what is meant by $\Delta u$?

Also I think it will be good if the online text is open to edit. Very often I can find such trivial typos that do not worth of posting here, but are nonetheless quite obstentious.

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##### APM346––Home Assignments / What is this?
« on: March 08, 2018, 08:45:09 AM »
$$F^*$$The one that suffices $FF^*=F^*F=I$?

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##### APM346––Home Assignments / bounded assumption
« on: March 06, 2018, 08:16:19 PM »
Do we not need to know that the function remains bounded in order to solve the first parts in chap 6, question 3,4?

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