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Topics - Victor Ivrii

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871
Term Test 2 / TT2--Problem 5
« on: November 15, 2012, 08:26:34 PM »
  Let $Q = \{(x,y)\in {\mathbb{R}}^2: |x|<1, |y|<1\}.$ Draw the set $Q.$ We define data $g$ on the boundary of $Q$:

Find the solution $u$ of the Dirichlet problem on $Q$:
\begin{equation*}
\Delta u=0 \qquad \text{for  }  (x,y) \in Q
\end{equation*}
with the boundary conditions
\begin{equation*}
u = \left\{\begin{aligned}
&y &&\text{as  }x=1,\\[3pt]
-&y &&\text{as }x=-1,\\[3pt]
&x &&\text{as   }y=1,\\[3pt]
-&x &&\text{as   }y=-1.
\end{aligned}\right.
\end{equation*}

Post after 22:30

872
Term Test 2 / TT2--Problem 4
« on: November 15, 2012, 08:23:51 PM »
Find Fourier transform of the  function
\begin{equation*}
f(x)= \left\{\begin{aligned}
&1-|x| &&|x|<1\\
&0 &&|x|>1.
\end{aligned}\right.
\end{equation*}
and write this function $f(x)$ as a Fourier integral.

Post after 22:30

873
Term Test 2 / TT2--Problem 3
« on: November 15, 2012, 08:22:48 PM »
Let $\phi: \mathbb{R} \rightarrow \mathbb{R}$ satisfy the following conditions:
  • $\phi$ is continuous.
  • $\phi'$ is continuous.
  • $\phi(x) = 0$ for all $|x|>1$.
Consider the integral
$$
I_\lambda = \int_{-\infty}^{+\infty} \phi (x) \cos (\lambda x)\, dx.
$$
Prove that $|I_\lambda| \rightarrow 0$ when $\lambda \rightarrow \infty$.


Post after 22:30

874
Term Test 2 / TT2--Problem 2
« on: November 15, 2012, 08:20:35 PM »
  Consider the diffusion equation
\begin{equation*}
u_t -ku_{xx}=0 \qquad \text{for} \quad t>0,\ x \in (0,2\pi)
\end{equation*}
with the boundary conditions
\begin{equation*}
u_x(0,t)=u_x(2\pi,t)=0
\end{equation*}
and the initial condition
\begin{equation*}
u(x,0)=|\sin (x)|.
\end{equation*}
  • (a) Write the associated eigenvalue problem.
  • (b) Find all  eigenvalues and corresponding eigenfunctions.
  • (c) Show that the eigenfunctions associated to 2 different eigenvalues are orthogonal.
  • (d) Write the solution in the form of  a series expansion.
  • (e) Write a formula for  the coefficients of the series expansion.

Post after 22:30

875
Term Test 2 / TT2--Problem 1
« on: November 15, 2012, 08:14:38 PM »
Let $f:{\mathbb{R}}\rightarrow {\mathbb{R}}$ be a continuous but non-differentiable function which satisfies $f(x)=0$ for all $|x| > 1.$ Let $g:{\mathbb{R}}\rightarrow {\mathbb{R}}$ be a continuous function which satisfies satisfies $g(x)=0$ for all $|x| > 2.$ Suppose further that derivative $g'$ and second derivative $g''$ are both continuous. The convolution $f*g$ of these two functions is defined by the formula
$$
(f*g) (x) = \int f(x-y)g(y)\,dy.
$$
  • (a) Prove that the function $f*g(x) =0$ for $|x|>3$.
  • (b) Prove that the derivative of the function $f*g$ is continuous.

post after 22:30

876
Term Test 2 / TT2 -- posting solutions
« on: November 14, 2012, 10:58:58 AM »
I will post problems Thu, Oct 15 at 20:30--20:45 and you may post solutions at 22:30.

877
APM346 Misc / HA4-5
« on: November 09, 2012, 12:33:29 PM »
One of TAs told me that they got very few HA4. I am concerned about this.

Currently there are only 22 marks for HA4 on BB. Meanwhile marks for HA5 are being entered.

Please advise me through poll if you submitted HA4 and got the mark. Please vote even if everything is normal (either got your mark or have not submitted the paper). Thanks!

879
APM346 Misc / Lecture notes
« on: November 03, 2012, 12:19:22 AM »
I slightly improved format of the last one (Lecture 21)—simply I decided to spend a bit more time, put anchors at subsections (and made a table of content).

Now one can refer (externally or internally) not only to the Lecture itself but to a specific section: section 21.3; ability to refer to specific numbered equation was here already:  (21.19) due to MathJax putting anchors automatically.

This improvements are only for this lecture: lack of time may prevent me to do the same in the next lectures leave alone to go backwards, but for the next class (Fall 2013)  this feature will be implemented for Lectures and Home Assignments and Tests (as they will be included—surely after the test).


P.S. We can end up with the creation of Open Textbook (and probably Open Source Textbook) but it will take several years (faster if we get some help). By no means there will be a conventional textbook or anything for profit.

880
APM346 Announcements / TT1 graded
« on: November 02, 2012, 02:01:08 PM »
Term Test 1 has been marked and marks should appear at BlackBoard soon. You can get it from TAs today, November 2, Friday, at 5pm in Math Lounge (Bahen, 6-th floor, across hallway from Southern group of elevators).

As of 15:15 methinks all marks has been entered.

Some marks (43) for HA3 are on Blackboard as well. The rest of marks, plus marks for HA4 and probably HA5 should be there by Monday

881
Home Assignment 5 / About HA5
« on: November 01, 2012, 05:10:42 AM »
Some of subproblems are just tricks:

Example: Decompose into full F.s. $\cos (mx)$. Answer: $\cos (mx)$ as it is one of the basic functions (however it would not be so if we decompose into $\sin$-F.s., or F.s with respect to $\sin((n+\frac{1}{2}x)$, $n=0,1,\ldots$.


What is the sum F.s. on $(-\infty,\infty) was discussed in http://www.math.toronto.edu/courses/apm346h1/20129/L16.html

Again see below:

882
General Discussion / Merry Halloween
« on: October 30, 2012, 01:30:28 PM »




Two last slides from my talk in MSRI Oct 31, 2008 (it was the last talk on the conference)

883
Home Assignment 6 / Bonus Web Problem--4
« on: October 27, 2012, 06:21:01 AM »
Don't post until November 7, 21:30

Prove the following properties of convolution:

  • (a) $f*g=g*f$
  • (b) $(f*g)'=f'*g = f*g'$, where $'$ denotes the derivative in one variable
  • (c) $f*(g*h)=(f*g)*h$
  • (d) Let $x_+^\lambda := x^\lambda$ as $x>0$ and $0$ as $x<0$. Prove that for $f$ which fast decays as $x\to -\infty$ and $n=1,2,\ldots$
    \begin{equation*}
    \frac{x_+^{n-1}}{(n-1)!} * f =\underbrace{\int_{-\infty}^x \int_{-\infty}^{x_1} \ldots \int_{-\infty}^{x_{n-1}}}_{\text{$n$ integrals}} f(x_n)\, dx_n \cdots dx_1
    \end{equation*}

884
Home Assignment 4 / Web Bonus Problem--3
« on: October 26, 2012, 09:14:14 AM »
Oscillations of the beam  (with left end clamped and right end free) are described by an equation
\begin{equation*}
u_{tt} + K u_{xxxx}=0, \qquad 0<x<l
\end{equation*}
with $K>0$ and the boundary conditions
\begin{equation*}
u(0,t)=u_{x}(0,t)=u_{xx}(l,t)=u_{xxx}(l,t)=0.
\end{equation*}

  • (a) Find  equation describing frequencies and corresponding  eigenfunctions
    (You may assume that all eigenvalues are real and positive).
  • (b) Solve  this equation graphically.
  • (c) Prove  that eigenfunctions corresponding to different eigenvalues are orthogonal.
  • (d) Bonus  Prove that eigenvalues are simple, i.e. all eigenfunctions corresponding to the same eigenvalue are proportional.

Compare with eigenvalues of Problem 2 of HA2

885
APM346 Announcements / Homework submission (HA4, HA5, HA6, HA7,....)
« on: October 24, 2012, 02:40:08 AM »
Its deadline today, Wed Oct 24.

I will be in my office HU1008 from 3:00 to 3:30 (office hours).

I hope that one of TAs will be in Bahen Math Lounge (Bahen 6th floor across Southern block of elevators) from 5:00 to 5:30 (or longer). -- I also be there from 5:00 (or several minutes after) to 5:25 (sharp! I cannot stay there longer as I have other commitments).

 Also I have a stapler in my office but not in Bahen lounge (and don't even think to pester secretaries of the Department asking for one!!!)

Don't leave in mailboxes or under the doors!!!


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