## Math 642-600 Assignments — Spring 2015

Assignment 1 - Due Wednesday, 1/28/2015.

• Do the following problems.
1. Problem 2, page 204 (§ 5.1).
2. Problem 7, page 204 (§ 5.1).
3. Hanging chain problem. A chain having uniform linear density ρ hangs between the points (0,0) and (1,0). The total mass m, which is fixed, and the total energy E of the chain are

Assuming that the chain hangs in a shape that minimizes the energy, find the shape of the hanging chain.
4. Consider the functional J(y) = ∫abF(x,y,y′)dx, where y ∈ C1 and y(a)=A, y(b)=B are fixed.
1. Let y extremize J, subject to all of the conditions above. Verify that if η ∈ C1 and η(a)=η(b)=0, then
Δ J = ∫ab (-g(x) + Fy′(x,y,y′))η′(x)dx, where g(x) = ∫axFy(u,y,y′)du.
2. Use (a) to show that -g+Fy′(x,y,y′)=c, a constant. From this, it follows that Fy′(x,y,y′)∈ C1. (Du Bois-Reymond, 1879. The point is that one need not assume that y′′ exists.)
5. Smoothing spline. Let J[y] = ∫01 (y′(x))2dx. Let the admissible set for J be all piecewise C1 curves on [0,1] that satisfy y(k/n) = yk , for k = 0, 1, 2, ..., n, with the discontinuities in y′ appearing only at the points xk = k/n. Use the previous problem to show that the minimizer for J is a linear spline passing through all of the points {(xk,yk), k = 0, ..., n}. (Hint: y(x) is in C1[xk, xk+1]).)
6. Let x = (x,y) and let J[x] = x2y(x2 + y2)− 1 if x0, J[0] = 0. Show that the Gâteaux derivative exists at at x = 0 for every direction η = (h,k). Also, show that the Fréchet derivative doesn't exist at x = 0.

Assignment 2 - Due Wednesday, 2/11/2015.

• Do the following problems.
1. Problem 2, page 205 (§ 5.2).
2. Problem 6, page 205 (§ 5.2).
3. Problem 7, page 206 (§ 5.2). (See Fig. 5.4, p. 205 for a diagram.)
4. Problem 8, page 206 (§ 5.2). Also, find the Hamiltonian of the system, and write out Hamilton's equations for it. Finding integrals (constants) of the motion should follow immediately from these equations. (Note: There is an error in the expression for L. The (dθ/dt)2 sin2θ) should be replaced by (dφ/dt)2 sin2θ.)
5. Consider the torus (surface of a "doughnut") obtained by rotating a circle with center $(R,0,0)$ and radius $r < R$ about the $z$ axis. Find the metric tensor for it and use this tensor to obtain the 2nd order differential equations for the geodesics.
6. Find the Legendre transformation H(p) for F(x) = xT A x, where x is in Rn and A is a symmetric, positive definite n×n matrix.

Assignment 3 - Due Wednesday, 2/18/2015.

• Do the following problems.
1. Problem 1, page 207 (§ 5.4). (Hint: Use Minimax Principle.)
2. Problem 4, page 208 (§ 5.4).
3. Problem 6, page 208 (§ 5.4). Take Ω to be a 2D disk centered at 0 and having radius r = a. How does the lowest eigenvalue change with a?
4. Use a quadratic polynomial to estimate the lowest eigenvalue for the Sturm-Liouville problem with Lu = − u′′ = λu, u(0) = 0, u′(1)+u(1) = 0.
5. In class we showed that the Hamiltonian $H=\frac{1}{2m} (p_r^2+r^{-2}p_\theta^2) - Kr^{-1}$, for a planet (mass $m$) in orbit around the sun (mass=$M$), where the constant $K=GmM$. In addition, we also found that $H = E$ and $p_\theta = \ell$ are constants of the motion.
1. Show that $p_r = \ell r^{-2} \frac{dr}{d\theta} = - \ell \frac{d}{d\theta}(r^{-1})$.
2. Let $u = r^{-1}- mK\ell^{-2}$. Show that $2m E\ell^{-2} + m^2K^2\ell^{-4} = \big(\frac{du}{d\theta}\big)^2 + u^2$. Solve this first order differential equation for u in terms of $\theta$.
3. Use your solution to the previous part to establish Kepler's laws of planetary motion:
1. The orbit of a planet is an ellipse, with the sun at one focus.
2. The radius vector from the sun to the planet sweeps out equal areas in equal times.
3. The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit. The proportionality constant is the same for all planets.

Assignment 4 - Due Friday, 2/27/2015.

• Do the following problems.
1. Section 6.2: 12, 13 (Assume $\alpha \ge 2$.)
2. Section 6.4: 2, 4, 9, 25
3. Let $\xi \ge 0$. Find $\hat f(\xi) := \frac{1}{2\pi}\int_{-\infty}^\infty \frac{e^{i\xi x}}{1+x^4}dx$.

Assignment 5 - Due Monday, 3/9/2015.

• Read sections 7.1 and 7.2.
• Do the following problems.
1. Problem 26, pg. 278 (§ 6.4). In this problem, assume that $|z f(z)| \le e^{k|z|}$ for all $z\in \mathbb C$ and, for all real $x$, $|x f(x)| \le M$, for some constant $M>0$.
2. Problem 2, page 279 (§ 6.5).
3. Problem 5(a), page 279 (§ 6.5).
4. Problem 9, page 280 (§ 6.5). (Do $J_{3/2}(z)$.)
5. Problem 10, page 280 (§ 6.5).

Assignment 6 - Due Monday, 4/6/2015.

• Do the following problems.
1. Section 7.1: 2(a), 2(b). For each operator in 2(a), 2(b), find the norm and adjoint $L^\ast$. You may use any of the problems below.

2. Let $L$ be a bounded linear operator on $\mathcal H$; that is, $L \in {\mathcal B}(\mathcal H)$. Show that $\rho(L)$ contains the exterior of the disk $\{\lambda \in {\mathbb C} \colon |\lambda| \le ||L||\}$. (Consequently, $\sigma(L)$ is contained in the disk.)

3. Let $K \in {\mathcal B}(\mathcal H)$ and let $L:D \to \mathcal H$ be a densely defined, closed linear operator.
1. Show that $L+K$ is defined on $D$ and is also closed.
2. Show that $(L+K)^\ast$ is defined on the domain of $L^\ast$ and that $(L+K)^\ast = L^\ast + K^\ast$.
3. Suppose that $0 \in \rho(L)$. Show that $0 \in \rho(L^\ast)$ and that $(L^{-1})^\ast = (L^\ast)^{-1}$.

4. Let $L = L^\ast$ be defined on a domain $D$. In class, we showed that $\sigma_r(L) = \emptyset$ and that $\sigma_d(L) \subset \mathbb R$. Show that $\sigma_c(L) \subset \mathbb R$. Thus, $\sigma(L)$ is a subset of the reals.

5. Let $L = L^\ast$ be defined on a domain $D$ and suppose that $L$ satisfies $\langle Lf,f \rangle \ge 0 \$, for all $f\in D$. Show that $\sigma(L) \subseteq [0, \infty)$.

Assignment 7 - Due Monday, 4/20/2015.

• Do the following problems.
1. Let $A\in \mathbb C^{n\times n}$, $A=A^\ast$. Let $C_\lambda$ be a simple closed curve that encircles the eigenvalues $\lambda_j < \lambda_{j+1} < \cdots <\lambda_k$, along with $\lambda$, where $\lambda_k < \lambda <\lambda_{k+1}$ or $\lambda > \lambda_n$.
1. Show that $E_\lambda = -\frac{1}{2\pi}\oint_{C_\lambda} R_z(A)dz = \sum_{\ell = j}^k E_j$, where $E_j$ is the orthogonal projection onto the eigenspace of $\lambda_j$.
2. If $m \in \mathbb Z$, $m \ge 1$, $j=1$ and $\lambda > \lambda_n$, show that $A^m = -\frac{1}{2\pi}\oint_{C_\lambda} z^m R_z(A)dz$. (Hint: we did the $m = 1$ case in class on 31 Mar. Use this case and induction to obtain the result.)
2. Adopt the notation and assumptions of the previous problem. Define $E_{\lambda^\pm} = \lim_{\varepsilon \downarrow 0} E_{\lambda \pm \varepsilon}\,$.
1. Show that, for $1 \le k\le n$, $E_{\lambda_k^+} = \sum_{j=1}^k E_j$ and $E_{\lambda_k^-}= \sum_{j=1}^{k-1} E_j$. (Note: $\sum_{j=1}^{0} E_j = 0$.)
2. Normalize $E_\lambda$ to be right continuous, so $E_{\lambda^+} = E_\lambda$. Suppose that the $\lambda_j$'s all have multiplicity 1 and $u_j$, $\|u_j\| = 1,$ is the eigenvector corresponding to $\lambda_j$. Show that if $\lambda \ge \lambda_k$, $k=1,\ldots, n$, then $E_\lambda v = \sum_{j=1}^k \langle v,u_j\rangle u_j$. Also, use your knowledge of matrices to show that when $\lambda \ge \lambda_n$, $E_\lambda = I$ and so $v = \sum_{j=1}^n \langle v,u_j\rangle u_j$.
Remark: Let $\hat v_j := \langle v,u_j\rangle$. Finding $\hat v_j$ is the analysis step in processing $v$. The synthesis step is then recovering $v$ via $v = \sum_{j=1}^n \hat v_j u_j$. As part of processing $v$, the $\hat v_j$'s may be altered — to remove noise, for example.
3. If $E_\lambda$ is right continuous, show that $\Delta E_\lambda := E_\lambda - E_{\lambda^-} = \begin{cases} E_k & \lambda = \lambda_k,\\ 0 & \lambda \ne \lambda \end{cases}.$

Assignment 8 - Due Wednesday, 4/29/2015.

• Do the following problems.
1. Prove the following theorems:
1. If $f,g \in L^1(\mathbb R) \cap L^2(\mathbb R)$, then $f\ast g\in L^1(\mathbb R) \cap L^\infty(\mathbb R)$.

2. If $f\in L^1(\mathbb R)$, then its Fourier transform $F$ is in $C_0(\mathbb R)$.

3. If $f,F,g,G$ are all in $L^1(\mathbb R) \cap L^2(\mathbb R)$, then, $\int_{-\infty}^\infty f(t)G(t)dt = \int_{-\infty}^\infty F(t)g(t)dt$.

2. Let $L=-d^2/dx^2$, $D(L) =\{f\in L^2[0,\infty) \colon Lf \in L^2[0,\infty) \text{and} \ f'(0)=0\}$. Use Stone's formula for the family of projections to obtain the Fourier cosine transform.

3. Consider the one dimensional heat equation, $u_t = u_{xx}$, with $u(x,0) = f(x)$, where $-\infty < x < \infty$ and $0 < t < \infty$. By taking the Fourier transform in $x$, show that the solution $u(x,t)$ is given by $u(x,t) = \int_{\mathbb R}K(x-y,t)f(y)dy,\ K(\xi,t) = (4\pi t)^{-1/2} e^{-\xi^2/4t}.$ (The function $K(\xi,t)$ is the one-dimensional heat kernel.)

4. Problems 7.2.6(a,c).

Assignment 9 - Not to be handed in.

• Suggested problems.

1. Let $f$ be in Schwartz space, $\mathcal S$, and let $g$ be $C^\infty$ and satisfy $|g^(m)(x)| \le c_m(1+t^2)^{n_m}$ for all nonnegative integers m. Here $c_m$ and $n_m$ depend on $g$ and $m$. Show that $fg$ is in Schwartz space. Use this result to explain how to define the product $g(x)T(x)$, where $T$ is a tempered distribution.

2. Let T be a tempered distribution. Find the Fourier transforms for T′, xT(x), T(x−a), and eibxT(x).

3. (Problem 1, p. 7, Feldman.) Suppose that $f$ is locally Lebesgue integrable on $\mathbb R$ and that there is a polynomial $P(x) = \sum_{k=N_1}^{N_2} a_kx^k$, where $0\le N_1 < N_2 <\infty$, such that $|f(x)| \le P(x)$ a.e. in $\mathbb R$.
1. Show that there is a constant $C>0$ such that $|f(x)|(1+x^2) \le C(|x|^{N_1} +|x|^{N_2})$.
2. Show that for all $\phi\in \mathcal S$ we have $\big|\langle f, \phi\rangle \big| \le \pi C\big( \|\phi\|_{N_1,0} + \| \phi \|_{N_2+2,0} \big)$.

4. Problem 7, page 464 (§10.3)

5. Problem 9, page 465 (§10.3)

6. Problem 11, page 465 (§10.3)

Updated 5/7/2015 (fjn)