Math 642-600 Assignments

Math 642-600 Assignments — Spring 2017

Assignment 1 - Due Wednesday, 1/25/2017.

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) = ∫_a^bF(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 = ∫_a^b (-g(x) + F_y′(x,y,y′))η′(x)dx, where g(x) = ∫_a^xF_y(u,y,y′)du.
  2. Use (a) to show that -g+F_y′(x,y,y′)=c, a constant. From this, it follows that F_y′(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) = y_k, for k = 0, 1, 2, ..., n, with the discontinuities in y′ appearing only at the points x_k = k/n. Use the previous problem to show that the minimizer for J is a linear spline passing through all of the points {(x_k,y_k), k = 0, ..., n}. (Hint: y(x) is in C¹[x_k, x_k+1]).)

Assignment 2 - Due Wednesday, 2/1/2017.

Do the following problems.
1. Problem 5, page 204 (§ 5.1).
2. Problem 9, page 204 (§ 5.1).
3. Problem 12, page 204 (§ 5.1).
4. 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.
5. Problem 6, page 205 (§ 5.2).
6. Find the Legendre transformation H(p) for F(x) = x^T A x, where x is in Rⁿ and A is a symmetric, positive definite n×n matrix.

Assignment 3 - Due Wednesday, 2/8/2016.

Do the following problems.
1. Problem 7, page 206 (§ 5.2). (See Fig. 5.4, p. 205 for a diagram.)
2. Problem 8, page 206 (§ 5.2). (Note: There is an error in the expression for L. The (dθ/dt)² sin²θ) should be replaced by (dφ/dt)² sin²θ.)
3. The Hamiltonian of a system having a Lagrangian $L(q_1,q_2,\ldots,q_n, \dot q_1,\dot q_2,\ldots,\dot q_n)$ is defined as the Legendre transform of $L$ with respect to the $\dot q_j$'s. (See pg. 187). In class we showed that, for a planet (mass $m$) in orbit around the sun (mass=$M$), where the constant $K=GmM$, the polar form of the Lagrangian is $ L=\frac{m}{2}\big(\dot r^2 + r^2 \dot \theta^2\big)+ Kr^{-1}$.
  1. Show that the Hamiltonian $H=\frac{1}{2m} (p_r^2+r^{-2}p_\theta^2) - Kr^{-1}$.
  2. Show that the angular momentum $p_\theta=\ell$ is a constant of the motion.
  3. Show that the Hamiltonian is a constant of the motion, $H=E$, where $E$ is the total energy of the system.
  4. Show that $p_r = \ell r^{-2} \frac{dr}{d\theta} = - \ell \frac{d}{d\theta}(r^{-1})$.
  5. 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 $\theta$.
  6. 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, 2/17/2017.

Read sections 6.1 and 6.2.
Do the following problems.
1. Problem 1, page 205 (§ 5.2).
2. Problem 3, page 205 (§ 5.2).
3. Problem 4, page 205 (§ 5.2). (Attaching a mass at the midpoint $x_c$ amounts to making changing the density from $\rho$ to $\rho+ m\delta(x-x_c)$.)
4. Problem 1, page 207 (§ 5.4). (Hint: Use Minimax Principle.)
5. 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.
6. 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? Explain.

Assignment 5 - Due Monday, 2/27/2017.

Read sections 6.2 and 6.4.
Do the following problems.
1. Do problems 4, 5 and 11 in the Exercises for Complex Variables
2. Find all of the Laurent expansions about $z=0$ for $f(z) = \frac{1}{(z^2-1)(z+2)}$
3. Section 6.1: 4, 6
4. Section 6.2: 6, 9, 10
5. Section 6.4: 2, 4, 9
6. 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 6 - Due Wednesday, 3/8/2017.

Read section 6.5 and 7.1.
Do the following problems.
1. Section 6.4: 3, 7, 20, 25 (You may use the version of the residue theorem proved in class on 2/27).
2. Section 6.5: 2
3. Suppose that $f(z)$ is analytic in and on a simple closed curve $C$, except for isolated singularities at $z_1,z_2,\ldots,z_n$ and one simple pole at a point $z_0$ on the boundary. If $z_0$ is on a corner of the curve where the angle between the new and old directions is $\alpha$, show that \[ \oint_Cf(z)dz= 2\pi i\sum_{j=1}^n Res_{z_j}(f) + i\alpha \,Res_{z_0}(f) \]

Assignment 7 - Due Monday, 4/10/2017.

Read section 7.2.1.
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 $\sigma_c(L) \subseteq \mathbb R$. 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 8 - Due Monday, 4/24/2017.

Read section 7.2.1.
Do the following problems.
1. Let $L$ be a closed, densely defined operator on a Hilbert space $\mathcal H$ and let $R_\lambda(L)$ be the resolvent operator. Prove this: Let $D_L$ be the domain of $L$. Then, on $D_L$, $LR_\lambda = R_\lambda L$.
2. Suppose that $L\in \mathcal B(\mathcal H)$ and that $C$ is a simple closed rectifiable curve in that encloses $\sigma(L)$. Let $P_C := -\frac{1}{2\pi i}\oint_C (L-\lambda I)^{-1}d \lambda= -\frac{1}{2\pi i}\oint_C R_\lambda(L) d\lambda$.
  1. Show that $P_c=I$.
  2. Show that for every polynomial $p(\lambda)$ one has \[ p(L) = -\frac{1}{2\pi i}\oint_C p(\lambda)R_\lambda(L) d\lambda \]
3. Let $L=L^\ast$ and let $C$ be a positively oriented, simple closed curve in the resolvent set, encircling $\sigma_0\subset \sigma(L)\in \mathbb R$ — but no other points in $\sigma(L)$. (Note: the analyticity of $R_\lambda(L)$ implies that $C$ may be deformed into any contour with the same properties.) Let $P_C := -\frac{1}{2\pi i}\oint_C R_\lambda(L) d\lambda$.
  1. Consider the curve $C'$ that is the complex conjugate of $C$. Show that $C'$ contains $\sigma_0$, that it is negatively oriented, and that $P_{-C'} =-\frac{1}{2\pi i}\oint_{-C'} R_\lambda d\lambda=P_C$.
  2. Use part (a) to show that $P_C^\ast = P_C$. (Since we know $P_C$ is a projection, this implies that it is an orthogonal projection.
4. Let $L=A$ be a self-adjoint, $n\times n$ matrix. Suppose that $A$ has $r$ distinct eigenvalues, arranged in the order $\lambda_1 < \lambda_2 < \cdots < \lambda_r$. In addition, let the orthogonal projections onto the eigenspaces be $Q_1,\ldots,Q_r$. Suppose $a < b$, and neither is an eigenvalue of $A$; let $C_\varepsilon$ be a positively oriented rectangular contour, with corners $a-i\varepsilon$, $b-i\varepsilon$, $b+i\varepsilon$, and $a+i\varepsilon$. Show that \[ \lim_{\varepsilon\downarrow 0}\,-\frac{1}{2\pi i} \oint_{C_\varepsilon} (A-\lambda I)^{-1}d\lambda = E_b-E_a, \] where $E_\lambda := \sum_{j=1}^r H(\lambda - \lambda_j)Q_j$ and $H(t)=\begin{cases} 1 & t\ge 0\\0 & t<0\end{cases}$.
5. Let $\mathcal H$ be a Hilbert space and suppose that $E$ and $F$ are orthogonal projections satisfying $E\ge F$ — i.e., $\langle Ef,f\rangle \ge \langle Ff,f\rangle$. Show that $EF=FE = F$. (Hint: Note that $\langle (E-F)f,f\rangle \ge 0$. Use $f=(I-E)g$, where $g\in \mathcal H$.)
6. Find the Green's function for the operator $Lu = -u''$, with $D_L=\{u,u''\in L^2[0,\infty): u'(0)=0\}$.

Assignment 9 - Not to be handed in.

Suggested problems (Fourier transform convention: $\hat f(\xi) = \int_{-\infty}^\infty f(x)e^{-i\xi x}dx$.)
1. Consider the Green's function that you found in problem 6, Assignment 8, for the operator $Lu- -u''$, $D_L=\{u,u''\in L^2[0,\infty)\colon u'(0)=0\}$. Use it and Stone's formula for the spectral family of $L$ to derive the Fourier cosine transform pair: \[ F(\xi)= \frac{2}{\pi}\int_0^\infty f(x)\cos(\xi x)dx \ \text{and}\ f(x)= \int_{0}^\infty F(\xi)\cos(\xi x)d\xi \]
2. 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 $\hat f$ is in $C_0(\mathbb R)$.
3. Consider the one dimensional heat equation, $u_t = u_{xx}$, with $u(x,0) = f(x)$, where $-\infty < x < \infty$ and $0 < t < \infty$. You are given that ${\mathcal F}[e^{-x^2}] =\sqrt{\pi}e^{-\xi^2/4}$. 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. Let T be a tempered distribution. Find the Fourier transforms for T′, xT(x), T(x−a), and e^ibxT(x).
5. Let $f$ be in Schwartz space, $\mathcal S$, and let $g$ be $C^\infty$ and satisfy $|g^{(m)}(x)| \le c_m(1+x^2)^{n_m}$ 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.
6. Section 10.3: 1, 3, 14.

Updated 5/3/2017 (fjn)