Exam
Department of Mathematics University of Utah\ 1.5 ] Written Qualifying Examination in APPLIED MATHEMATICS\ 0.5 ] September 9, 1996
Fall 1996 · 12 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question A.1
State and prove the Fredholm Alternative Theorem for linear operators \(L:\mathbb {R}^n\to \mathbb {R}^m\).
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Question A.2
Given a set of linearly independent vectors \(u_1,u_2,\ldots ,u_n\)
- Describe the Gram-Schmidt procedure to produce an orthogonal set of vectors from the original set.
- Suppose \(U\) is a matrix with linearly independent columns. Describe the \(QR\) factorization of \(U\) using Householder transformations.
- In what sense are these two procedures equivalent? Prove your answer.
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Question A.3
Let \(A\) be an \(m\times n\) matrix and let the \(n\times m\) matrix \(A'\) denote the Least Square Pseudo-Inverse of \(A\).
- State the conditions that define \(A'\).
- Show that \(A'\) always exists.
- Construct \(A'\) explicitly for the \(2\times 2\) matrix \[ A=\begin {pmatrix} 1 & 1\\ 0.5 & 0.5 \end {pmatrix}. \]
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Question A.4
Let \(Lu(x)=\int_0^x(t-x)u(t)\,dt\). **(a)** Find the resolvent operator associated with the integral equation \(u=f(x)+Lu\). **(b)** Solve the integral equation \(u=1+Lu\). **(c)** Show that the integral equation \(u=1+Lu\) is equivalent to a differential equation and give the equation.
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Question B.1
- Find the extended operator for the boundary value problem (BVP): \begin{equation} y''-y=f(x),\qquad ay(0)+y'(0)=0,\qquad y'(1)=a. \end{equation}
- Solve the differential equation in the sense of distributions: \begin{equation} xu'=\delta (4x^2-1),\qquad x\in [-1,1],\qquad x(-1)=1,\qquad x(1)=0. \end{equation} Discuss why the first order differential equation satisfies two boundary conditions.
- Question B.2
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Question B.3
Compute the Euler equation for the minimizer of the following variational problem. Check Legendre’s necessary condition. Discuss the dependence of the minimizer on the parameter \(a\). \begin{equation} \min _{w\in W}\int _0^1\left ((xw')^2+aw^2w'-x\ln w\right )\,dx, \qquad W=\{w:\ w\in H_2(0,1);\ w(0)=1\}. \end{equation}
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Question B.4
Compute the following integral. Use the Jordan lemma. \[ \int_0^\infty \frac{x\sin x}{x^2+1}\,dx. \] **Applied Math Qualifying Exam Part C** Do two out of the four problems for full credit on Part C.
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Question C.1
Suppose \(f(t)\) is a band limited function with Fourier Transform \(F(\mu )=0\) for \(|\mu |>\pi /h\). Define the sinc function \(S_0(t)\) by \begin{equation} S_0(t)=\frac {\sin (\pi t/h)}{\pi t/h}. \end{equation} Show that \begin{equation} f(t)=\sum _{k=-\infty }^{\infty } f(kh)S_0(t-kh). \end{equation}
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Question C.2
Suppose \(f(\theta )\) is a \(2\pi \) periodic function. Define its Hilbert Transform as follows: Let \(u(r,\theta )\) be the solution of Laplace’s equation on the unit circle satisfying Neumann data \(u_r(1,\theta )=f(\theta )\). The Hilbert transform of \(f\) is \(H(f)=u_\theta (1,\theta )\).
Use separation of variables to find the Hilbert Transform of \(f\) in terms of its Fourier coefficients.
- Question C.3
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Question C.4
A uniform spherical potato at temperature \(T_0\) is placed in the center of a large oven that is kept at a uniform temperature \(T_1\). Estimate how long it takes for the temperature at the center of the potato to reach \(\frac{1}{2}(T_0+T_1)\). Hint: The Laplacian in three dimensions with spherical symmetry is \[ \nabla^2u=\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{du}{dr}\right). \]