PhD Preliminary Qualifying Examination: Applied Mathematics (6710/20) August 12th, 2002

Consider the error function \[ E(\mathbf {w})=\sum _{p=1}^{P}(y_p-\mathbf {w}\cdot \mathbf {x}_p)^2 \] where \(\mathbf {w}\) and \(\mathbf {x}_p\) are \(N\)-component vectors, \(N\ge P\), and \(y_p,\mathbf {x}_p\) are prescribed.

1. Show that if \(\mathbf {w}_0\) is a minimizer of \(E\) then \[ [X^TX]\mathbf {w}_0=X^T\mathbf {y} \] where \(X_{pj}=[\mathbf {x}_p]_j\), \(j=1,\ldots ,N\), \(X^T\) is the transpose of the matrix \(X\), and \(\mathbf {y}=(y_1,\ldots ,y_P)^T\).
2. Suppose that the weight vector \(\mathbf {w}\) is updated at each time step \(m\) using the gradient decent algorithm \[ w_j(m+1)=w_j(m)-\eta \frac {\partial E(\mathbf {w}(m))}{\partial w_j(m)}. \] Show that \(\mathbf {w}(m)\to \mathbf {w}_0\) as \(m\to \infty \) provided that the rate parameter \(\eta \) satisfies \[ |1-\eta \lambda _j|<1,\qquad j=1,\ldots ,N \] where \(\lambda _j\) are the eigenvalues of the symmetric \(N\times N\) matrix \(X^TX\). [Hint: Explicitly calculate \(\partial E/\partial w_j\), use part (a) to rewrite the difference equation in terms of \(\Delta \mathbf {w}(m)=\mathbf {w}(m)-\mathbf {w}_0\), and then diagonalize. Assume that \(X^TX\) is invertible so that \(\mathbf {w}_0\) is unique].

Consider a linear chain of \(2N\) atoms with the same spacing \(a\) and force constant \(\beta \) but of two different masses \(m,M\), \(M>m\), placed alternately. Label the light atoms by even integers \(2n\), \(n=0,\ldots ,N-1\) and the heavy atoms by odd integers \(2n-1\), \(n=1,\ldots ,N\). Denoting their displacements from equilibrium by the variables \(U_{2n}\) and \(V_{2n-1}\), Newton’s law of motion gives \[ \begin {aligned} m\ddot {U}_{2n} &= \beta [V_{2n-1}+V_{2n+1}-2U_{2n}],\\ M\ddot {V}_{2n-1} &= \beta [U_{2n}+U_{2n-2}-2V_{2n-1}]. \end {aligned} \] Determine the modes of vibration of the system and sketch the resulting pair of dispersion curves. [Assume periodic boundary conditions \(U_0=U_{2N}\) and \(V_1=V_{2N+1}\).]

Find the eigenvalues and eigenfunctions of the integral operator \[ (Ku)(x)=\int _0^1 k(x,y)u(y)\,dy \] where \[ k(x,y)= \begin {cases} x(1-y), & 0\le x<y\le 1,\\ y(1-x), & 0\le y<x\le 1. \end {cases} \] Solve the corresponding integral equation \[ u(x)-\lambda \int _0^1 k(x,y)u(y)\,dy=f(x) \] using the eigenfunctions of \(K\) assuming that \(\lambda \ne n^2\pi ^2\) for any integer \(n\) (i.e. determine the resolvent operator). Write down the Fredholm alternative condition for \(f\) when \(\lambda =m^2\pi ^2\) for some integer \(m\).

Suppose that the real scaling function \(\phi \) satisfies the dilation equation \[ \phi (x)=\sum _k c_k\phi (2x-k). \] Also assume that \(\{\phi (x-k),\, k\in \mathbb {Z}\}\) forms an orthonormal set with respect to the inner product \(\langle u,v\rangle =\int _{-\infty }^{\infty }u(x)v(x)\,dx\). Define the corresponding wavelet function according to \[ \psi (x)=\sum _k(-1)^k c_{1-k}\phi (2x-k). \]

1. Prove that \[ \sum _{k\in \mathbb {Z}} c_kc_{k-2p}=2\delta _{0,p}. \]
2. Show that \(\{2^{j/2}\psi (2^jx-k),\, k\in \mathbb {Z}\}\) forms an orthonormal set.
3. Briefly explain signal decomposition and reconstruction within the framework of multiresolution analysis.

Consider the differential operator \(Lu=u''+k^2u\), \(k\ne 0\), subject to homogeneous boundary conditions \(u(0)=u(1)=0\). Show by direct calculation that the associated Green’s function is \[ g(x,y)= \begin {cases} \dfrac {\sin kx\,\sin k(y-1)}{k\sin k}, & 0\le x<y\le 1,\\[1.2ex] \dfrac {\sin ky\,\sin k(x-1)}{k\sin k}, & 0\le y<x\le 1, \end {cases} \] provided \(\sin k\ne 0\). Using the fact that \(L\) is self-adjoint, show that the solution to the equation \[ u''(x)+k^2u(x)=f(x) \] with inhomogeneous boundary data \(u(0)=\alpha \), \(u(1)=\beta \) is \[ u(x)=\int _0^1 g(x,y)f(y)\,dy-\frac {\alpha }{\sin k}\sin k(x-1) +\frac {\beta }{\sin k}\sin kx \] provided \(\sin k\ne 0\). What happens when \(k=m\pi \)?

1. Using Jordan’s lemma show that \[ \int _0^\infty \frac {\cos ax}{1+x^2}\,dx=\frac {\pi }{2}e^{-a}. \]
2. Show that \[ \int _0^{2\pi }\frac {d\theta }{A+B\sin \theta }= \frac {2\pi }{\sqrt {A^2-B^2}} \] where \(A^2>B^2\), \(A>0\).

Use contour integration to show that \[ \int _0^\infty \frac {x^\alpha }{(1+x)^2}\,dx=\frac {\pi \alpha }{\sin \pi \alpha } \] where \(|\operatorname {Re}\alpha |<1\).

Discuss the flow pattern around a circular obstacle associated with the complex potential \[ \Omega (z)=u_0\left (z+\frac {a^2}{z}\right )+\frac {i\gamma }{2\pi }\log z. \] Show that \(r=a\) is a streamline. Determine the asymptotic velocity when \(z\to \infty \). Calculate the stagnation points. Sketch the flow when \(\gamma =0\). Use the Blasius formula \[ F_x-iF_y=\frac {1}{2}i\rho \int _C\left (\frac {d\Omega }{dz}\right )^2\,dz \] to determine the lift on the obstacle when \(\gamma \ne 0\).

Use Fourier transforms to show that the solution of the diffusion equation \[ \frac {\partial \phi }{\partial t}=\frac {\partial ^2\phi }{\partial x^2},\qquad -\infty <x<\infty \] with initial data \(\phi (x,0)=h(x)\) takes the form \[ \phi (x,t)=\int _{-\infty }^{\infty } g(x-x',t)h(x')\,dx' \] where \[ g(x,t)=\frac {e^{-x^2/4Dt}}{2\sqrt {\pi Dt}}. \] Determine \(\lim _{t\to 0}g(x,t)\).

Show that the conformal transformation \[ w=\frac {z-a}{z+a},\qquad a=\sqrt {c^2-\rho ^2} \] where \(0<\rho <c\), maps the domain bounded by the circle \(|z-c|=\rho \) and the imaginary axis onto an annular region centered about the origin with outer radius \(|w|=1\) and inner radius \(|w|=\delta \). Find the radius \(\delta \).