PhD Preliminary Qualifying Examination: Applied Mathematics (6710/20) August 16th, 2004

The dilation equation is \[ \phi (x)=\sum _k c_k\phi (2x-k) \] and the associated wavelet is \[ W(x)=\sum _k (-1)^k c_{1-k}\phi (2x-k). \]

1. Show that the function \[ N_2(x)= \begin {cases} x, & 0<x<1,\\ 2-x, & 1\le x<2,\\ 0, & \text {elsewhere} \end {cases} \] satisfies the dilation equation and determine the values of \(c_k\)? Construct the associated wavelet.
2. Prove that if the functions \(\{\phi (x-k)\}_k\) form an orthonormal set then the functions \(\{2^{j/2}W(2^jx-k)\}_k\) form an orthonormal set for each fixed integer \(j\).

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 \[ k(x,y)=\min \{x,y\}. \] 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 \(\lambda \ne 1/a_n^2\) for any integer \(n\), where \(a_n=(2n+1)\pi /2\) (i.e. determine the resolvent operator). Write down the Fredholm alternative condition for \(f\) when \(\lambda =1/a_n^2\).

1. Prove that the eigenvalues of a self-adjoint linear operator acting in a Hilbert space \(H\) are real and that the eigenfunctions of distinct eigenvalues are orthogonal.
2. Prove that if \(L\) is a bounded linear operator in \(H\) then the adjoint \(L^*\) is also a bounded linear operator.

Construct the Green’s function for the equation \[ u''(x)+\alpha ^2u(x)=f(x),\qquad u(0)=u(1),\qquad u'(0)=u'(1) \] and express the solution in terms of the Green’s function. For what values of \(\alpha \) does the Green’s function fail to exist?

Show that \[ \int _0^\infty \frac {\cosh ax}{\cosh \pi x}\,dx=\frac {1}{2}\sec \left (\frac {a}{2}\right ), \qquad |a|<\pi \] Use a rectangular contour with corners at \(\pm R\) and \(\pm R+i\).

Obtain the inverse Laplace transform of the function \[ \widehat {F}(s)=s^{-a},\qquad 0<a<1 \] by evaluating the associated Bromwich contour integral with a branch cut along the negative real axis. Express the answer in terms of the gamma function \[ \Gamma (z)=\int _0^\infty t^{z-1}e^{-t}\,dt. \]

Consider the flow around a thin airfoil of length \(2a\) that is inclined at an angle \(\alpha \) to the \(x\)-axis. Assume that there is uniform flow at \(\infty \) with velocity \(U_x=U\), \(U_y=0\).

1. Show that the conformal map \[ z=\frac {ae^{-i\alpha }}{2}\left (\omega +\frac {1}{\omega }\right ) \] maps the unit circle in the \(\omega \)-plane to the airfoil in the \(z\)-plane.
2. Consider the complex potential \[ \Gamma (\omega )=\frac {aU}{2}\left (e^{-i\alpha }\omega +\frac {e^{i\alpha }}{\omega }\right ) -\frac {i\gamma }{2\pi }\log \omega . \] Calculate the corresponding potential \(\Omega (z)=\Gamma (\omega (z))\) where \(\omega (z)\) is the inverse of the conformal map in part (a). What does the term in \(\gamma \) represent?
3. Calculate \(\Omega '(z)\) and show that it satisfies the correct asymptotic flow. Determine what happens at the end points of the airfoil, and use this to calculate \(\gamma \).

Find the solution of Laplace’s equation on the circular domain \(0\le r\le R\), \(0\le \theta <2\pi \), with Dirichlet boundary data \(u(R,\theta )=f(\theta )\).

Liouville’s theorem states that a function analytic and bounded in the complex plane is necessarily a constant.

Prove this theorem. Using it prove the Fundamental theorem of algebra: Every polynomial \(P(z)\) which is not a constant has at least one complex root.