PhD Preliminary Qualifying Examination Applied Mathematics Tuesday, August 16, 2016

Consider the operator \(T:\ell ^2\to \ell ^2\) defined for \(x=(\xi _j)\in \ell ^2\) by \[ Tx=(0,\xi _1\alpha _1,\xi _2\alpha _2,\ldots ), \] for some sequence \((\alpha _j)\) with \(\alpha _j\to 0\). Show that \(T\) is compact.

Let \(X\) be a Banach space and consider the non-linear mapping \(F:X\to X\) defined by \[ F(x)=y-\alpha \lVert x\rVert x, \] where \(y\in X\) is fixed and \(\alpha \) is a scalar. Show that there is a constant \(C>0\) such that for any \(\lvert \alpha \rvert <C\), \(F\) is a contraction on the open ball of radius \(1\) centered at \(y\).

Let \(M\) be a subset of a Hilbert space \(H\). Assume \(M\) is such that for any \(v,w\in H\) for which the equality \[ \langle v,x\rangle =\langle w,x\rangle \] holds for all \(x\in M\), we must have \(v=w\). Show that \(M^\perp =\{0\}\).

Let \(X\) be a real Banach space. Assume \(f\in X^*\) has a closed nullspace \(\mathcal {N}(f)\). The goal of this problem is to show that \(f\) must be a bounded linear functional.

1. Let \(x_0\in X\) be such that \(f(x_0)=1\). Explain why there is an \(\epsilon >0\) for which the ball \[ B(x_0,\epsilon )\equiv \{x\in X\mid \lVert x-x_0\rVert <\epsilon \} \] satisfies \(B(x_0,\epsilon )\subset X-\mathcal {N}(f)\).
2. Prove that \(f(x)>0\) for all \(x\in B(x_0,\epsilon )\), where \(\epsilon \) is as in part (a). To prove this, you may assume for contradiction that there is some \(y\in B(x_0,\epsilon )\) with \(f(y)<0\).
3. Any \(x\in B(x_0,\epsilon )\) can be written as \(x=x_0+\epsilon u\) for some \(u\) with \(\lVert u\rVert <1\). Use the result of part (b) to show that \(\lvert f(u)\rvert <1/\epsilon \).
4. To conclude, give an upper bound for \(\lVert f\rVert \).

Let \(T:X\to X\) be a bounded linear operator on a complex Banach space \(X\). Prove that \(\sigma (T)\) lies in the complex plane disk: \[ \{\lambda \in \mathbb {C}\mid \lvert \lambda \rvert \le \lVert T\rVert \}. \]

The following function \(f(x)\) [\(x\) is a real variable] can be represented by a series (Taylor or Laurent) in powers of \((x-7)\). Find the radius of convergence of the series in each case \[ \begin {array}{rcl} \text {(a)}\quad f(x)&=&e^{(x-7)^{10}},\\[0.6ex] \text {(b)}\quad f(x)&=&\left (\dfrac {\sin x-3}{\sin x-2}\right )^2,\\[1.4ex] \text {(c)}\quad f(x)&=&\dfrac {\sin x-2}{x^2-49}. \end {array} \] [You do not need to find the series themselves.]

1. Prove: If a function is analytic then its real and imaginary parts are harmonic.
2. Prove: If a function \(u(x,y)\) is harmonic in a domain \(D\), then \(u(x,y)\) cannot attain a strict local maximum in \(D\). [Hint: If \(f(z)=u+iv\), then function \(\phi (z)=e^{f(z)}\) has absolute value \(\lvert \phi (z)\rvert =e^u\).]

Integrate

1. \[ \int _0^\infty \frac {\sin \alpha x}{x}\,dx, \]
2. \[ \int _0^\infty \frac {x^\alpha }{1+x}\,dx, \]
3. \[ \int _0^\infty \sin x^2\,dx \]

[Explain the logic (in particular, the choice of contour), but do not worry about getting the exact numbers; \(\alpha \) is a real parameter.]

1. Consider a 2-dimensional map \((x,y)\to (u,v)\). Explain: If the map is given by an analytic function [\(u+iv=f(x+iy)\)] and at some point \(z_0=x_0+iy_0\) the derivative \(f'(z_0)\ne 0\), then this map preserves small shapes in the vicinity of \(z_0\).
2. Is a square conformally equivalent to a circle?
3. Explain: A map by analytic function preserves Laplace’s equation.

Solve Laplace’s equation \[ u_{xx}+u_{yy}=0 \quad \text {in the domain}\quad D=\{(x,y)\in \mathbb {R}^2:\ (x-1)^2+y^2>1\ \& \ (x-2)^2+y^2<4\} \] subject to the boundary condition \[ u(x,y)=a\text { when }(x-1)^2+y^2=1 \qquad \text {and}\qquad u(x,y)=b\text { when }(x-2)^2+y^2=4 \] [\(a\) and \(b\) are real parameters; the two circles touch each other; express your answer in terms of the original real variables \(x,y\)].