UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Applied Mathematics January 4th, 2017.

If \((x_n)\) is a weakly convergent sequence in a normed space \(X\), say, \(x_n\to x_0\) weakly, show that there is a sequence \((y_n)\) of linear combinations of elements of \((x_n)\) which converges strongly to \(x_0\). Hint: First prove that \(x_0\in \overline {\operatorname {span}(x_n)}\).

Let \(f\) be a real-valued and twice continuously differentiable on an interval \([a,b]\). Let \(\hat {x}\) be a simple zero of \(f\), i.e. \(f'(\hat {x})\ne 0\). Use the Banach Fixed Point Theorem to show that Newton’s method, defined by \[ x_{n+1}=x_n-\frac {f(x_n)}{f'(x_n)} \] converges to \(\hat {x}\) for any \(x_0\) sufficiently close to \(\hat {x}\).

Consider the multiplication operator \(T:L^2[0,1]\to L^2[0,1]\) defined by \(Tx(t)=tx(t)\).

1. Show that \(T\) is a bounded, self-adjoint linear operator.
2. Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuous spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\), and resolvent \(\rho (T)\).
3. Show that the corresponding spectral family \((E_\lambda )_{\lambda \in \mathbb {R}}\) is given by \[ E_\lambda x= \begin {cases} 0, & \lambda <0,\\ v_\lambda , & 0\le \lambda \le 1,\\ x, & \lambda >1, \end {cases} \qquad \text {where}\qquad v_\lambda (t)= \begin {cases} x(t), & 0\le t\le \lambda ,\\ 0, & \lambda <t\le 1. \end {cases} \]

Let \(X,Y\) be Hilbert spaces and let \((e_n)\subset X\) be a total orthonormal sequence. Recall that an operator \(T:X\to Y\) is said to be Hilbert-Schmidt if the Hilbert-Schmidt norm \[ \lVert T\rVert _{\mathrm {HS}}:=\left (\sum _{n=1}^{\infty }\lVert Te_n\rVert ^2\right )^{1/2} \] is finite.

1. Show that the Hilbert-Schmidt norm is independent of the sequence \((e_n)\) chosen. Hint: Use Parseval’s identity.
2. Show that a Hilbert-Schmidt operator is compact.
3. For a Hilbert-Schmidt operator, \(T\), show that \(\lVert T\rVert \le \lVert T\rVert _{\mathrm {HS}}\), where \(\lVert \cdot \rVert \) denotes the operator norm.

Let \(T:H\to H\) be a compact, self-adjoint linear operator defined on a separable Hilbert space \(H\). Let \((v_n)\) be a complete orthonormal sequence of eigenvectors of \(T\) and let \((\lambda _n)\) be the corresponding eigenvalues.

1. Let \(\lambda \in \rho (T)\). Show that the resolvent of \(T\) at \(\lambda \), acting on some \(x\in H\) can be written as \[ R_\lambda x=\sum _{n=1}^{\infty }\frac {1}{\lambda _n-\lambda }\langle x,v_n\rangle v_n. \]
2. Use part (a) to show that if \(\lambda ,\mu \in \rho (T)\) then the first resolvent identity holds: \[ R_\lambda -R_\mu =(\lambda -\mu )R_\lambda R_\mu . \]

1. Find all values of \(i^i\).
2. Find all solutions of the equation \(\cos z=2\).
3. Find all solutions of the equation \(\tan z=i\).

The following function \(f(x)\) [\(x\) is a real variable] can be represented by a series (Taylor or Laurent) in powers of \(x\). Find the radius of convergence of the series in each case \[ \begin {array}{rcl} \text {(a)}\quad f(x)&=&\left (\dfrac {\sin x}{x}\right )^2,\\[1.4ex] \text {(b)}\quad f(x)&=&\dfrac {\sin x}{x^2},\\[1.4ex] \text {(c)}\quad f(x)&=&\left (\dfrac {x}{\sin x}\right )^2,\\[1.4ex] \text {(d)}\quad f(x)&=&\dfrac {x^2}{\sin x},\\[1.4ex] \text {(e)}\quad f(x)&=&\dfrac {\sin x}{x^2+1}. \end {array} \] [You do not need to find the series themselves.]

Integrate \[ \begin {array}{rcl} \text {(a)}&\displaystyle \int _0^{2\pi }\frac {d\phi }{(p+q\cos \phi )^2} &\text {(parameters }p>q>0\text {)},\\[2.2ex] \text {(b)}&\displaystyle \int _{-\infty }^{\infty }\frac {e^{\alpha x}}{1+e^x}\,dx &\text {(parameter }\alpha \in (0,1)\text {)},\\[2.2ex] \text {(c)}&\displaystyle \int _0^{\infty }\sin (a^2x^2)\,dx &\text {(positive parameter }a\text {)}. \end {array} \]

Prove that if all singularities of an analytic function \(f(z)\) in the extended complex plane \(\overline {\mathbb {C}}\) are poles, then \(f(z)\) is a rational function (i.e. the ratio of two polynomials).

Prove that a bilinear map takes all circles into circles.