UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Applied Mathematics

Let \((u_n)\) and \((v_n)\) be two orthonormal sequences in a Hilbert space \(H\). Assume that \[ \sum _{n=1}^{\infty }\lVert u_n-v_n\rVert <1. \] Show that if \((u_n)\) is a total orthonormal sequence, then \((v_n)\) is also a total orthonormal sequence.

Hint: Assume for contradiction that \((v_n)\) is not total.

Let \((e_n)\) be a total orthonormal sequence in a Hilbert space \(H\). Let \((x_n)\subset H\) be a sequence. Prove that the two following statements are equivalent.

1. \(x_n\to x\) weakly.
2. The sequence \((x_n)\) is bounded and for any \(j\ge 1\), \(\langle x_n,e_j\rangle \to \langle x,e_j\rangle \) as \(n\to \infty \).

Hint: Show that if (ii) holds and \(y_N\in \operatorname {span}\{e_1,\ldots ,e_N\}\) then \(\langle x_n,y_N\rangle \to \langle x,y_N\rangle \) as \(n\to \infty \).

Consider the nonlinear integral equation \[ u(x)-\lambda \int _a^b w(x,y)\sin (u(y))\,dy=\phi (x), \] where \(\lambda \in \mathbb {R}\), \(\phi \) is continuous on the interval \([a,b]\) and \(w(x,y)\) is continuous on \([a,b]\times [a,b]\). Find a constant \(C>0\) such that \(\lvert \lambda \rvert <C\) implies that the nonlinear integral equation has a unique solution \(u\).

Hint: You may assume the inequality \(\lvert \sin (x)-\sin (y)\rvert \le \lvert x-y\rvert \), for any \(x,y\in \mathbb {R}\).

Consider the left shift operator \(L:\ell ^2\to \ell ^2\), defined for a sequence \((x_n)\in \ell ^2\) by \[ L(x_1,x_2,x_3,\ldots )=(x_2,x_3,x_4,\ldots ). \]

1. Prove that \(\lVert L\rVert =1\).
2. Explain why if \(\lambda \in \mathbb {C}\) with \(\lvert \lambda \rvert >1\), then \(\lambda \in \rho (L)\), the resolvent set of \(L\).
3. Prove that that if \(\lambda \in \mathbb {C}\) with \(\lvert \lambda \rvert <1\), then \(\lambda \in \sigma _p(L)\), the point spectrum of \(L\).
4. Deduce what is \(\sigma (L)\), the spectrum of \(L\).

Let \(L\) be a partial differential operator of the form: \[ L=\sum _{j=0}^{m-1} c_j\partial ^j, \] where the \(c_j\) are arbitrary constants. Show that \(x^m(L\delta )=0\) as distributions in \(\mathcal {D}'(\mathbb {R})\). Here \(\delta \in \mathcal {D}'(\mathbb {R})\) is the usual “delta function”.

If a complex function is differentiable and has constant modulus, show that the function itself is constant.

Let \(f(z)\) be an entire function such that there exists a constant \(M\), an \(R>0\) and an integer \(n\ge 1\) with \(\lvert f(z)\rvert \le M\lvert z\rvert ^n\) for \(\lvert z\rvert >R\). Show that \(f(z)\) is a polynomial of degree at most \(n\).

1. Find the Laurent expansion of \[ f(z)=\frac {z}{(z+i)(z-2)} \] for \(1<\lvert z\rvert <2\).
2. Classify the singularities of the function \[ f(z)=\frac {z+1}{z\sin (z)}. \]

Evaluate \[ I=\int _{-\infty }^{\infty }\frac {\cos (2x)}{(x+1)^2+4}\,dx. \] You must clearly show the contour of integration and treat integration over each contour separately.

Find the leading non constant behavior as \(x\to 0^+\) of \[ \int _1^\infty \cos (xt)t^{-1}\,dt. \] You must express the constant term in terms of known quantities and/or constants.