PhD Preliminary Qualifying Examination Applied Mathematics August, 2013

Suppose that \(E\) is a bounded linear operator on a Hilbert space \(H\), with operator norm \(\lVert E\rVert =r<1\). Show that the operator \(I-E\), where \(I\) is the identity, is bounded and invertible. Give an estimate for \(\lVert (I-E)^{-1}\rVert \), and a convergent series representation for \((I-E)^{-1}\).

Suppose that the square real matrix \(A=(a_{ij})\) satisfies \(a_{ij}=a_{ji}\) for all \(i,j\).

1. Show that every eigenvalue \(\lambda \) of \(A\) must be real.
2. Show that for any eigenpairs \((\lambda _1,u_1)\), \((\lambda _2,u_2)\), with \(\lambda _1\ne \lambda _2\), the eigenvectors \(u_1\) and \(u_2\) must be orthogonal.
3. Let \[ c=\min _{\lVert u\rVert =1}u^TAu. \] Prove that if \(c>0\), then \(A^{-1}\) exists. In this case, find an estimate for the matrix operator norm \(\lVert A^{-1}\rVert \).

Consider the problem \(Lu=1\) on \((0,\pi )\), where \(Lu=u''+u\) with boundary conditions \(u'(0)=u'(\pi )=0\).

1. Write down the weak formulation of the problem.
2. Find the \(L^2[0,\pi ]\) adjoint of \(L\), and determine whether or not \(L\) is self-adjoint.
3. Find the nullspace of \(L\). Use Fredholm theory to characterize the set of functions \(f\) such that \(Lu=f\) has a solution. Verify that \(f=1\) satisfies your condition, and find an infinite set of solutions to the problem \(Lu=1\).

Define the operator \(K\) on \(L^2[0,1]\) by \[ Ku(x)=\int _0^1 k(x,y)u(y)\,dy, \] for \(x\in [0,1]\), where \(k(x,y)\) is a real-valued, bounded, continuous function, with \(k(x,y)>0\).

1. Find the \(L^2\) adjoint \(K^*\) for \(K\), and find a condition on \(k\) which guarantees that \(K\) is self-adjoint.
2. Describe the properties of the set \[ \{\lambda \in \mathbb {C}:(K-\lambda I)u=0\text { for some }u\ne 0\} \] (cardinality, bounds, etc..).
3. Consider the problem \(Ku=f\). Let \(X\) be the set of all \(f\in L^2[0,1]\) such that a unique solution \(u=K^{-1}f\) exists. Can we be assured that there is a constant \(C\) such that \(\lVert K^{-1}f\rVert \le C\lVert f\rVert \), for all \(f\in X\)? Why or why not?

Let \(f(x)=|x|-1\) on the interval \([-1,1]\).

1. Treating \(f\) as a distribution, calculate \[ \int _{-1}^{1} f'(x)f'(x)\,dx \quad \text {and}\quad \int _{-1}^{1} f(x)f''(x)\,dx. \]
2. Explain why \[ \int _{-1}^{1} f'(x)f''(x)\,dx \quad \text {and}\quad \int _{-1}^{1} f(x)f'''(x)\,dx \] do not make sense.
3. In general, what property is required of \(g(x)\) so that \[ \int _{-1}^{1} g(x)f^{(n)}(x)\,dx \] is well-defined for a given \(n\)th derivative?

Find the radius of convergence of the Taylor series for the function \[ \frac {1}{2+\cos x} \qquad (x\text { is a real variable}) \] around the origin (\(x_0=0\)). {You do not need to find the Taylor series itself.}

1. Show that any bilinear map takes circles into circles.
2. Find the image of the triangle \[ \{z=x+iy:\ x\ge 0,\ y\ge 0,\ x+y\le 1\} \] under the mapping \(w=1/z\).

Evaluate the integrals:

1. \[ I=\int _0^\infty \frac {\sin x}{x}\,dx, \]
2. \[ I=\int _0^\infty \frac {x^\alpha }{2+x}\,dx \qquad (-1<\alpha <1). \]

Formulate and derive the uncertainty principle. Is its inequality optimal? Explain.

Find the leading behavior, as \(s\to +\infty \), of the integral

1. \[ I=\int _0^3\frac {1}{\sqrt {x^2+2x}}e^{-sx}\,dx, \]
2. \[ I=\int _0^{\pi /2} e^{is\cos x}\,dx. \]