We present a quantum-mechanical theory of the cooling of a cantilever coupled via radiation pressure to an illuminated optical cavity. Applying the quantum noise approach to the fluctuations of the radiation pressure force, we derive the optomechanical cooling rate and the minimum achievable phonon number. We find that reaching the quantum limit of arbitrarily small phonon numbers requires going into the good-cavity (resolved phonon sideband) regime where the cavity linewidth is much smaller than the mechanical frequency and the corresponding cavity detuning. This is in contrast to the common assumption that the mechanical frequency and the cavity detuning should be comparable to the cavity damping.

Abstract. We extend Feynman's analysis of an infinite ladder circuit to fractal circuits, providing examples in which fractal circuits constructed with purely imaginary impedances can have characteristic impedances with positive real part. Using (weak) self-similarity of our fractal structures, we provide algorithms for studying the equilibrium distribution of energy on these circuits. This extends the analysis of self-similar resistance networks introduced by Fukushima, Kigami, Kusuoka, and more recently studied by Strichartz et al.

We identify a collection of periodic billiard orbits in a self-similar Sierpinski carpet billiard table Ω(Sa). Based on a refinement of the result of Durand-Cartagena and Tyson regarding nontrivial line segments in Sa, we construct what is called an eventually constant sequence of compatible periodic orbits of prefractal Sierpinski carpet billiard tables Ω(Sa,n). The trivial limit of this sequence then constitutes a periodic orbit of Ω(Sa). We also determine the corresponding translation surface S(Sa,n) for each prefractal table Ω(Sa,n), and show that the genera {gn} ∞ n=0 of a sequence of translation surfaces {S(Sa,n)} ∞ n=0 increase without bound. Various open questions and possible directions for future research are offered.

Abstract. We investigate the spectrum of the self-similar Laplacian, which generates the socalled "pq random walk" on the integer half-line Z + . Using the method of spectral decimation, we prove that the spectral type of the Laplacian is singularly continuous whenever p = 1 2 . This serves as a toy model for generating singularly continuous spectrum, which can be generalized to more complicated settings. We hope it will provide more insight into Fibonacci-type and other weakly self-similar models.