The Karhunen-Loève spectrum of a discrete function can vary depending on how symmetry group objects are mapped to function minterms. For completely specified functions with a range space smaller than the cardinality of a symmetry group, we propose a technique to assign the symmetry group don’t cares to maximize the number of zero-valued KL spectral coefficients of the function. To evaluate thes...

Symmetry is a fundamental organizational concept in art as well as science. To develop and exploit this concept to its fullest, it must be given a precise mathematical formulation. This has been a primary motivation for developing the branch of mathematics known as “group theory.” There are many kinds of symmetry, but the symmetries of rigid bodies are the most important and useful, because the...

Motivated by geometric problems in signal processing, computer vision, and structural biology, we study a class of orbit recovery problems where we observe noisy copies of an unknown signal, each acted upon by a random element of some group (such as Z/p or SO(3)). The goal is to recover the orbit of the signal under the group action. This generalizes problems of interest such as multi-reference...

Abstract: We show that the concept of complete symmetry group introduced by Krause (J Math Phys 35 (1994) 5734-5748) in the context of the Newtonian Kepler problem has wider applicability, extending to the relativistic context of the Einstein equations describing spherically symmetric bodies with certain conformal Killing symmetries. We also provide a simple demonstration of the nonuniqueness o...

We introduce the concept of algebra eigenstates which are defined for an arbitrary Lie group as eigenstates of elements of the corresponding complex Lie algebra. We show that this concept unifies different definitions of coherent states associated with a dynamical symmetry group. On the one hand, algebra eigenstates include different sets of Perelomov’s generalized coherent states. On the other...

I describe the current status of a collaboration with J. D. Romano, R. H. Price, and W. Krivan to model the geometry of and gravitational radiation emitted by a binary system of compact objects in the regime where nonperturbative gravitational effects exist, but the rate of inspiral is still small relative to the orbital frequency. The method of looking for a stationary spacetime which approxim...

Symmetry is a fundamental organizational concept in art as well as science. To develop and exploit this concept to its fullest, it must be given a precise mathematical formulation. This has been a primary motivation for developing the branch of mathematics known as “group theory.” There are many kinds of symmetry, but the symmetries of rigid bodies are the most important and useful, because the...

Choptuik has demonstrated that naked singularities can arise in gravitational collapse from smooth, asymptotically flat initial data, and that such data have codimension one in spherical symmetry. Here we show, for perfect fluid matter with equation of state p = ρ/3, by perturbing around spherical symmetry, that such data have in fact codimension one in the full phase space, at least in a neigh...

We formulate a general framework for the construction of hyperbolic billiards. Spherical symmetry is exploited for a simple treatment of billiards with spherical caps and soft billiards in higher dimensions. Other examples include the Papenbrock stadium.

We construct a quantum system of spherical spins with a continuous local symmetry. The model is exactly soluble in the thermodynamic limit and exhibits a number of interesting properties. We show that the local symmetry is spontaneously broken at finite as well as zero temperatures, implying the existence of classical and quantum phase transitions with a nontrivial critical behavior. The dynami...