Exterior differential systems, Lie algebra cohomology, and the rigidity of homogenous varieties

These are expository notes from the 2008 Srni Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of partial differential equations, and (2) to give an exposition of recent work (joint with C. Robles) on the study of the Fubini-Griffiths-Harris rigidity of rational homogeneous varieties, which also involves an advance in the EDS technology.Comment: To appear in the proceedings of the 2008 Srni Winter School on Geometry and Physic

An N-body Integrator for Gravitating Planetary Rings, and the Outer Edge of Saturn's B Ring

A new symplectic N-body integrator is introduced, one designed to calculate the global 360 degree evolution of a self-gravitating planetary ring that is in orbit about an oblate planet. This freely-available code is called epi_int, and it is distinct from other such codes in its use of streamlines to calculate the effects of ring self-gravity. The great advantage of this approach is that the perturbing forces arise from smooth wires of ring matter rather than discreet particles, so there is very little gravitational scattering and so only a modest number of particles are needed to simulate, say, the scalloped edge of a resonantly confined ring or the propagation of spiral density waves. The code is applied to the outer edge of Saturn's B ring, and a comparison of Cassini measurements of the ring's forced response to simulations of Mimas' resonant perturbations reveals that the B ring's surface density at its outer edge is 195+-60 gm/cm^2 which, if the same everywhere across the ring would mean that the B ring's mass is about 90% of Mimas' mass. Cassini observations show that the B ring-edge has several free normal modes, which are long-lived disturbances of the ring-edge that are not driven by any known satellite resonances. Although the mechanism that excites or sustains these normal modes is unknown, we can plant such a disturbance at a simulated ring's edge, and find that these modes persist without any damping for more than ~10^5 orbits or ~100 yrs despite the simulated ring's viscosity of 100 cm^2/sec. These simulations also indicate that impulsive disturbances at a ring can excite long-lived normal modes, which suggests that an impact in the recent past by perhaps a cloud of cometary debris might have excited these disturbances which are quite common to many of Saturn's sharp-edged rings.Comment: 55 pages, 13 figures, accepted for publication in the Astrophysical Journa

Belief propagation decoding of quantum channels by passing quantum messages

Belief propagation is a powerful tool in statistical physics, machine learning, and modern coding theory. As a decoding method, it is ubiquitous in classical error correction and has also been applied to stabilizer-based quantum error correction. The algorithm works by passing messages between nodes of the factor graph associated with the code and enables efficient decoding, in some cases even up to the Shannon capacity of the channel. Here we construct a belief propagation algorithm which passes quantum messages on the factor graph and is capable of decoding the classical-quantum channel with pure state outputs. This gives explicit decoding circuits whose number of gates is quadratic in the blocklength of the code. We also show that this decoder can be modified to work with polar codes for the pure state channel and as part of a polar decoder for transmitting quantum information over the amplitude damping channel. These represent the first explicit capacity-achieving decoders for non-Pauli channels.Comment: v3: final version for publication; v2: improved discussion of the algorithm; 7 pages & 2 figures. v1: 6 pages, 1 figur

The Unification of Germany: What Would Jhering Say?

Two of Jhering\u27s ideas are crucial to understanding the problems besetting the merger of East and West Germany. They are (a) the centrality of the notion of private property as the foundation, not only of property rights, but of personal rights as well; and (b) his notion of rechtsgefühl, translated clumsily as a feeling of legal right, but implying the pain and irritation a person feels when he has been put upon. [FN8] It is my thesis that a fundamental difference between the way these two concepts are viewed in the former East and West Germanies is a sword in the bed, presenting a fierce obstacle to the union that both desire

A BEAUTY THAT SAVES: DOSTOEVSKY’S THEOLOGY OF BEAUTY THE IDIOT

This paper examines Dostoevsky’s understanding of beauty and its place in The Idiot. Examining the historical and immediate environment in which Dostoevsky wrote the novel provides crucial insights into his conception of beauty. It is argued that the beauty Dostoevsky encountered in Florence colored his use of beauty in The Idiot. The use recent popes have made of Dostoevsky’s works also underscore the Christian theology of his ideal. From the post-Vatican II pontiffs and from Dostoevsky’s own writing it becomes clear that Dostoevsky’s view of beauty flows from the Christian belief that Christ is the Supreme Beauty. The beauty of Prince Myshkin and other characters all flow from this Beauty by reflecting Him in different ways. Ultimately, however, Myshkin fails to bring about salvation. He lacks the perfect beauty and goodness of Christ, the only one who can save. This salvific beauty is noticeably missing from the novel. The Hans Holbein painting of “Christ in the Tomb,” it is argued, lacks beauty because it fails to show forth the Incarnation. Instead, it depicts Christ without any hint of his divinity and without any hope of resurrection. Through this examination of Dostoevsky’s context and theology, it is concluded that the lack of a beauty that saves in The Idiot is meant to underscore mankind’s inability to save itself