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Completely Prime Maximal Ideals and Quantization
Let [Fraktur lowercase]g be a complex simple Lie algebra of classical type, [italic capital]U([Fraktur lowercase]g) its enveloping algebra. We classify the completely prime maximal spectrum of [italic capital]U([Fraktur lowercase]g). We also construct some interesting algebra extensions of primitive quotients of [italic capital]U([Fraktur lowercase]g), and compute their Goldie ranks, lengths as bimodules, and characteristic cycles. Finally, we study the relevance of these algebras to D. Vogan's program of "quantizing" covers of nilpotent orbits [script]O in [Fraktur lowercase]g[superscript]*.
Deformation Quantization for Actions of $R^d$
This work describes a general construction of a deformation quantization for any Poisson bracket on a manifold which comes from an action of R ]d on that manifold. These deformation quantizations are strict, in the sense that the deformed product of any two functions is again a function and that there are corresponding involutions and operator norms. Many of the techniques involved are adapted from the theory of pseudo-differential operators. The construction is shown to have many favorable properties. A number of specific examples are described, ranging from basic ones such as quantum disks, quantum tori, and quantum spheres, to aspects of quantum groups.
On the Martingale Problem for Interactive Measure-Valued Branching Diffusions
This book develops stochastic integration with respect to ``Brownian trees'' and its associated stochastic calculus, with the aim of proving pathwise existence and uniqueness in a stochastic equation driven by a historical Brownian motion. Perkins uses these results and a Girsanov-type theorem to prove that the martingale problem for the historical process associated with a wide class of interactive branching measure-valued diffusions (superprocesses) is well-posed. The resulting measure-valued processes will arise as limits of the empirical measures of branching particle systems in which particles interact through their spatial motions or, to a lesser extent, through their branching rates.
On the Coefficients of Cyclotomic Polynomials
Let [italic]a([italic]m, [italic]n) denote the [italic]mth coefficient of the [italic]nth cyclotomic polynomial [capital Greek]Phi[subscript italic]n([italic]z), and let [italic]a([italic]m) = max[subscript italic]n [conditional event/restriction/such that] |[italic]a([italic]m, [italic]n)[conditional event/restriction/such that] |. Our principal result is an asymptotic formula for log [italic]a([italic]m) that improves over a recent estimate of Montgomery and Vaughan.
Behavior of Distant Maximal Geodesics in Finitely Connected Complete 2-dimensional Riemannian Manifolds
This monograph studies the topological shapes of geodesics outside a large compact set in a finitely connected, complete, and noncompact surface admitting total curvature. When the surface is homeomorphic to a plane, all such geodesics behave like those of a flat cone. In particular, the rotation numbers of the geodesics are controlled by the total curvature. Accessible to beginners in differential geometry, but also of interest to specialists, this monograph features many illustrations that enhance understanding of the main ideas.
Textile Systems for Endomorphisms and Automorphisms of the Shift
We introduce the notion of a textile system. Using this, we study the dynamical properties of endomorphisms and automorphisms of topological Markov shifts including one-sided ones. The dynamical properties of automorphisms of sofic systems are also studied.
Chapter 16 of Ramanujan's Second Notebook: Theta-Functions and $q$-Series
The first part of Chapter 16 in Ramanujan's second notebook is devoted to q-series. Several of the results obtained by Ramanujan are classical, but many are new. In particular, certain elegant q-continued fraction expansions have not appeared heretofore in print. In the remainder of this chapter, Ramanujan develops the theory of the classical theta-functions in a manner different from his nineteenth century predecessors such as Jacobi. Although many of Ramanujan's discoveries about theta-functions are well-known, several new results are also to be found.
Some Special Properties of the Adjunction Theory for $3$-Folds in $\mathbb P^5$
This work studies the adjunction theory of smooth 3-folds in P]5. Because of the many special restrictions on such 3-folds, the structure of the adjunction theoretic reductions are especially simple, e.g. the 3-fold equals its first reduction, the second reduction is smooth except possibly for a few explicit low degrees, and the formulae relating the projective invariants of the given 3-fold with the invariants of its second reduction are very explicit. Tables summarizing the classification of such 3-folds up to degree 12 are included. Many of the general results are shown to hold for smooth projective n-folds embedded in P]N with N 2n -1.
