On the posterior distribution of classes of random means

The study of properties of mean functionals of random probability measures is an important area of research in the theory of Bayesian nonparametric statistics. Many results are now known for random Dirichlet means, but little is known, especially in terms of posterior distributions, for classes of priors beyond the Dirichlet process. In this paper, we consider normalized random measures with independent increments (NRMI's) and mixtures of NRMI. In both cases, we are able to provide exact expressions for the posterior distribution of their means. These general results are then specialized, leading to distributional results for means of two important particular cases of NRMI's and also of the two-parameter Poisson--Dirichlet process.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ200 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A rigged configuration model for B(∞)B(\infty)

We describe a combinatorial realization of the crystals $B(\infty)$ and $B(\lambda)$ using rigged configurations in all symmetrizable Kac-Moody types up to certain conditions. This includes all simply-laced types and all non-simply-laced finite and affine types

A differential-geometric approach to deformations of pairs (X,E)(X,E)

This article gives an exposition of the deformation theory for pairs $(X, E)$, where $X$ is a compact complex manifold and $E$ is a holomorphic vector bundle over $X$, adapting an analytic viewpoint \`{a} la Kodaira-Spencer. By introducing and exploiting an auxiliary differential operator, we derive the Maurer--Cartan equation and differential graded Lie algebra (DGLA) governing the deformation problem, and express them in terms of differential-geometric notions such as the connection and curvature of $E$, obtaining a chain level refinement of the classical results that the tangent space and obstruction space of the moduli problem are respectively given by the first and second cohomology groups of the Atiyah extension of $E$ over $X$. As an application, we give examples where deformations of pairs are unobstructed.Comment: 28 pages; v4: title changed, to appear in Complex Manifold

Asymmetric double-quantum-well phase modulator using surface acoustic waves

An AlGaAs-GaAs asymmetric double-quantum-well (DQW) optical phase modulator using surface acoustic waves is investigated theoretically. The optimization steps of the DQW structure, which so far have not been reported in detail, are discussed here. The optimized phase modulator structure is found to contain a five-period QDW active region. A surface acoustic wave induces a potential field which provides the phase modulation. Analysis of the modulation characteristics show that by using the asymmetric DQW, the large change of the induced potential at the surface and thus large modification of the quantum-well (QW) structure can be utilized. The modification of each QW structure is consistent, although this consistency is not always preserved in typical surface acoustic wave devices. Consequently, the change of refractive index in each of the five DQW's is almost identical. Besides, the change of effective refractive index is ten times larger here in comparison to a modulator with a five-period single QW as the active region and thus produces a larger phase modulation. In addition, a long wavelength and a low surface acoustic wave power required here simplify the fabrication of surface acoistic wave transducer and the acoustooptic phase modulator.published_or_final_versio

Computational Arithmetic Geometry I: Sentences Nearly in the Polynomial Hierarchy

We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, consider the following: (I) Given a polynomial f in Z[v,x,y], decide the sentence \exists v \forall x \exists y f(v,x,y)=0, with all three quantifiers ranging over N (or Z). (II) Given polynomials f_1,...,f_m in Z[x_1,...,x_n] with m>=n, decide if there is a rational solution to f_1=...=f_m=0. We show that, for almost all inputs, problem (I) can be done within coNP. The decidability of problem (I), over N and Z, was previously unknown. We also show that the Generalized Riemann Hypothesis (GRH) implies that, for almost all inputs, problem (II) can be done via within the complexity class PP^{NP^NP}, i.e., within the third level of the polynomial hierarchy. The decidability of problem (II), even in the case m=n=2, remains open in general. Along the way, we prove results relating polynomial system solving over C, Q, and Z/pZ. We also prove a result on Galois groups associated to sparse polynomial systems which may be of independent interest. A practical observation is that the aforementioned Diophantine problems should perhaps be avoided in the construction of crypto-systems.Comment: Slight revision of final journal version of an extended abstract which appeared in STOC 1999. This version includes significant corrections and improvements to various asymptotic bounds. Needs cjour.cls to compil

Dirac cohomology and Euler-Poincar\'e pairing for weight modules

Let $\mathfrak{g}$ be a reductive Lie algebra over $\mathbb{C}$. For any simple weight module of $\mathfrak{g}$ with finite-dimensional weight spaces, we show that its Dirac cohomology is vanished unless it is a highest weight module. This completes the calculation of Dirac cohomology for simple weight modules since the Dirac cohomology of simple highest weight modules was carried out in our previous work. We also show that the Dirac index pairing of two weight modules which have infinitesimal characters agrees with their Euler-Poincar\'{e} pairing. The analogue of this result for Harish-Chandra modules is a consequence of the Kazhdan's orthogonality conjecture which was settled by the first named author and Binyong Sun