Modular invariant partition functions in the quantum Hall effect

We study the partition function for the low-energy edge excitations of the incompressible electron fluid. On an annular geometry, these excitations have opposite chiralities on the two edges; thus, the partition function takes the standard form of rational conformal field theories. In particular, it is invariant under modular transformations of the toroidal geometry made by the angular variable and the compact Euclidean time. The Jain series of plateaus have been described by two types of edge theories: the minimal models of the W_1+∞ algebra of quantum area-preserving diffeomorphisms, and their non-minimal version, the theories with affine algebra. We find modular invariant partition functions for the latter models. Moreover, we relate the Wen topological order to the modular transformations and the Verlinde fusion algebra. We find new, non-diagonal modular invariants which describe edge theories with extended symmetry algebra; their Hall conductivities match the experimental values beyond the Jain series.     

Relations between the Coulomb gas picture and conformal invariance of two-dimensional critical models

Partition functions of critical 2D models on a torus can be derived from their microscopic formulation and their free field representation in the continuum limit. This is worked out explicitly for theO(n) andQ-state Potts model. Forn orQ integer we recover results obtained from conformal invariance, but our procedure also extends to nonintegral values. In the latter case the expansion on characters of the Virasoro algebra involves real coefficients of either sign. The operator content of both models is discussed in detail.     

A tale of two conformally invariant metrics

The Harnack metric is a conformally invariant metric defined in quite general domains that coincides with the hyperbolic metric in the disk. We prove that the Harnack distance is never greater than the hyperbolic distance and if the two distances agree for one pair of distinct points, then either the domain is simply connected or it is conformally equivalent to the punctured disk.     

Conditionally Optimal Weights of Evidence

A weight of evidence is a calibrated statistic whose values in [0,1] indicate the degree of agreement between the data and either of two hypothesis, one being treated as the null (H_0) and the other as the alternative (H_1). A value of zero means perfect agreement with the null, whereas a value of one means perfect agreement with the alternative. The optimality we consider is minimal mean squared error (MSE) under the alternative while keeping the MSE under the null below a fixed bound. This paper studies such statistics from a conditional point of view, in particular for location and scale models.     

Nonsmooth analysis on smooth manifolds

We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function.

A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on manifolds. We also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions.

     

Extreme Value Asymptotics for Multivariate Renewal Processes

For a sequence of partial sums ofd-dimensional independent identically distributed random vectors a corresponding multivariate renewal process is defined componentwise. Via strong invariance together with an extreme value limit theorem for Rayleigh processes, a number of weak asymptotic results are established for thed-dimensional renewal process. Similar theorems for the estimated version of this process are also derived. These results are suggested to serve as simultaneous asymptotic testing devices for detecting changes in the multivariate setting.     

生存分析中乘积限估计的大样本性质

生存分析中，人们关心的问题之一是利用不完全的寿命调查数据估计生物折寿命分布。在实际问题中，比较常见的不完全数据包括右删失数据，左截断数据和左截断右删失数据。利用这三种数据估计寿命分布时，常用的统计量是乘积限估计。于是，乘积限估计的大样本性质的研究一直受到关注。本文就这方面的研究近况做一比较系统的论述。     

Charge-conserving electron density averaging for a set of nuclear configurations

The Löwdin-Inverse Löwdin Transformation (LILT) technique combined with the Diophantine Density Matrix Purification (DDMP) method is suggested for a charge-preserving density averaging approach for a range of nuclear configurations.     

Invariance and independence in multivariate distribution theory

Several general results are presented whereby various properties of independence or conditional independence between certain random variables may be deduced from the symmetries enjoyed by their joint distributions. These are applied to the distributions of sample correlation and canonical correlation coefficients when the underlying data-distribution has suitable orthogonal invariance. A typical result is that, for a random sample of observations on three independent normal variables, r₁₂, r₁₃, and r_23.1 are mutually independent.     

Asymptotic expansions in functional limit theorems

Asymptotic expansions for a class of functional limit theorems are investigated. It is shown that the expansions in this class fit into a common scheme, defined by a sequence of functions h_n (ε₁,…, ε_n), n ≥ 1, of “weights” (for n observations), which are smooth, symmetric, compatible and have vanishing first derivatives at zero. Then $\text{h}{}_{\mathrm{n}}\text{(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{,…, n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$ admits an asymptotic expansion in powers of $\text{n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$. Applications to quadratic von Mises functionals, the C.L.T. in Banach spaces, and the invariance principle are discussed.