Improving the Data Augmentation Algorithm in the Two-Block Setup

The data augmentation (DA) approach to approximate sampling from an intractable probability density f_X is based on the construction of a joint density, f_{X, Y}, whose conditional densities, f_X|Y and f_Y|X, can be straightforwardly sampled. However, many applications of the DA algorithm do not fall in this “single-block” setup. In these applications, X is partitioned into two components, X = (U, V), in such a way that it is easy to sample from f_Y|X, f_{U|V, Y}, and f_{V|U, Y}. We refer to this alternative version of DA, which is effectively a three-variable Gibbs sampler, as “two-block” DA. We develop two methods to improve the performance of the DA algorithm in the two-block setup. These methods are motivated by the Haar PX-DA algorithm, which has been developed in previous literature to improve the performance of the single-block DA algorithm. The Haar PX-DA algorithm, which adds a computationally inexpensive extra step in each iteration of the DA algorithm while preserving the stationary density, has been shown to be optimal among similar techniques. However, as we illustrate, the Haar PX-DA algorithm does not lead to the required stationary density f_X in the two-block setup. Our methods incorporate suitable generalizations and modifications to this approach, and work in the two-block setup. A theoretical comparison of our methods to the two-block DA algorithm, a much harder task than the single-block setup due to nonreversibility and structural complexities, is provided. We successfully apply our methods to applications of the two-block DA algorithm in Bayesian robit regression and Bayesian quantile regression. Supplementary materials for this article are available online.     

Switching wetting morphologies in triangular grooves

Static wetting morphologies in grooves with triangular cross section are studied here. Using the electrowetting effect, we can switch between different equilibrium morphologies and transport liquid along prefabricated grooves. The particular filling and drainage behavior in triangular grooves is discussed in terms of the groove geometry.     

Hurst exponents for interacting random walkers obeying nonlinear Fokker-Planck equations

Anomalous diffusion of random walks has been extensively studied for the case of non-interacting particles. Here we study the evolution of nonlinear partial differential equations by interpreting them as Fokker-Planck equations arising from interactions among random walkers. We extend the formalism of generalized Hurst exponents to the study of nonlinear evolution equations and apply it to several illustrative examples. They include an analytically solvable case of a nonlinear diffusion constant and three nonlinear equations which are not analytically solvable: the usual Fisher equation which contains a quadratic nonlinearity, a generalization of the Fisher equation with density-dependent diffusion constant, and the Nagumo equation which incorporates a cubic rather than a quadratic nonlinearity. We estimate the generalized Hurst exponents.     

Study of magnetotransport in double-layered La1.4Ca1.6Mn2O7 manganite: Presence of nano-ferromagnetic domains in paramagnetic matrix

Double-layered manganite La_1.4Ca_1.6Mn₂O₇ has been synthesized using the solid-state reaction method. It had a metal-to-insulator transition at temperature T_M1≈127 K. The temperature dependence of ac susceptibility showed a broad ferromagnetic transition. The two-dimensional (2D)-ferromagnetic ordering temperature (T_C2) was observed as ≈245 K. The temperature dependence of its low-field magnetoresistance has been studied. The low-field magnetoresistance of double-layered manganite, in the temperature regions between T_M1 and T_C2, has been found to follow 1/T⁵. The observed behaviour of temperature dependence of resistivity and low-field magnetoresistance has been explained in terms of two-phase model where ferromagnetic domains exist in the matrix of paramagnetic regions in which spin-dependent tunneling of charge carriers occurs between the ferromagnetic correlated regions. Based on the two-phase model, the dimension of these ferromagnetic domains inside the paramagnetic matrix has been estimated as ∼12 Å.     

Serre’s modularity conjecture (I)

This paper is the first part of a work which proves Serre’s modularity conjecture. We first prove the cases \(p\not=2\) and odd conductor, and p=2 and weight 2, see Theorem 1.2, modulo Theorems 4.1 and 5.1. Theorems 4.1 and 5.1 are proven in the second part, see Khare and Wintenberger (Invent. Math., doi: 10.1007/s00222-009-0206-6, 2009). We then reduce the general case to a modularity statement for 2-adic lifts of modular mod 2 representations. This statement is now a theorem of Kisin (Invent. Math., doi: 10.1007/s00222-009-0207-5, 2009).     

Vector spaces as unions of proper subspaces

In this note, we find a sharp bound for the minimal number (or in general, indexing set) of subspaces of a fixed (finite) codimension needed to cover any vector space V over any field. If V is a finite set, this is related to the problem of partitioning V into subspaces.     

Temperature and magnetic field induced structural transformation in Si-doped CeFe2: An in-field X-ray diffraction study

Using the X-ray powder diffraction technique at various temperatures and applied magnetic fields, we have studied the magnetostructural properties of Ce(Fe_0.95Si_0.05)₂. The X-ray diffraction data establish quantitative relationships between bulk magnetization and the evolution of structurally distinct phases with magnetic field and temperature, and confirm the distinct features of a first-order phase transition such as supercooling and superheating, metastability, and phase co-existence of different structural polymorphs. We observe the lattice volume mismatch across the structural phase transition, which appears to be the cause for the step behavior of the magnetization isotherms at low temperatures. The present study shows that the lattice distortion has to be treated explicitly, like spin, along with the effects of lattice–spin coupling to account for the magnetization behavior of this system. This structure template can resolve the issue of kinetics in this material as observed in different time scale measurements and with different experimental protocols.     

Current-induced effect on resistivity and magnetoresistance of La0.67Ba0.33MnO3 manganite

The influence of dc biasing current on temperature dependence of resistivity and low-field magnetoresistance (MR) of La_0.67Ba_0.33MnO₃ bulk sample is reported. A prominent finding is the change in resistivity around the insulator-to-metal transition temperature (T_IM) and the change in MR around the ferromagnetic transition temperature (T_C). The decrease in MR around T_C at higher biasing current indicates a strong interaction between carrier spin and spin of Mn ions resulting in a higher alignment of Mn ion spins. Change in resistivity around T_IM is interpreted in the framework of percolative conduction model based on the mixed phase of itinerant electrons and localized magnetic polarons.     

Conditional Entropy and the Rokhlin Metric on an Orthomodular Lattice with Bayessian State

The present paper deals with the study of conditional entropy and its properties in a quantum space (L,s), where L is an orthomodular lattice and s is a Bayessian state on L. First, we obtained a pseudo-metric on the family of all partitions of the couple (B,s), where B is a Boolean algebra and s is a state on B. This pseudo-metric turns out to be a metric (called the Rokhlin metric) by using a new notion of s-refinement and by identifying those partitions of (B,s) which are s-equivalent. The present theory has then been extended to the quantum space (L,s), where L is an orthomodular lattice and s is a Bayessian state on L. Applying the theory of commutators and Bell inequalities, it is shown that the couple (L,s) can be equivalently replaced by a couple (B,s ₀), where B is a Boolean algebra and s ₀ is a state on B.