Singular Masas and Measure-Multiplicity Invariant

In this paper we study relations between the \emph{left-right-measure} and properties of singular masas. Part of the analysis is mainly concerned with masas for which the \emph{left-right-measure} is the class of product measure. We provide examples of Tauer masas in the hyperfinite $\rm{II}_{1}$ factor whose \emph{left-right-measure} is the class of Lebesgue measure. We show that for each subset $S\subseteq \mathbb{N}$, there exist uncountably many pairwise non conjugate singular masas in the free group factors with \emph{Puk\'{a}nszky invariant} $S\cup\{\infty\}$.Comment: 24 pages, to appear in Houston. J. Mat

Fast and Accurate Bilateral Filtering using Gauss-Polynomial Decomposition

The bilateral filter is a versatile non-linear filter that has found diverse applications in image processing, computer vision, computer graphics, and computational photography. A widely-used form of the filter is the Gaussian bilateral filter in which both the spatial and range kernels are Gaussian. A direct implementation of this filter requires $O(\sigma^2)$ operations per pixel, where $\sigma$ is the standard deviation of the spatial Gaussian. In this paper, we propose an accurate approximation algorithm that can cut down the computational complexity to $O(1)$ per pixel for any arbitrary $\sigma$ (constant-time implementation). This is based on the observation that the range kernel operates via the translations of a fixed Gaussian over the range space, and that these translated Gaussians can be accurately approximated using the so-called Gauss-polynomials. The overall algorithm emerging from this approximation involves a series of spatial Gaussian filtering, which can be implemented in constant-time using separability and recursion. We present some preliminary results to demonstrate that the proposed algorithm compares favorably with some of the existing fast algorithms in terms of speed and accuracy.Comment: To appear in the IEEE International Conference on Image Processing (ICIP 2015

Inequality and Growth in a Knowledge Economy

We develop a two sector growth model to understand the relation between inequality and growth. Agents, who are endowed with different levels of knowledge, select either into a retail or a manufacturing sector. Agents in the manufacturing sector match to carry out production. A by-product of production is creation of ideas that spill over to the retail sector and improve productivity, thereby causing growth. Ideas are generated according to an idea production function that takes the knowledge of all the agents in a firm as arguments. We go on to study how an increase in the inequality of the knowledge distribution affects the growth rate. A change in the distribution not only affects the occupational choice of agents, but also the way agents match within the manufacturing sector. We show that if the idea generation function is sufficiently convex, an increase in inequality raises the growth rate of the economy.Inequality, growth, idea generation, matching, knowledge