An extension of Chebfun to two dimensions

An object-oriented MATLAB system is described that extends the capabilities of Chebfun to smooth functions of two variables defined on rectangles. Functions are approximated to essentially machine precision by using iterative Gaussian elimination with complete pivoting to form “chebfun2” objects representing low rank approximations. Operations such as integration, differentiation, function evaluation, and transforms are particularly efficient. Global optimization, the singular value decomposition, and rootfinding are also extended to chebfun2 objects. Numerical applications are presented

Stationary viscoelastic wave fields generated by scalar wave functions

The usual Helmholtz decomposition gives a decomposition of any vector valued function into a sum of gradient of a scalar function and rotation of a vector valued function under some mild condition. In this paper we show that the vector valued function of the second term i.e. the divergence free part of this decomposition can be further decomposed into a sum of a vector valued function polarized in one component and the rotation of a vector valued function also polarized in the same component. Hence the divergence free part only depends on two scalar functions. Further we show the so called completeness of representation associated to this decomposition for the stationary wave field of a homogeneous, isotropic viscoelastic medium. That is by applying this decomposition to this wave field, we can show that each of these three scalar functions satisfies a Helmholtz equation. Our completeness of representation is useful for solving boundary value problem in a cylindrical domain for several partial differential equations of systems in mathematical physics such as stationary isotropic homogeneous elastic/viscoelastic equations of system and stationary isotropic homogeneous Maxwell equations of system. As an example, by using this completeness of representation, we give the solution formula for torsional deformation of a pendulum of cylindrical shaped homogeneous isotropic viscoelastic medium

A Re-scaled Version of the Foster-Greer-Thorbecke Poverty Indices based on an Association with the Minkowski Distance Function

This note advances a family of poverty measures, ??, which are derived as simple, normalized Minkowski distance functions. The ?? indices turn out to be the ?th roots of the corresponding Foster, Greer and Thorbecke P? indices. The re-calibration of P? terms of ?? could have certain possible advantages, which are reviewed in the note. While the ?? indices are not decomposable in the ordinarily understood sense of that term, they are amenable to the completely general decomposition procedure advanced by Shorrocks (?Decomposition Procedures for Distributional Analysis: A Unified Framework Based on the Shapley Value?) and discussed, here, as an application in the poverty context.poverty measures, decomposition