Small Winding-Number Expansion: Vortex Solutions at Critical Coupling

We study an axially symmetric solution of a vortex in the Abelian-Higgs model at critical coupling in detail. Here we propose a new idea for a perturbative expansion of a solution, where the winding number of a vortex is naturally extended to be a real number and the solution is expanded with respect to it around its origin. We test this idea on three typical constants contained in the solution and confirm that this expansion works well with the help of the Pad\'e approximation. For instance, we analytically reproduce the value of the scalar charge of the vortex with an error of $O(10^{-6})$. This expansion is also powerful even for large winding numbers.Comment: 38 pages,48 figure

"How to Measure the Outcome of Innovations: Application to Product Innovations"

This paper provided a conceptual framework to quantify consumer gains from product innovations. Using an example of VCRs, we illustrate a model that is useful to measure consumer welfare, and describe what data are to be used for the analysis. We also make a selective survey of papers that measure the consumer welfare of product innovations and discuss advantages and limitations of the analytical framework.