Quasi-isometric embedding from the generalised Thompson's group TnT_n to TT

Brown has defined the generalised Thompson's group $F_n$, $T_n$, where $n$ is an integer at least $2$ and Thompson's groups $F= F_2$ and $T =T_2$ in the 80's. Burillo, Cleary and Stein have found that there is a quasi-isometric embedding from $F_n$ to $F_m$ where $n$ and $m$ are positive integers at least 2. We show that there is a quasi-isometric embedding from $T_n$ to $T_2$ for any $n \geq 2$ and no embeddings from $T_2$ to $T_n$ for $n \geq 3$

The Economic Cost of CO2 Emission Cuts

We follow Schmalensee, Stoker, and Judson (1998) to forecast CO2 emissions based on the environmental Kuznets curve (EKC). Our findings suggest that the EKC will not lead to significant decreases in CO2 emissions even by 2050 for countries with the highest incomes. Therefore, mandatory emissions cuts are required to limit climate change. In the same spirit of Horowitz (2009) and Ng and Zhao (2010), we then use a reduced-form approach to estimate the economic costs of mandatory emission cuts. Based on our parameter estimates, we find that a 25% mandatory deduction in CO2 emissions from 1990 will lead to a 5.63% decrease in the combined GDP of the 19 OECD countries, and a 40% deduction will result in a 12.92% loss in income (holding other relevant variables constant)! Our estimates are substantially higher than those in Paltsev, Reillya, Jacobya, and Morris (2009) and Dellink, Briner and Clapp (2010), and suggest that the economic cost to limit climate change as envisioned in the Copenhagen Accord may be substantial and more research should be done before mandatory emission cuts are implemented.Environmental Kuznets Curve, Carbon Dioxide Emissions, Economic Cost, Climate Change, Environmental Economics and Policy,

The Vanishing Moment Method for Fully Nonlinear Second Order Partial Differential Equations: Formulation, Theory, and Numerical Analysis

The vanishing moment method was introduced by the authors in [37] as a reliable methodology for computing viscosity solutions of fully nonlinear second order partial differential equations (PDEs), in particular, using Galerkin-type numerical methods such as finite element methods, spectral methods, and discontinuous Galerkin methods, a task which has not been practicable in the past. The crux of the vanishing moment method is the simple idea of approximating a fully nonlinear second order PDE by a family (parametrized by a small parameter \vepsi) of quasilinear higher order (in particular, fourth order) PDEs. The primary objectives of this book are to present a detailed convergent analysis for the method in the radial symmetric case and to carry out a comprehensive finite element numerical analysis for the vanishing moment equations (i.e., the regularized fourth order PDEs). Abstract methodological and convergence analysis frameworks of conforming finite element methods and mixed finite element methods are first developed for fully nonlinear second order PDEs in general settings. The abstract frameworks are then applied to three prototypical nonlinear equations, namely, the Monge-Amp\`ere equation, the equation of prescribed Gauss curvature, and the infinity-Laplacian equation. Numerical experiments are also presented for each problem to validate the theoretical error estimate results and to gauge the efficiency of the proposed numerical methods and the vanishing moment methodology.Comment: 141 pages, 16 figure