The nonuniqueness of the tangent cones at infinity of Ricci-flat manifolds

It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise an example of the Ricci-flat manifold implying that the assumption for the volume growth in the above result is essential. More precisely, we construct a complete Ricci-flat manifold of dimension 4 with non-Euclidean volume growth who has at least two distinct tangent cones at infinity and one of them has a smooth cross section

Uniqueness of fixed point of a two-dimensional map obtained as a generalization of the renormalization group map associated to the self-avoiding paths on gaskets

Let $W(x,y) = a x^3 + b x^4 + f_5 x^5 + f_6 x^6 + (3 a x^2)^2 y + g_5 x^5 y + h_3 x^3 y^2 + h_4 x^4 y^2 + n_3 x^3 y^3 + a_{24} x^2 y^4 + a_{05} y^5 + a_{15} x y^5 + a_{06} y^6$, and $X=\frac{\partial W}{\partial x}$, $Y=\frac{\partial W}{\partial y}$, where the coefficients are non-negative constants, with $a>0$, such that $X^{2}(x,x^{2})-Y(x,x^{2})$ is a polynomial of $x$ with non-negative coefficients. Examples of the 2 dimensional map $\Phi: (x,y)\mapsto (X(x,y),Y(x,y))$ satisfying the conditions are the renormalization group (RG) map (modulo change of variables) for the restricted self-avoiding paths on the 3 and 4 dimensional pre-gaskets. We prove that there exists a unique fixed point $(x_f,y_f)$ of $\Phi$ in the invariant set $\{(x,y)\in R^2\mid x^2\ge y\}\setminus\{0\}$.Comment: LaTeX2e, 12 pages, no figure

Policy and Product Differentiations Encourage the International Transfer of Environmental Technologies

This paper investigates the welfare effects of international transfers of environmental technologies in open economies with international oligopoly and transboundary pollution, and shows that policy differentiation between the donor and recipient countries and/or product differentiation between the donor and recipient firms play a critical role in obtaining a bilateral agreement on the transfer policy between nations. The results arise from the fact that policy differentiation weakens the strategic relationships in environmental policy setting between governments and that product differentiation weakens the strategic relationships in quantity choices between firms.Technology Transfer; Environmental Tax; Oligopoly