Minimal generating and separating sets for O(3)-invariants of several matrices

Given an algebra $F[H]^G$ of polynomial invariants of an action of the group $G$ over the vector space $H$, a subset $S$ of $F[H]^G$ is called separating if $S$ separates all orbits that can be separated by $F[H]^G$. A minimal separating set is found for some algebras of matrix invariants of several matrices over an infinite field of arbitrary characteristic different from two in case of the orthogonal group. Namely, we consider the following cases: 1) $GL(3)$-invariants of two matrices; 2) $O(3)$-invariants of $d>0$ skew-symmetric matrices; 3) $O(4)$-invariants of two skew-symmetric matrices; 4) $O(3)$-invariants of two symmetric matrices. A minimal generating set is also given for the algebra of orthogonal invariants of three $3\times 3$ symmetric matrices.Comment: 11 page

ENVIRONMENTAL PROTECTION AND ECONOMIC GROWTH

This paper explores the link between an environmental policy and economic growth employing an extension of the Neoclassical Growth Model. We include a state equation to renewable natural resources, and consider natural resources as a component of the aggregate productivity. It is assumed that the change of the environmental regulations induces costs and that economic agents also derive some utility from stock capital accumulation vis-`a-vis environment. Using the Hopf bifurcation theorem, it can be shown that cyclical environmental policy strategies are optimal, providing a theoretical support to the Environmental Kuznets Curve.

Environmental protection and economic growth

This paper explores the link between environmental policy and economic growth by employing an extension of the AK Growth Model. We include a state equation for renewable natural resources. We assume that the change in environmental regulations induces costs and that economic agents also derive some utility from capital stock accumulation vis-`a-vis the environment. Using the Hopf bifurcation theorem, we show that cyclical environmental policy strategies are optimal, providing theoretical support for the Environmental Kuznets Curve.

Optimal Environmental Protection and Environmental Kuznets Curve

This paper explores the link between an environmental policy and economic growth employing an extension of the Neoclassical Growth Model. We include a state equation to renewable natural resources, and consider natural resources as a component of the aggregate productivity. It is assumed that the change of the environmental regulations induces costs and that economic agents also derive some utility from stock capital accumulation via-a-vis environment. Using the Hopf bifurcation theorem, it can be shown that cyclical environmental policy strategies are optimal, providing a theoretical support to the Environmental Kuznets CurveNeoclassical Growth Model, Environmental Kuznets Curve, Hopf Bifurcation Theorem, Limit Cycles.