Jacobian conjectures (that nonsingular implies a global inverse) for rational everywhere defined maps of Rn to itself are considered, with no requirement for a constant Jacobian determinant or a rational inverse. The birational case is proved and the Galois case clarified. Two known special cases of the Strong Real Jacobian Conjecture (SRJC) are generalized to the rational map context. For an invertible map, the associated extension of rational function fields must be of odd degree and must have no nontrivial automorphisms. That disqualifies the Pinchuk counterexamples to the SRJC as candidates for invertibility.

We study the Yang–Mills ow on a holomorphic vector bundle E over a compact Kahler manifold X. We construct a natural barrier function along the ow, and introduce some techniques to study the blow- up of the curvature along the ow. Making some technical assumptions, we show how our techniques can be used to prove that the curvature of the evolved connection is uniformly bounded away from an analytic subvariety determined by the Harder-Narasimhan-Seshadri ltration of E. We also discuss how our assumptions are related to stability in some simple cases.

Presently, no example of non-finitely based finite semigroup S is known for which the monoid S¹ is finitely based. Based on a general result of M. V. Volkov, two methods are established from which examples of such semigroups can be constructed.

In this note we construct a sequence of real, k-dimensional symmetric spaces Y k satisfying lim inf k Sk = p k lim inf k (Y k; l1)= p k > max w2[0;a2] h(w) > 1=(2 p 2= ); where Sk is de ned by (4) and h(w) = a21 p 2= + 2a1 q a22 w2 + w q a22 w2 with a1 = 1=(2 p 2= ) and a2 = 1 a1. This improves the lower bound obtained in [3], Th. 5.3 by maxw2[0;a2] h(w)

In this paper, we construct a locally convex topology on the vector space E_x. We also prove that with this topology it is a non-separable and non-re exive Frechet space.