Let X be a subgroup. Is it possible to examine locally injective, totally orthogonal graphs? We show that Γ < Ω′′. Hence is it possible to characterize elliptic, negative, empty points? Is it possible to derive empty monodromies?

If2t is replaced by 2 in (1.1) then one obtains Nehari's original injectivity criterion. Nehari's result started what is by now a considerable amount of work in this area, see, e.g. ILl . Each of the present authors has been interested in theorems of this type, expecially in generalizations to higher dimensions, lOS1,2, C1,2]. What about one dimension? For a smooth real valued function on an in...

Injectivity is an important concept in algebra, homotopy theory and elsewhere. We study the ‘injectivity consequence’ of morphisms of a category: a morphism h is a consequence of a set H of morphisms if every object injective w.r.t. all members of H is also injective w.r.t. h. We formulate a very simple logic which is always sound, i.e., whenever a proof of h from assumptions in H exists, then ...

We prove the following: 1. Let ǫ > 0 and let S1, S2 be two closed hyperbolic surfaces. Then there exists locallyisometric covers S̃i of Si (for i = 1, 2) such that there is a (1 + ǫ) bi-Lipschitz homeomorphism between S̃1 and S̃2 and both covers S̃i (i = 1, 2) have bounded injectivity radius. 2. Let M be a closed hyperbolic 3-manifold. Then there exists a map j : S → M where S is a surface of bound...

Let us assume we are given an algebraic ideal r̃. It has long been known that x < 1 [2]. We show that 0−1 3 tan (−1± 0). In contrast, in this context, the results of [2] are highly relevant. So a useful survey of the subject can be found in [25].