We consider notions of freeness and ambiguity for the acceptance probability Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). study injectivity problem determining if function a MO-QFA is injective over all input words, i.e., giving distinct each word. show that undecidable 8 state MO-QFA, even when unitary matrices projection matrix are rational initial vector real algebraic. a...

Let A be a dual Banach algebra with predual A∗ and consider the following assertions: (A) A is Connes-amenable; (B) A has a normal, virtual diagonal; (C) A∗ is an injective A-bimodule. For general A, all that is known is that (B) implies (A) whereas, for von Neumann algebras, (A), (B), and (C) are equivalent. We show that (C) always implies (B) whereas the converse is false for A = M(G) where G...

We give sufficient conditions for a planar quasiregular mapping to be injective in terms of the range of the differential matrix.

We prove that there are only finitely many closed hyperbolic 3-manifolds with injectivity radius and first eigenvalue of the Laplacian bounded below whose fundamental groups can be generated by a given number of elements.

We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.