Fundamental theorem of geometry without the 1-to-1 assumption

Fundamental Theorem of Geometry without the 1-to-1 Assumption

It is proved that any mapping of an n-dimensional affine space over a division ring D onto itself which maps every line into a line is semiaffine, if n ∈ {2, 3, . . . } and D 6= Z2. This result seems to be new even for the real affine spaces. Some further generalizations are also given. The paper is self-contained, modulo some basic terms and elementary facts concerning linear spaces and also –...

Theorem 1 : Under assumption ( A 1 ) ( W̃ ?

This paper provides the duality structure of the optimal two-block H∞ problem. The dual description leads naturally to a numerical solution based on convex programming for LTI (including infinite dimensional) systems. Alignment conditions are obtained and show that the optimal solution is flat in general, and unique in the SISO case. It is also proved that under specific conditions a wellknown ...

The fundamental theorem of projective geometry

We prove the fundamental theorem of projective geometry. In addition to the usual statement, we also prove a variant in the presence of a symplectic form.

The Fundamental Theorem of q-Clan Geometry

The Fundamental Theorem of Algebra: a Constructive Development without Choice

Is it reasonable to do constructive mathematics without the axiom of countable choice? Serious schools of constructive mathematics all assume it one way or another, but the arguments for it are not compelling. The fundamental theorem of algebra will serve as an example of where countable choice comes into play and how to proceed in its absence. Along the way, a notion of a complete metric space...