In this paper, we describe an example of a hyperkähler quotient of a Banach manifold by a Banach Lie group. Although the initial manifold is not diffeomorphic to a Hilbert manifold (not even to a manifold modelled on a reflexive Banach space), the quotient space obtained is a Hilbert manifold, which can furthermore be identified either with the cotangent space of a connected component Gr res of...

For the case of a first-class constrained system with equivariant momentum map, we study the conditions under which the double process of reducing to the constraint surface and dividing out by the group of gauge transformations G is equivalent to the single process of dividing out the initial phase space by the complexification GC of G. For the particular case of a phase space action that is th...

We want to describe a procedure for enlarging real vector spaces to complex vector spaces in a natural way. For instance, the natural complex analogues of Rn, Mn(R), and R[X] are Cn, Mn(C) and C[X]. Why do we want to complexify real vector spaces? One reason is related to solving equations. If we want to prove theorems about real solutions to a system of real linear equations or a system of rea...

Although the growth of complexity during evolution seems obvious to most observers, it has recently been questioned whether such increase objectively exists. The present paper tries to clarify the issue by analysing the concept of complexity as a combination of variety and dependency. It is argued that variation and selection automatically produce differentiation (variety) and integration (depe...

Two major goals in machine learning are the discovery of complex multidimensional solutions and continual improvement of existing solutions. In this paper, we argue that complexification, i.e. the incremental elaboration of solutions through adding new structure, achieves both these goals. We demonstrate the power of complexification through the NeuroEvolution of Augmenting Topologies (NEAT) me...

In this paper we study weighted versions of Fourier algebras compact quantum groups. We focus on the spectral aspects these Banach in two different ways. first investigate their Gelfand spectrum, which shows a connection to maximal classical closed subgroup and its complexification. Secondly, specific finite dimensional representations coming from complexification underlying group. demonstrate ...