Dynamics on noncommutative spaces has long been expected to solve the problem of infinities in quantum field theory. While this hope remains unconfirmed to date, surprising effects emerged, testing some of the limits of present theories. Noncommutative theories enlarge by necessity fundamental concepts like gauge invariance, (non)locality, causality, or dimensional reduction. This makes them an excellent theoretical laboratory, in which fundamental geometric concepts and quantization can be extended in a new but controllable and often quite suggestive set-up, situated ''just beyond'' the current paradigm. On the more practical side, noncommutativity offers some unexpected technical solutions, for instance it provides a natural discretization of space~-- which takes care of \(classical\) field theory divergences~-- and a simple way to put fermions on a lattice. Noncommutativity appears in two different forms, called here ''particle noncommutativity'' and ''field noncommutativity''. This work reviews the case of ''particle noncommutativity'' and then shows how fields are associated with nonconstant commutators.

Dynamics on noncommutative spaces has long been expected to solve the problem of infinities in quantum field theory. While this hope remains unconfirmed to date, surprising effects emerged, testing some of the limits of present theories. Noncommutative theories enlarge by necessity fundamental concepts like gauge invariance, (non)locality, causality, or dimensional reduction. This makes them an excellent theoretical laboratory, in which fundamental geometric concepts and quantization can be extended in a new but controllable and often quite suggestive set-up, situated ''just beyond'' the current paradigm. On the more practical side, noncommutativity offers some unexpected technical solutions, for instance it provides a natural discretization of space-- which takes care of \(classical\) field theory divergences-- and a simple way to put fermions on a lattice. This second part will review the drastic implications of ''field noncommutativity''.

The evolution of the semimajor axes of the planets of the solar system is calculated from linear combinations of 434 distinct frequencies showing that the semimajor axes are bounded. Those of the giant planets vary by less than $\pm0.1\%$. Within four million years, the semimajor axes of Mars and Earth change by less than $\pm2\%$, that of Venus varies by less than $\pm5\%$ and that of Mercury ranges from $-22\%$ to $13\%$ of its average. Since the semimajor axes as well as the other orbital elements of the planets are bounded, it is concluded that the solar planetary system is stable.

The paper treats pseudo-Riemannian spaces admitting generalized \(\varphi\)(Ric)-vector fields. We investigate the fundamental equations governing the existence of such vector fields, which are expressed as closed linear partial equations of the Cauchy type. In particular, we examine these equations within the context of constant curvature and Kähler spaces.

We prove that a commutation relation of the electric flux and the magnetic flux operators is proportional to the crossing number of the surfaces defining the fluxes. To prove it we use Dirac's prescription for quantization of constraint system and de Rham's current forms which are geometric generalizations of so-called delta function.