Abstract:
We propose a generic mechanism for the formation of narrow rings in rotating systems. For this purpose we use a system of discs rotating about a common center lying well outside the discs. A discussion of this system shows that narrow rings occur, if we assume non-interacting particles. A saddle-center bifurcation is responsible for the relevant appearance of elliptic regions in phase space, that will generally assume ring shapes in the synodic frame, which will suffer a precession in the sidereal frame. Finally we discuss possible applications of this mechanism and find that it may be relevant for planetary rings as well as for semi-classical considerations.

Abstract:
In this paper, we discuss general relativistic, self-gravitating and uniformly rotating perfect fluid bodies with a toroidal topology (without central object). For the equations of state describing the fluid matter we consider polytropic as well as completely degenerate, perfect Fermi gas models. We find that the corresponding configurations possess similar properties to the homogeneous relativistic Dyson rings. On the one hand, there exists no limit to the mass for a given maximal mass-density inside the body. On the other hand, each model permits a quasistationary transition to the extreme Kerr black hole.

Abstract:
Rotating-wave approximation and its validity in multi-state quantum systems are studied through analytic approach. Their applicability is also verified from the viewpoint of generic states by the use of direct numerical integrations of the Schroedinger equation. First, we introduce an extension of the rotating-wave approximation for multi-state systems. Under an assumption that a smooth transition is induced by the optimal field, we obtain three types of analytic control fields and demonstrate their validity and deficiency for generic systems represented by random matrices. Through the comparison, we conclude that the analytic field based on our coarse-grained approach outperforms the other ones for generic quantum systems with a large number of states. Finally, the further extension of the analytic field is introduced for realistic chaotic systems and its validity is shown in banded random matrix systems.

Abstract:
A Roche model for describing uniformly rotating rings is presented and the results are compared with numerical solutions to the full problem for polytropic rings. In the thin ring limit, the surfaces of constant pressure including the surface of the ring itself are given in analytic terms, even in the mass-shedding case.

Abstract:
Let $A$ be a local Noetherian domain of Krull dimension $d$. Heinzer, Rotthaus and Sally have shown that if the generic formal fiber of $A$ has dimension $d-1$, then $A$ is birationally dominated by a one-dimensional analytically ramified local Noetherian ring having residue field finite over the residue field of $A$. We explore further this correspondence between prime ideals in the generic formal fiber and one-dimensional analytically ramified local rings. Our main focus is on the case where the analytically ramified local rings are stable, and we show that in this case the embedding dimension of the stable ring reflects the embedding dimension of a prime ideal maximal in the generic formal fiber, thus providing a measure of how far the generic formal fiber deviates from regularity. A number of characterizations of analytically ramified local stable domains are also given.

Abstract:
In this paper, we construct new solutions describing rotating black rings on Taub-NUT using the inverse-scattering method. These are five-dimensional vacuum space-times, generalising the Emparan-Reall and extremal Pomeransky-Sen'kov black rings to a Taub-NUT background space. When reduced to four dimensions in Kaluza-Klein theory, these solutions describe (possibly rotating) electrically charged black holes in superposition with a finitely separated magnetic monopole. Various properties of these solutions are studied, from both a five- and four-dimensional perspective.

Abstract:
We consider a circle of ideas involving differential algebra, local Noetherian rings, and their generic formal fibers. Connecting these ideas gives rise to what we term a "twisted" subring $R$ of a ring $S$. Each such subring $R$ arises as a pullback of a derivation taking values in an $S$-module $K$. The twisting relationship proves to be a kind of inversion of Nagata idealization: whereas idealization extends $S$ to the larger ring $S \star K$, twisting produces a subring of $S$ which behaves much like the ring $S \star K$. The rings produced in this manner exhibit pathological features, such as failing to have finite normalization, but in spite of this they are quite tractable and conceptually (if not practically) easy to locate, and in this way provide a rich but manageable source of non-standard Noetherian rings. The theory developed to support this construction involves extensive use of both Noetherian and non-Noetherian commutative ring theory, as well as differential algebra.

Abstract:
Highly accurate numerical solutions to the problem of Black Holes surrounded by uniformly rotating rings in axially symmetric, stationary spacetimes are presented. The numerical methods developed to handle the problem are discussed in some detail. Related Newtonian problems are described and numerical results provided, which show that configurations can reach an inner mass-shedding limit as the mass of the central object increases. Exemplary results for the full relativistic problem for rings of constant density are given and the deformation of the event horizon due to the presence of the ring is demonstrated. Finally, we provide an example of a system for which the angular momentum of the central Black Hole divided by the square of its mass exceeds one.

Abstract:
We discuss gravitomagnetism in connection with rotating cylindrical systems. In particular, the gravitomagnetic clock effect is investigated for the exterior vacuum field of an infinite rotating cylinder. The dependence of the clock effect on the Weyl parameters of the stationary Lewis metric is determined. We illustrate our results by means of the van Stockum spacetime.

Abstract:
Electrodynamics of rotating systems is expected to exhibit novel nonlocal features that come about when acceleration-induced nonlocality is introduced into the special relativity theory in conformity with the Bohr-Rosenfeld principle. The implications of nonlocality for the amplitude and frequency of electromagnetic radiation received by uniformly rotating observers are investigated.