Shaft vibrations in turbomachinery excited by cracks

During the past years the dynamic behavior of rotors with cracks has been investigated mainly theoretically. This paper deals with the comparison of analytical and experimental results of the dynamics of a rotor with an artificial crack. The experimental results verify the crack model used in the analysis. They show the general possibility to determine a crack by extended vibration control

Modular classes revisited

We present a graded-geometric approach to modular classes of Lie algebroids and their generalizations, introducing in this setting an idea of relative modular class of a Dirac structure for a certain type of Courant algebroids, called projectable. This novel approach puts several concepts related to Poisson geometry and its generalizations in a new light and simplifies proofs. It gives, in particular, a nice geometric interpretation of modular classes of twisted-Poisson structures on Lie algebroids.Comment: 10 pages, slightly revised version to appear in Int. J. Geom. Meth. Mod. Phy

Irrational l2-invariants arising from the lamplighter group

We show that the Novikov-Shubin invariant of an element of the integral group ring of the lamplighter group Z_2 \wr Z can be irrational. This disproves a conjecture of Lott and Lueck. Furthermore we show that every positive real number is equal to the Novikov-Shubin invariant of some element of the real group ring of Z_2 \wr Z. Finally we show that the l2-Betti number of a matrix over the integral group ring of the group Z_p \wr Z, p>1, can be irrational, and so the groups Z_p \wr Z become the simplest known groups which give rise to irrational l2-Betti numbers.Comment: 26 pages, 11 figures, v4: changes suggested by a referee (including fixing the proof of Lemma 11); To appear in Groups Geom. Dy

On Lie induction and the exceptional series

Lie bialgebras occur as the principal objects in the infinitesimalization of the theory of quantum groups — the semi-classical theory. Their relationship with the quantum theory has made available some new tools that we can apply to classical questions. In this paper, we study the simple complex Lie algebras using the double-bosonization construction of Majid. This construction expresses algebraically the induction process given by adding and removing nodes in Dynkin diagrams, which we call Lie induction. We first analyze the deletion of nodes, corresponding to the restriction of adjoint representations to subalgebras. This uses a natural grading associated to each node. We give explicit calculations of the module and algebra structures in the case of the deletion of a single node from the Dynkin diagram for a simple Lie (bi-)algebra. We next consider the inverse process, namely that of adding nodes, and give some necessary conditions for the simplicity of the induced algebra. Finally, we apply these to the exceptional series of simple Lie algebras, in the context of finding obstructions to the existence of finite-dimensional simple complex algebras of types E9, F5 and G3. In particular, our methods give a new point of view on why there cannot exist such an algebra of type E9

Brackets

We review origins and main properties of the most important bracket operations appearing canonically in differential geometry and mathematical physics in the classical, as well as the supergeometric setting. The review is supplemented by a few new concepts and examples.Comment: 40 pages, minor corrections, to appear in IJGMM

An introduction to loopoids

We discuss a concept of loopoid as a non-associative generalization of (Brandt) groupoid. We introduce and study also an interesting class of more general objects which we call semiloopoids. A differential version of loopoids is intended as a framework for Lagrangian discrete mechanics.Comment: 9 pages, proceedings of LOOPS'1