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

Local Lie algebra determines base manifold

It is proven that a local Lie algebra in the sense of A. A. Kirillov determines the base manifold up to a diffeomorphism provided the anchor map is nowhere-vanishing. In particular, the Lie algebras of nowhere-vanishing Poisson or Jacobi brackets determine manifolds. This result has been proven for different types of differentiability: smooth, real-analytic, and holomorphic.Comment: 13 pages, minor corrections, to appear in "From Geometry to Quantum Mechanics, in Honor of Hideki Omori", Y.Maeda et al., eds., Progress in Math. 252, Birkhaeuser 200

A note on the longest common Abelian factor problem

Abelian string matching problems are becoming an object of considerable interest in last years. Very recently, Alatabbi et al. \cite{AILR2015} presented the first solution for the longest common Abelian factor problem for a pair of strings, reaching $O(\sigma n^2)$ time with $O(\sigma n \log n)$ bits of space, where $n$ is the length of the strings and $\sigma$ is the alphabet size. In this note we show how the time complexity can be preserved while the space is reduced by a factor of $\sigma$, and then how the time complexity can be improved, if the alphabet is not too small, when superlinear space is allowed.Comment: v3 is vastly different to the previous on

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

Isomorphisms of algebras of smooth functions revisited

It is proved that isomorphisms between algebras of smooth functions on Hausdorff smooth manifolds are implemented by diffeomorphisms. It is not required that manifolds are second countable nor paracompact. This solves a problem stated by A. Wienstein. Some related results are discussed as well.Comment: 6 pages, minor changes, the final version to appear in Archiv der Mathemati

Group ring elements with large spectral density

Given an arbitrary d>0 we construct a group G and a group ring element S in Z[G] such that the spectral measure mu of S has the property that mu((0,eps)) > C/|log(eps)|^(1+d) for small eps. In particular the Novikov-Shubin invariant of any such S is 0. The constructed examples show that the best known upper bounds on mu((0,eps)) are not far from being optimal.Comment: 19 pages, v3: Changes suggested by a referee; Essentially this is the version published in Math. An

Courant-Nijenhuis tensors and generalized geometries

Nijenhuis tensors $N$ on Courant algebroids compatible with the pairing are studied. This compatibility condition turns out to be of the form $N+N^*=aI$ for irreducible Courant algebroids, in particular for the extended tangent bundles $TM\oplus T^*M$. It is proved that compatible Nijenhuis tensors on irreducible Courant algebroids must satisfy quadratic relations $N^2-aN+bI=0$, so that the corresponding hierarchy is very poor. The particular case $N^2=-I$ is associated with Hitchin's generalized geometries and the cases $N^2=I$ and $N^2=0$ -- to other "generalized geometries". These concepts find a natural description in terms of supersymplectic Poisson brackets on graded supermanifolds.Comment: 10 page

Vanishing of l^2-cohomology as a computational problem

We show that it is impossible to algorithmically decide if the l^2-cohomology of the universal cover of a finite CW complex is trivial, even if we only consider complexes whose fundamental group is equal to the elementary amenable group (Z_2 \wr Z)^3. A corollary of the proof is that there is no algorithm which decides if an element of the integral group ring of the group (\Z_2 \wr Z)^4 is a zero-divisor. On the other hand, we show, assuming some standard conjectures, that such an algorithm exists for the integral group ring of any group with a decidable word problem and a bound on the sizes of finite subgroups.Comment: 18 pages; rewritten following referee's reports; to appear in Bulletin of LM

Graded cluster algebras

In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We transfer a definition of Gekhtman, Shapiro and Vainshtein to the algebraic setting, yielding the notion of a multi-graded cluster algebra. We then study gradings for finite type cluster algebras without coefficients, giving a full classification. Translating the definition suitably again, we obtain a notion of multi-grading for (generalised) cluster categories. This setting allows us to prove additional properties of graded cluster algebras in a wider range of cases. We also obtain interesting combinatorics---namely tropical frieze patterns---on the Auslander--Reiten quivers of the categories.Comment: 23 pages, 6 figures. v2: Substantially revised with additional results. New section on graded (generalised) cluster categories. v3: added Prop. 5.5 on relationship with Grothendieck group of cluster categor

My Mark Twain: Old Man River

Flowing across his pages, the Mississippi River inextricably winds itself through Mark Twain’s canon. Therefore, it comes as no surprise that my image of Clemens, my Mark Twain, is as a personification of his beloved river. Twain draws his readers to the water’s edge, seduces readers to stare into his depths, and reflects the achievements and failings of humanity. Furthermore, like the Mississippi River, Twain embeds himself in the American psyche