Hamiltonian submanifolds of regular polytopes

We investigate polyhedral $2k$-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\it $k$-Hamiltonian} if it contains the full $k$-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called {\it super-neighborly triangulations}) we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the $d$-dimensional cross polytope. These are the "regular cases" satisfying equality in Sparla's inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of 7 copies of $S^2 \times S^2$. By this example all regular cases of $n$ vertices with $n < 20$ or, equivalently, all cases of regular $d$-polytopes with $d\leq 9$ are now decided.Comment: 26 pages, 4 figure

Combinatorial properties of the K3 surface: Simplicial blowups and slicings

The 4-dimensional abstract Kummer variety K^4 with 16 nodes leads to the K3 surface by resolving the 16 singularities. Here we present a simplicial realization of this minimal resolution. Starting with a minimal 16-vertex triangulation of K^4 we resolve its 16 isolated singularities - step by step - by simplicial blowups. As a result we obtain a 17-vertex triangulation of the standard PL K3 surface. A key step is the construction of a triangulated version of the mapping cylinder of the Hopf map from the real projective 3-space onto the 2-sphere with the minimum number of vertices. Moreover we study simplicial Morse functions and the changes of their levels between the critical points. In this way we obtain slicings through the K3 surface of various topological types.Comment: 31 pages, 3 figure

On Instability of Certain Bi-Metric and Massive-Gravity Theories

Stability about cosmological background solutions to the bi-metric Hassan-Rosen theory is studied. The results of this analysis are presented, and it is shown that a large class of cosmological backgrounds is classically unstable. This sets serious doubts on the physical viability of the Hassan-Rosen theory - and in turn also of the de Rham-Gadabaze-Tolley model, to which the mentioned theory is parent. A way to overcome this instability by means of curvature-type deformations is discussed.Comment: 4 pages, 2 figures; v2: minor changes to match PRD versio

Long-range correlated random field and random anisotropy O(N) models: A functional renormalization group study

We study the long-distance behavior of the O(N) model in the presence of random fields and random anisotropies correlated as ~1/x^{d-sigma} for large separation x using the functional renormalization group. We compute the fixed points and analyze their regions of stability within a double epsilon=d-4 and sigma expansion. We find that the long-range disorder correlator remains analytic but generates short-range disorder whose correlator develops the usual cusp. This allows us to obtain the phase diagrams in (d,sigma,N) parameter space and compute the critical exponents to first order in epsilon and sigma. We show that the standard renormalization group methods with a finite number of couplings used in previous studies of systems with long-range correlated random fields fail to capture all critical properties. We argue that our results may be relevant to the behavior of He-3A in aerogel.Comment: 8 pages, 3 figures, revtex

Productive Government Expenditure and Economic Growth

We provide a comprehensive survey of the recent literature on the link between productive government expenditure and economic growth. Starting with the seminal paper of Robert Barro (1990) we show that an understanding of the core results of the ensuing contributions can be gained from the study of their respective Euler equations. We argue that the existing literature incorporates many relevant aspects, however, policy recommen- dations tend to hinge on several knife-edge assumptions. Therefore, future research ought to focus more on idea-based endogenous growth models to check the robustness of policy recommendations. Moreover, the inclusion of hitherto unexplored types of government expenditure, e. g., on the "rule of law", would be desirable.Economic Growth, Government Expenditure, Public Goods, Fiscal Policy

Partitioning the triangles of the cross polytope into surfaces

We present a constructive proof that there exists a decomposition of the 2-skeleton of the k-dimensional cross polytope $\beta^k$ into closed surfaces of genus $g \leq 1$, each with a transitive automorphism group given by the vertex transitive $\mathbb{Z}_{2k}$-action on $\beta^k$. Furthermore we show that for each $k \equiv 1,5(6)$ the 2-skeleton of the (k-1)-simplex is a union of highly symmetric tori and M\"obius strips.Comment: 13 pages, 1 figure. Minor update. Journal-ref: Beitr. Algebra Geom. / Contributions to Algebra and Geometry, 53(2):473-486, 201

Vegetation, Ökosystemdynamik und Renaturierung von zentralasiatischen Flussauen am Beispiel des Tarim in Xinjiang, NW-China

Naturally, the floodplains of Central Asian rivers harbour riparian, so-called ‘Tugai’ forests, reeds with Phragmites australis, and shrub communities which form a mosaic depending on the variety of available ground water. In recent decades, these natural ecosystems have been strongly altered anthropogenically or even completely destroyed. In order to restore those ecosystems, knowledge on vegetation, ecosystem dynamics, and natural regeneration processes is essential. In our study, we present results of ecological investigations at the Tarim River. We gathered comprehensive data on soil, vegetation, forest stand age, tree vitality, river course dynamics, and land use and brought it to the landscape level. Thus, recommendations are derived for the maintenance of these floodplain ecosystems, in particular with regard to their biological diversity.Entlang der Flussauen Zentralasiens findet sich natürlicherweise ein Mosaik aus Auenwäldern (‚Tugai-Wäldern‘), Schilfröhrichten mit Phragmites australis und Sträuchern, welches von der Verfügbarkeit des Grundwassers abhängt. In den vergangenen Jahrzehnten wurden diese natürlichen Ökosysteme durch den Menschen stark beeinträchtigt bis hin zu völlig zerstört. Um diese Ökosysteme wiederherzustellen, sind genaue Kenntnisse über die Vegetation, die Ökosystemdynamik und natürliche Regenerationsprozesse unabdingbar. In der vorliegenden Studie berichten wir über Ergebnisse unserer langjährigen ökologischen Untersuchungen am Tarim-Fluss. Diese umfassen Untersuchungen des Bodens, der Vegetation, der Altersstruktur und Vitalität der Tugai-Wälder, der Flusslaufdynamik und der Landnutzung, welche auf Landschaftsebene ausgewertet wurden. Auf dieser Grundlage leiten wir Empfehlungen zum dauerhaften Erhalt dieser Flussauenökosystem ab unter besonderer Berücksichtigung der biologischen Vielfalt