Embedding Stacked Polytopes on a Polynomial-Size Grid

A stacking operation adds a $d$-simplex on top of a facet of a simplicial $d$-polytope while maintaining the convexity of the polytope. A stacked $d$-polytope is a polytope that is obtained from a $d$-simplex and a series of stacking operations. We show that for a fixed $d$ every stacked $d$-polytope with $n$ vertices can be realized with nonnegative integer coordinates. The coordinates are bounded by $O(n^{2\log(2d)})$, except for one axis, where the coordinates are bounded by $O(n^{3\log(2d)})$. The described realization can be computed with an easy algorithm. The realization of the polytopes is obtained with a lifting technique which produces an embedding on a large grid. We establish a rounding scheme that places the vertices on a sparser grid, while maintaining the convexity of the embedding.Comment: 22 pages, 10 Figure

Assessing old-age long-term care using the concepts of healthy life expectancy and care duration: the new parameter "Long-Term Care-Free Life-Expectancy (LTCF)"

Achieving old ages is also connected with prevalence of illness and long-term care. With the introduction of the statutory long-term care insurance in 1996 and the long-term care statistics in 1999 research data of about 2.3 million people receiving long-term care benefits is available. Average life expectancy can be qualitatively divided into lifetime spent in good health and lifetime spent in long-term care dependence (average care duration). In Germany women’s and men’s average care duration amount 3.6 years respectively 2.1 years.Germany, ageing, laboratories, life expectancy

Mott law as lower bound for a random walk in a random environment

We consider a random walk on the support of a stationary simple point process on $R^d$, $d\geq 2$ which satisfies a mixing condition w.r.t.the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott's law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem

Dynamical Ne K Edge and Line Variations in the X-Ray Spectrum of the Ultra-compact Binary 4U 0614+091

We observed the ultra-compact binary candidate 4U 0614+091 for a total of 200 ksec with the high-energy transmission gratings onboard the \chandra X-ray Observatory. The source is found at various intensity levels with spectral variations present. X-ray luminosities vary between 2.0$\times10^{36}$ \ergsec and 3.5$\times10^{36}$ \ergsec. Continuum variations are present at all times and spectra can be well fit with a powerlaw component, a high kT blackbody component, and a broad line component near oxygen. The spectra require adjustments to the Ne K edge and in some occasions also to the Mg K edge. The Ne K edge appears variable in terms of optical depths and morphology. The edge reveals average blue- and red-shifted values implying Doppler velocities of the order of 3500 \kms. The data show that Ne K exhibits excess column densities of up to several 10$^{18}$ cm$^{-2}$. The variability proves that the excess is intrinsic to the source. The correponding disk velocities also imply an outer disk radius of the order of $< 10^9$ cm consistent with an ultra-compact binary nature. We also detect a prominent soft emission line complex near the \oviii L$\alpha$ position which appears extremely broad and relativistic effects from near the innermost disk have to be included. Gravitationally broadened line fits also provide nearly edge-on angles of inclination between 86 and 89$^{\circ}$. The emissions appear consistent with an ionized disk with ionization parameters of the order of 10$^4$ at radii of a few 10$^7$ cm. The line wavelengths with respect to \oviiia\ are found variably blue-shifted indicating more complex inner disk dynamics.Comment: 24 pages, 8 figures, submitted to the Astrophyscial Main Journa

Bounds on the maximum multiplicity of some common geometric graphs

We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, non-weighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of n points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits {\Omega}(8.65^n) different triangulations. This improves the bound {\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by Aichholzer et al. (ii) We present a new lower bound of {\Omega}(12.00^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.62^n) non-crossing spanning cycles. (iv) We derive lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). We show that the number of shortest non-crossing tours can be exponential in n. Likewise, we show that both the number of longest non-crossing tours and the number of longest non-crossing perfect matchings can be exponential in n. Moreover, we show that there are sets of n points in convex position with an exponential number of longest non-crossing spanning trees. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(nlog n) time algorithm for computing them

Counting Carambolas

We give upper and lower bounds on the maximum and minimum number of geometric configurations of various kinds present (as subgraphs) in a triangulation of $n$ points in the plane. Configurations of interest include \emph{convex polygons}, \emph{star-shaped polygons} and \emph{monotone paths}. We also consider related problems for \emph{directed} planar straight-line graphs.Comment: update reflects journal version, to appear in Graphs and Combinatorics; 18 pages, 13 figure

Stripes on a 6-Leg Hubbard Ladder

While DMRG calculations find stripes on doped n-leg t-J ladders, little is known about the possible formation of stripes on n-leg Hubbard ladders. Here we report results for a 7x6 Hubbard model with 4 holes. We find that a stripe forms for values of U/t ranging from 6 to 20. For U/t ~ 3-4, the system exhibits the domain wall feature of a stripe, but the hole density is very broadened.Comment: 4 pages, 5 figure