Strong inapproximability of the shortest reset word

The \v{C}ern\'y conjecture states that every $n$-state synchronizing automaton has a reset word of length at most $(n-1)^2$. We study the hardness of finding short reset words. It is known that the exact version of the problem, i.e., finding the shortest reset word, is NP-hard and coNP-hard, and complete for the DP class, and that approximating the length of the shortest reset word within a factor of $O(\log n)$ is NP-hard [Gerbush and Heeringa, CIAA'10], even for the binary alphabet [Berlinkov, DLT'13]. We significantly improve on these results by showing that, for every $\epsilon>0$, it is NP-hard to approximate the length of the shortest reset word within a factor of $n^{1-\epsilon}$. This is essentially tight since a simple $O(n)$-approximation algorithm exists.Comment: extended abstract to appear in MFCS 201

Early Term Effects of rhBMP-2 on Pedicle Screw Fixation in a Sheep Model: Histomorphometric and Biomechanical Analyses

Background: The effects of recombinant human bone morphogenetic protein-2 (rhBMP-2) on pedicle screw pullout force and its potential to improve spinal fixation have not previously been investigated. rhBMP-2 on an absorbable collagen sponge (ACS) carrier was delivered in and around cannulated and fenestrated pedicle screws in a sheep lumbar spine instability model. Two control groups (empty screw and ACS with buffer) were also evaluated. We hypothesized that rhBMP-2 could stimulate bone growth in and around the cannulated and fenestrated pedicle screws to improve early bone purchase. Methods: Eight skeletally mature sheep underwent destabilizing laminectomies at L2–L3 and L4–L5 followed by stabilization with pedicle screw and rod constructs. An ACS carrier was used to deliver 0.15 mg of rhBMP-2 within and around the cannulated and fenestrated titanium pedicle screws. Biomechanics and histomorphometry were used to evaluate the early term results at 6 and 12 postoperative weeks. Results: rhBMP-2 was unable to improve bony purchase of the cannulated and fenestrated pedicle screws compared to both control groups. Although rhBMP-2 groups had pullout forces that were less than both control groups, both rhBMP-2 groups had pullout force values exceeding 2,000 N, which was comparable to previously published results for unmodified pedicle screws. Significant differences in the percentages of bone in peri-screw tissues was not observed amongst the four treatment groups. Microradiography and quantitative histomorphometry showed that at 6 weeks, rhBMP-2 induced peri-screw remodeling regions containing peri-implant bone which was hypodense with respect to surrounding native trabeculae. A moderate correlation between biomechanical pullout variables and histomorphometry data was observed. Conclusions: The design of the cannulated and fenestrated pedicle screw was able to facilitate new bone formation to achieve high pullout forces. However, delivery of rhBMP-2 should be carefully controlled to prevent excessive bone remodeling which could cause early screw loosening

Growth kinetics effects on self-assembled InAs/InP quantum dots

A systematic manipulation of the morphology and the optical emission properties of MOVPE grown ensembles of InAs/InP quantum dots is demonstrated by changing the growth kinetics parameters. Under non-equilibrium conditions of a comparatively higher growth rate and low growth temperature, the quantum dot density, their average size and hence the peak emission wavelength can be tuned by changing efficiency of the surface diffusion (determined by the growth temperature) relative to the growth flux. We further observe that the distribution of quantum dot heights, for samples grown under varying conditions, if normalized to the mean height, can be nearly collapsed onto a single Gaussian curve.Comment: 2 figure

Implementation of higher-order absorbing boundary conditions for the Einstein equations

We present an implementation of absorbing boundary conditions for the Einstein equations based on the recent work of Buchman and Sarbach. In this paper, we assume that spacetime may be linearized about Minkowski space close to the outer boundary, which is taken to be a coordinate sphere. We reformulate the boundary conditions as conditions on the gauge-invariant Regge-Wheeler-Zerilli scalars. Higher-order radial derivatives are eliminated by rewriting the boundary conditions as a system of ODEs for a set of auxiliary variables intrinsic to the boundary. From these we construct boundary data for a set of well-posed constraint-preserving boundary conditions for the Einstein equations in a first-order generalized harmonic formulation. This construction has direct applications to outer boundary conditions in simulations of isolated systems (e.g., binary black holes) as well as to the problem of Cauchy-perturbative matching. As a test problem for our numerical implementation, we consider linearized multipolar gravitational waves in TT gauge, with angular momentum numbers l=2 (Teukolsky waves), 3 and 4. We demonstrate that the perfectly absorbing boundary condition B_L of order L=l yields no spurious reflections to linear order in perturbation theory. This is in contrast to the lower-order absorbing boundary conditions B_L with L<l, which include the widely used freezing-Psi_0 boundary condition that imposes the vanishing of the Newman-Penrose scalar Psi_0.Comment: 25 pages, 9 figures. Minor clarifications. Final version to appear in Class. Quantum Grav