Minimizing the Continuous Diameter when Augmenting Paths and Cycles with Shortcuts

We seek to augment a geometric network in the Euclidean plane with shortcuts to minimize its continuous diameter, i.e., the largest network distance between any two points on the augmented network. Unlike in the discrete setting where a shortcut connects two vertices and the diameter is measured between vertices, we take all points along the edges of the network into account when placing a shortcut and when measuring distances in the augmented network. We study this network augmentation problem for paths and cycles. For paths, we determine an optimal shortcut in linear time. For cycles, we show that a single shortcut never decreases the continuous diameter and that two shortcuts always suffice to reduce the continuous diameter. Furthermore, we characterize optimal pairs of shortcuts for convex and non-convex cycles. Finally, we develop a linear time algorithm that produces an optimal pair of shortcuts for convex cycles. Apart from the algorithms, our results extend to rectifiable curves. Our work reveals some of the underlying challenges that must be overcome when addressing the discrete version of this network augmentation problem, where we minimize the discrete diameter of a network with shortcuts that connect only vertices

A Note on the Unsolvability of the Weighted Region Shortest Path Problem

Let S be a subdivision of the plane into polygonal regions, where each region has an associated positive weight. The weighted region shortest path problem is to determine a shortest path in S between two points s, t in R^2, where the distances are measured according to the weighted Euclidean metric-the length of a path is defined to be the weighted sum of (Euclidean) lengths of the sub-paths within each region. We show that this problem cannot be solved in the Algebraic Computation Model over the Rational Numbers (ACMQ). In the ACMQ, one can compute exactly any number that can be obtained from the rationals Q by applying a finite number of operations from +, -, \times, \div, \sqrt[k]{}, for any integer k >= 2. Our proof uses Galois theory and is based on Bajaj's technique.Comment: 6 pages, 1 figur

Bioactive coatings of glass-ceramics on metals

To combine the mechanical properties of high-strength base metals with the biological properties of bioactive materials, coatings of BIOVERIT® glasses and glass-ceramics on CoCr alloys and titanium have been produced. Different kinds of coating processes have been used: dipping, sputtering, plasma spraying, sintering and pasting. Dipping and pasting seem not to be suitable to produce layers because of cracks and low adhesion strength of the coatings (dipping) and the limitations in relation to the implant shape and the thickness of the layers (pasting). Using sputter techniques it is possible to produce dense layers ( < 10µim) with a high adhesion strength. Plasma spraying resuhed in layers with a thickness of 50 to 300µim, but during the plasma spraying process there is a partial evaporation of the highly volatile glass components. Sintering processes are very suitable to produce layers with a high adhesion strength; these layers are long term stable in simulated body fluid

Wavelength dependent ac-Stark shift of the 1S0 - 3P1 transition at 657 nm in Ca

We have measured the ac-Stark shift of the 4s2 1S0 - 4s4p 3P1 line in 40Ca for perturbing laser wavelengths between 780 nm and 1064 nm with a time domain Ramsey-Borde atom interferometer. We found a zero crossing of the shift for the mS = 0 - mP = 0 transition and \sigma polarized perturbation at 800.8(22) nm. The data was analyzed by a model deriving the energy shift from known transition wavelengths and strengths. To fit our data, we adjusted the Einstein A coefficients of the 4s3d 3D - 4s4p 3P and 4s5s 3S - 4s4p 3P fine structure multiplets. With these we can predict vanishing ac-Stark shifts for the 1S0 m = 0 - 3P1 m = 1 transition and \sigma- light at 983(12) nm and at 735.5(20) nm for the transition to the 3P0 level.Comment: 8 pages, 5 figures, 2 table

Effectiveness of a low‐dose mindfulness‐based intervention for alleviating distress in young unemployed adults

While the effectiveness of mindfulness‐based interventions (MBIs) with respect to distress has been widely researched, unemployed individuals, who often suffer from high levels of distress, have largely been neglected in MBI research. The present study aimed to investigate the effects of a low‐dose MBI on distress in a sample of young unemployed adults. The sample included 239 young unemployed adults enrolled for a 6‐week long employability‐related training camp. Participants were allocated into an intervention group that received weekly 1‐hour mindfulness training over 4 weeks, and a control group. Dispositional mindfulness, distress and well‐being were assessed in the entire sample prior to the start and upon completion of the mindfulness training. A mixed model ANCOVA showed that distress was inversely and significantly predicted by baseline levels of mindfulness and well‐being. After accounting for the baseline levels of mindfulness and well‐being, a significant effect of the mindfulness intervention was evident. This result shows that a low‐dose MBI can decrease distress in a sample of young unemployed adults and its effectiveness is positively associated with initial levels of dispositional mindfulness and well‐being

Network Farthest-Point Diagrams

Consider the continuum of points along the edges of a network, i.e., an undirected graph with positive edge weights. We measure distance between these points in terms of the shortest path distance along the network, known as the network distance. Within this metric space, we study farthest points. We introduce network farthest-point diagrams, which capture how the farthest points---and the distance to them---change as we traverse the network. We preprocess a network G such that, when given a query point q on G, we can quickly determine the farthest point(s) from q in G as well as the farthest distance from q in G. Furthermore, we introduce a data structure supporting queries for the parts of the network that are farther away from q than some threshold R > 0, where R is part of the query. We also introduce the minimum eccentricity feed-link problem defined as follows. Given a network G with geometric edge weights and a point p that is not on G, connect p to a point q on G with a straight line segment pq, called a feed-link, such that the largest network distance from p to any point in the resulting network is minimized. We solve the minimum eccentricity feed-link problem using eccentricity diagrams. In addition, we provide a data structure for the query version, where the network G is fixed and a query consists of the point p.Comment: A preliminary version of this work was presented at the 24th Canadian Conference on Computational Geometr