Spreading speeds in reducible multitype branching random walk

This paper gives conditions for the rightmost particle in the $n$th generation of a multitype branching random walk to have a speed, in the sense that its location divided by n converges to a constant as n goes to infinity. Furthermore, a formula for the speed is obtained in terms of the reproduction laws. The case where the collection of types is irreducible was treated long ago. In addition, the asymptotic behavior of the number in the nth generation to the right of na is obtained. The initial motive for considering the reducible case was results for a deterministic spatial population model with several types of individual discussed by Weinberger, Lewis and Li [J. Math. Biol. 55 (2007) 207-222]: the speed identified here for the branching random walk corresponds to an upper bound for the speed identified there for the deterministic model.Comment: Published in at http://dx.doi.org/10.1214/11-AAP813 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Survival, extinction and approximation of discrete-time branching random walks

We consider a general discrete-time branching random walk on a countable set X. We relate local, strong local and global survival with suitable inequalities involving the first-moment matrix M of the process. In particular we prove that, while the local behavior is characterized by M, the global behavior cannot be completely described in terms of properties involving M alone. Moreover we show that locally surviving branching random walks can be approximated by sequences of spatially confined and stochastically dominated branching random walks which eventually survive locally if the (possibly finite) state space is large enough. An analogous result can be achieved by approximating a branching random walk by a sequence of multitype contact processes and allowing a sufficiently large number of particles per site. We compare these results with the ones obtained in the continuous-time case and we give some examples and counterexamples.Comment: 32 pages, a few misprints have been correcte

Enhancing university student engagement using online multiple choice questions and answers

For many education providers, student engagement can be a major issue. Given the positive correlation between engagement and good performance, providers are continually looking for ways to engage students in the learning process. The growth of student digital literacy, the wide proliferation of online tools and the understanding of why online gaming can be addictive have combined to create a set of tools that providers can leverage to enhance engagement. One such tool is Peerwise, https://peerwise.cs.auckland.ac.nz/, an online, multiple choice question (MCQ) and answer tool in which students create questions that are answered by other students. Why use MCQs? Using MCQs tests knowledge, provides reassurance of learning, identifies gaps and makes this data available to student and provider. Students use this information to focus their time on areas requiring additional work [1], benefiting from the early feedback provided. Formative assess- ments using MCQs are beneficial in preparing students for summative testing and are appreciated and liked by students [2]. Providers can use this information to determine how the material is being received and react accordingly. Students use Peerwise to create MCQs that are answered, rated and commented on by their peers. Students’ engagement in Peerwise earns trophies for contributing regular use and for providing feedback, all of which act to stimulate further engagement, using the principles of gamification. Bournemouth University, a public university in the UK with over 18,000 students, has been embedding Peerwise in under-graduate and post-graduate units since 2014. The results experienced by Bournemouth University have been beneficial and correlate with other studies of using Peerwise [3] [4]. A statistically significant improvement was seen by one cohort of students compared to the previous year where Peerwise was not used. However, no correlation was found between Peerwise participation and a student’s unit mark. The processes followed by Bournemouth University and the advantages and disadvantages, backed by qualitative and quantitative data, will be presented so that other institutions can gain an informed view of the merits of Peerwise for their own teaching and learning environments

Comment on "A centrosome-independent role for gamma-TuRC proteins in the spindle assembly checkpoint"

Müller et al. (Reports, 27 October 2006, p. 654) showed that inhibition of the γ-tubulin ring complex (γ-TuRC) activates the spindle assembly checkpoint (SAC), which led them to suggest that γ-TuRC proteins play molecular roles in SAC activation. Because γ-TuRC inhibition leads to pleiotropic spindle defects, which are well known to activate kinetochore-derived checkpoint signaling, we believe that this conclusion is premature

Metric mechanics with nontrivial topology: Actuating irises, cylinders, and evertors.

Liquid crystal elastomers contract along their director on heating and recover on cooling, offering great potential as actuators and artificial muscles. If a flat sheet is programed with a spatially varying director pattern, then it will actuate into a curved surface, allowing the material to act as a strong machine such as a grabber or lifter. Here we study the actuation of programed annular sheets which, owing to their central hole, can sidestep constraints on area and orientation. We systematically catalog the set of developable surfaces encodable via axisymmetric director patterns and uncover several qualitatively new modes of actuation, including cylinders, irises, and everted surfaces in which the inner boundary becomes the outer boundary after actuation. We confirm our designs with a combination of experiments and numerics. Many of our actuators can reattain their initial inner or outer radius upon completing actuation, making them particularly promising, as they can avoid potentially problematic stresses in their activated state even when fixed onto a frame or pipe

Spatially configuring wrinkle pattern and multiscale surface evolution with structural confinement

© 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. Surface elastic instabilities, such as wrinkling and creasing, can enable a convenient strategy to impart reversible patterned topography to a surface. Here the classic system of a stiff layer on a soft substrate is focused, which famously produces parallel harmonic wrinkles at modest uniaxial compression that period-double repeatedly at higher compressions and ultimately evolve into deep folds and creases. By introducing micrometer-scale planar Bravais lattice holes to spatially pattern the substrate, these instabilities are guided into a wide variety of different patterns, including wrinkling in parallel bands and star shape bands, and radically reduce the threshold compression. The experimental patterns and thresholds are enabled to understand by considering a simple plane-strain model for the patterned substrate-deformation, decorated by wrinkling on the stiff surface layer. The experiments also show localized wrinkle-crease transitions at modest compression, yielding a hierarchical surface with different generations of instability mixed together. By varying the geometrical inputs, control over the stepwise evolution of surface morphologies is demonstrated. These results demonstrate considerable control over both the patterns and threshold of the surface elastic instabilities, and have relevance to many emerging applications of morphing surfaces, including in wearable/flexible electronics, biomedical systems, and optical devices

Long and short paths in uniform random recursive dags

In a uniform random recursive k-dag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If S_n is the shortest path distance from node n to the root, then we determine the constant \sigma such that S_n/log(n) tends to \sigma in probability as n tends to infinity. We also show that max_{1 \le i \le n} S_i/log(n) tends to \sigma in probability.Comment: 16 page