Let $\gamma(G)$ and $${\gamma _ \circ }(G)$$ denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if G is an n-vertex graph of minimum degree at least d, then

$$\begin{equation}\gamma(G) \leq \frac{n}{d}(\log d + 1).\end{equation}$$

In this paper the main result is that if G is any n-vertex d-regular graph of girth at least five, then

$$\begin{equation}\gamma_(G) \leq \frac{n}{d}(\log d + c)\end{equation}$$

$d \rightarrow \infty$

$$\begin{equation}\gamma_(G) \sim \frac{n}{d}\log d.\end{equation}$$

Furthermore, if G is a disjoint union of ${n}/{(2d)}$ complete bipartite graphs $K_{d,d}$, then ${\gamma_\circ}(G) = \frac{n}{2}$. We also prove that there are n-vertex graphs G of minimum degree d and whose maximum degree grows not much faster than d log d such that ${\gamma_\circ}(G) \sim {n}/{2}$ as $d \rightarrow \infty$. Therefore both the girth and regularity conditions are required for the main result.

Personalized PageRank has found many uses in not only the ranking of webpages, but also algorithmic design, due to its ability to capture certain geometric properties of networks. In this paper, we study the diffusion of PageRank: how varying the jumping (or teleportation) constant affects PageRank values. To this end, we prove a gradient estimate for PageRank, akin to the Li–Yau inequality for positive solutions to the heat equation (for manifolds, with later versions adapted to graphs).

Objective: To investigate the effects of methylphenidate on long-term executive and neuropsychological functioning in children with attention problems following TBI, as well as the relationship between methylphenidate associated changes in lab-based neuropsychological measures of attentional control, processing speed, and executive functioning and parent- or self-report measures of everyday executive functioning. Method: 26 children aged 6–17 years, who were hospitalized for moderate-to-severe blunt head trauma 6 or more months previously, were recruited from a large children’s hospital medical center. Participants were randomized into a double-masked, placebo-controlled cross-over clinical trial. Participants completed a comprehensive neuropsychological battery and parent- and self-report ratings of everyday executive functioning at baseline, and at 4 weeks and 8 weeks following upward titration of medication to an optimal dose or while administered a placebo. Results: Methylphenidate was associated with significant improvements in processing speed, sustained attention, and both lab-based and everyday executive functioning. Significant treatment-by-period interactions were found on a task of sustained attention. Participants who were randomized to the methylphenidate condition for the first treatment period demonstrated random or erratic responding, with slower and more variable response times when given placebo during the second period. Conclusion: Results indicate that methylphenidate treatment is associated with positive outcomes in processing speed, sustained attention, and both lab-based and everyday measures of executive functioning compared to placebo group. Additionally, results suggest sustained attention worsens when discontinuing medication. (JINS, 2019, 25, 740–749)

Older care home residents are excluded from the sexual imaginary. Based on a consultative study involving interviews with three residents, three female spouses of residents and two focus groups of care home staff (N = 16), making an overall sample of 22 study participants, we address the neglected subject of older residents' sexuality and intimacy needs. Using thematic analysis, we highlight how residents’ and spouses’ accounts of sexuality and intimacy can reflect an ageist erotophobia occurring within conditions of panoptical control that help construct residents as post-sexual. However, not all accounts contributed to making older residents’ sexuality appear invisible or pathological. Some stories indicated recuperation of identities and the normalisation of relationships with radically changed individuals, e.g. because of a dementia. We also examine care home staff accounts of the discursive obstacles that frustrate meeting residents’ needs connected with sexuality and intimacy. Simultaneously, we explore staffs’ creative responses to dilemmas which indicate approaches to sexuality driven more by observed needs than erotophobic anxiety and governance, as well as panoptical surveillance.

Sexuality and intimacy in care homes for older people are overshadowed by concern with prolonging physical and/or psychological autonomy. When sexuality and intimacy have been addressed in scholarship, this can reflect a sexological focus concerned with how to continue sexual activity with reduced capacity. We review the (Anglophone) academic and practitioner literatures bearing on sexuality and intimacy in relation to older care home residents (though much of this applies to older people generally). We highlight how ageism (or ageist erotophobia), which defines older people as post-sexual, restricts opportunities for the expression of sexuality and intimacy. In doing so, we draw attention to more critical writing that recognises constraints on sexuality and intimacy and indicates solutions to some of the problems identified. We also highlight problems faced by lesbian, gay, bisexual and trans (LGB&T) residents who are doubly excluded from sexual/intimate citizenship because of ageism combined with the heterosexual assumption. Older LGB&T residents/individuals can feel obliged to deny or disguise their identity. We conclude by outlining an agenda for research based on more sociologically informed practitioner-led work.

Prospective cohort studies have shown inverse associations between fibre intake and CVD, possibly mediated by blood pressure (BP). However, little is known about the impact of types of fibre on BP. We examined cross-sectional associations with BP of total, insoluble and soluble fibre intakes. Data were used from the INTERnational study on MAcro/micronutrients and blood Pressure (INTERMAP) study, including 2195 men and women aged between 40 and 59 years from the USA. During four visits, eight BP, four 24 h dietary recalls and two 24 h urine samples were collected. Linear regression models adjusted for lifestyle and dietary confounders to estimate BP differences per 2 sd higher intakes of total and individual types of fibre were calculated. After multivariable adjustment, total fibre intake higher by 6·8 g/4184 kJ (6·8 g/1000 kcal) was associated with a 1·69 mmHg lower systolic blood pressure (SBP; 95 % CI −2·97, −0·41) and attenuated to −1·01 mmHg (95 % CI −2·35, 0·34) after adjustment for urinary K. Insoluble fibre intake higher by 4·6 g/4184 kJ (4·6 g/1000 kcal) was associated with a 1·81 mmHg lower SBP (95 % CI −3·65, 0·04), additionally adjusted for soluble fibre and urinary K excretion, whereas soluble fibre was not associated with BP. Raw fruit was the main source of total and insoluble fibre, followed by whole grains and vegetables. In conclusion, higher intakes of fibre, especially insoluble, may contribute to lower BP, independent of nutrients associated with higher intakes of fibre-rich foods.

Microlensing searches for planets are sensitive to small, cold exoplanets from 1–6 AU from their host stars and therefore probe an important part of parameter space. Other techniques would require many years of observations, often from space, to detect similar systems. Microlensing events can be characterised from only ground-based observations over a relatively short (≤100d) timescales. LCOGT and SUPA/St Andrews are building a robotic global network of telescopes that will be well suited to follow these events. Here we present preliminary results of the Galactic Bulge observing season 2010 March–October.

The dynamics of strongly driven phase transitions at surfaces are studied by ultra-fast time-resolved reflection high energy electron diffraction. The surfaces are excited by an intense fs-laser pulse (pump) and probed by an ultra-short electron pulse with variable time delay. The order-disorder phase transition from a c(4×2) to a (2×1) of the bare Si(001) surface shows a transient decrease of the intensity of the c(4×2) spots which recovers on a time scale of a few hundred picoseconds indicating the excitation of the phase transition. On Si(111) a monolayer of Indium induces a (4×1) reconstruction which undergoes a Peierls like phase transition to a (8ד2”) reconstruction below 100 K. Upon laser excitation at a temperature of 40 K the phase transition was strongly driven. The (8ד2”)-diffraction spots instantaneously disappears, while the intensity of the (4×1)-spots increases. This increase of the (4×1) spot intensity excludes an explanation by the Debye-Waller-Effect and is evidence for a true structural phase transition at a surface.

In natural convection in a porous medium heated from below, the convective flow in two dimensions becomes unsteady above a certain critical Rayleigh number and exhibits a fluctuating or oscillatory behaviour (depending on the confinement in the horizontal dimension). This fluctuating behaviour is due to a combination of the instability of the thermal boundary layers at horizontal boundaries together with a ‘triggering’ effect of earlier disturbances. The point of the origin of the instability of the thermal boundary layer appears to play a dominant role in determining the regularity of the fluctuating flow. This numerical study investigates the importance of this point of evolution and concludes that there may exist more than one oscillatory mode of convection, depending on its position. The investigation focuses on the symmetry of the flow and demonstrates that with stable and accurate numerical schemes, an artificial symmetry may be imposed in the absence of realistic physical noise. If an initially symmetric perturbation is imposed the flow retains an essentially symmetric flow pattern with a high degree of regularity in the oscillatory behaviour. The imposition of an asymmetric perturbation results in a degradation of regularity. The appearance of the symmetric, regularly oscillatory flow is characterized by a symmetric (and stationary) arrangement of the points of origin of the instability of the upper and lower thermal boundary layers; in the case of the irregular oscillations the points of origin are not symmetric and their locations are not fixed.

We describe a new method of orbital light curve decomposition which is applicable to long-term photometry of interacting binaries. This method determines the orbital dependence (including eclipses) of the mean light, secular variability and RMS flickering. We identify the true line of centres of RW Tri and find that the accretion disc is a location of secular variability and a source of flickering.