Some remarks on the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians

We consider the problem of minimising the $k$th eigenvalue, $k \geq 2$, of the ($p$-)Laplacian with Robin boundary conditions with respect to all domains in $\mathbb{R}^N$ of given volume $M$. When $k=2$, we prove that the second eigenvalue of the $p$-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For $p=2$ and $k \geq 3$, we prove that in many cases a minimiser cannot be independent of the value of the constant $\alpha$ in the boundary condition, or equivalently of the volume $M$. We obtain similar results for the Laplacian with generalised Wentzell boundary conditions $\Delta u + \beta \frac{\partial u}{\partial \nu} + \gamma u = 0$.Comment: 16 page

Factors associated with changes of state of foot conformation and lameness in a flock of sheep

The aim of this research was to investigate transitions between foot conformation, lameness and footrot in sheep. Data came from one lowland flock of approximately 700 ewes studied for 18 months. Multilevel multistate analyses of transitions between good and poor foot conformation states in ewes, and lame and non-lame states in ewes and lambs were conducted. Key results were that the longer sheep had feet in good conformation, the more likely they were to stay in this state; similarly, the longer a ewe was not lame the more likely she was not to become lame. Ewes with poor foot conformation were more likely to become lame (OR: 1.83 (1.24-2.67)) and to be > 4 years (OR: 1.50(1.09-2.05)). Ewes with footrot were less likely to move to good foot conformation (OR: 0.48 (0.31-0.75)) and were more likely to become lame (OR: 3.81(2.60-5.59)). Ewes lame for > 4 days and not treated with parenteral antibacterials had a higher risk of developing (OR: 2.00 (1.08-3.61)), or remaining in (OR: 0.49 (0.29-0.95)) poor foot conformation compared with ewes never lame. Treatment of ewes lame with footrot with parenteral antibacterials increased the probability of transition from a lame to a non-lame state (OR: 1.46 (1.05-2.02)) and these ewes, even if lame for > 4 days, were not more likely to develop poor foot conformation. The risk of a ewe becoming lame increased when at least one of her offspring was lame (OR: 2.03 (1.42-2.92)) and when the prevalence of lameness in the group was ≥ 5% (OR: 1.42 (1.06-1.92)). Lambs were at increased risk of becoming lame when they were male (OR: 1.42 (1.01-2.01)), single (OR: 1.86 (1.34-2.59)) or had a lame dam or sibling (OR: 3.10 (1.81-5.32)). There were no explanatory variables associated with lambs recovering from lameness. We conclude that poor foot conformation in ewes increases the susceptibility of ewes to become lame and that this can arise from untreated footrot. Treatment of ewes lame with footrot with parenteral antibacterials leads to recovery from lameness and prevents or resolves poor foot conformation which then reduces the susceptibility to further lameness with footrot

Quantum models of classical mechanics: maximum entropy packets

In a previous paper, a project of constructing quantum models of classical properties has been started. The present paper concludes the project by turning to classical mechanics. The quantum states that maximize entropy for given averages and variances of coordinates and momenta are called ME packets. They generalize the Gaussian wave packets. A non-trivial extension of the partition-function method of probability calculus to quantum mechanics is given. Non-commutativity of quantum variables limits its usefulness. Still, the general form of the state operators of ME packets is obtained with its help. The diagonal representation of the operators is found. A general way of calculating averages that can replace the partition function method is described. Classical mechanics is reinterpreted as a statistical theory. Classical trajectories are replaced by classical ME packets. Quantum states approximate classical ones if the product of the coordinate and momentum variances is much larger than Planck constant. Thus, ME packets with large variances follow their classical counterparts better than Gaussian wave packets.Comment: 26 pages, no figure. Introduction and the section on classical limit are extended, new references added. Definitive version accepted by Found. Phy

A within farm clinical trial to compare two treatments (parenteral antibacterials and hoof trimming) for sheep lame with footrot

From observational studies, farmers who use parenteral antibacterials to promptly treat all sheep with footrot (FR) or interdigital dermatitis (ID) have a prevalence of lameness of <2% compared with a prevalence of 9% lameness reported by farmers who treat lame sheep by trimming affected feet. We tested the hypothesis that prompt treatment of sheep lame with naturally developing FR or ID with parenteral and topical antibacterials reduces the prevalence and incidence of lameness with these conditions compared with less frequent treatment with trimming of hoof horn and applying topical antibacterials. A further hypothesis was that reduction of ID and FR would improve productivity. A lowland sheep flock with 700 ewes was used to test these hypotheses in an 18-month within farm clinical trial with four groups of ewes: two intervention and two control. The duration and severity of lameness was used to categorise sheep into three weighted scores of lameness (WLS): never lame (WLS0), mildly lame/lame for <6 days (WLS1) and severely or chronically lame (WLS2). The intervention reduced the prevalence of lameness due to FR and ID in ewes and lambs and the incidence of lameness in ewes. The WLS was also significantly lower in sheep in the intervention groups. Ewes with a higher WLS were subsequently significantly more likely to have a body condition score <2.5 and to have lame lambs. Significantly more ewes lambed and successfully reared more lambs that were ready for slaughter at a younger age in the intervention versus control groups. There was an increase in the gross margin of £630/100 ewes mated in the intervention group, including the cost of treatment of £150/100 ewes mated. We conclude that prompt parenteral and topical antibacterial treatment of sheep lame with ID and FR reduced the prevalence and incidence of these infectious conditions and led to improved health, welfare and productivity

Bohmian mechanics, the quantum-classical correspondence and the classical limit: the case of the square billiard

Square billiards are quantum systems complying with the dynamical quantum-classical correspondence. Hence an initially localized wavefunction launched along a classical periodic orbit evolves along that orbit, the spreading of the quantum amplitude being controlled by the spread of the corresponding classical statistical distribution. We investigate wavepacket dynamics and compute the corresponding de Broglie-Bohm trajectories in the quantum square billiard. We also determine the trajectories and statistical distribution dynamics for the equivalent classical billiard. Individual Bohmian trajectories follow the streamlines of the probability flow and are generically non-classical. This can also hold even for short times, when the wavepacket is still localized along a classical trajectory. This generic feature of Bohmian trajectories is expected to hold in the classical limit. We further argue that in this context decoherence cannot constitute a viable solution in order to recover classicality.Comment: Figures downgraded to low resolution; To be published in Found. Phys. (2009)

Billing practices among us tobacco use treatment providers

and improving coordination between intensive therapies validated in research and "real-world" logistics.Objectives: The US Affordable Care Act (ACA) now requires almost all health insurance plans to cover tobacco use treatment (TUT), but TUT remains underutilized. Methods: We conducted an anonymous online survey of US TUT providers in 2016 regarding their billing practices. Results: Participants (n131) provided services primarily in medical and behavioral health settings and were from a variety of professions. Most provided intensive individual (>15 minutes per session) and/or group counseling. Although most reported that their organization accepted at least 1 form of insurance, only 34% reported that TUT services were billed, with about equal proportions endorsing billing under their own independent tax ID and "incident to" billing under a supervisor. Half of billers (52%) reported using at least 1 Current Procedural Terminology code. The most common codes were 99406 and 99407, but 18 unique codes were specified. Themes of qualitative responses (n101) included concern about how to initiate and sustain adequate reimbursement, and experiences with billing not being "worth" the time or effort. Conclusions: Overall, results demonstrate a need for providers, administrators, and billing managers to work collaboratively. Even with the ACA mandate, and consistent with prior reports, reimbursement rates may be inadequate for intensive counseling. Areas for advocacy include recognizing that TUT requires similar intensity, expertise, and reimbursement as other substance use disorders and chronic medical conditionsgiving Tobacco Treatment Specialists the ability to bill independentl

On Uniqueness of the Jump Process in Quantum Measurement Theory

We prove that, contrary to the standard quantum theory of continuous observation, in the formalism of Event Enhanced Quantum Theory the stochastic process generating individual sample histories of pairs (observed quantum system, observing classical apparatus) is unique. This result gives a rigorous basis to the previous heuristic argument of Blanchard and Jadczyk. Possible implications of this result are discussed.Comment: 31 pages, LaTeX, article; e-mail contact [email protected]

The contribution of previous lameness events and body condition score to the occurrence of lameness in dairy herds: a study of 2 herds

It has been demonstrated that low body condition and previous occurrence of lameness increase the risk of future lameness in dairy cows. To date the population attributable fraction (PAF), which provides an estimate of the contribution that a risk factor makes toward the total number of disease events in a population, has not been explored for lameness using longitudinal data with repeated measures. Estimation of PAF helps to identify control measures that could lead to the largest improvements on-farm. The aim of this study was to use longitudinal data to evaluate the proportion of lameness that could be avoided in 2 separate herds (2 populations) through (1) reduced recurrence of previous lameness events, (2) and moving body condition score (BCS) into more optimal ranges. Data were obtained from 2 UK dairy herds: herd A, a 200-cow herd with 8 yr of data from a total of 724 cows where lameness events were based on weekly locomotion scores (LS; 1 to 5 scale), and herd B, a 600-cow herd with data recorded over 44 mo from a total of 1,040 cows where treatment of clinical cases was used to identify lameness events. The PAF for categories of BCS were estimated using a closed equation appropriate for multiple exposure categories. Simulation models were used to explore theoretical scenarios to reflect changes in BCS and recurrence of previous lameness events in each herd. For herd A, 21.5% of the total risk periods (cow-weeks) contained a lameness event (LS 3, 4, or 5), 96% of which were repeat events and 19% were recorded with BCS 16 wk before a risk period. The median PAF estimated for changes in BCS were in the region of 4 to 11%, depending on severity of lameness. Repeated bouts of lameness made a very large contribution to the total number of lameness events. This could either be because certain cows are initially susceptible and remain susceptible, due to the increased risk associated with previous lameness events, or due to interactions with environmental factors. This area requires further research

Low body condition predisposes cattle to lameness: An 8-year study of one dairy herd

Lameness in dairy cows is a multifactorial and progressive disease with complex interactions between risk factors contributing to its occurrence. Detailed records were obtained from one United Kingdom dairy herd over an 8-yr period. Weekly locomotion scores were used to classify cows as not lame (score 1 to 2), mildly lame (score 3) and severely lame (score 4 to 5). These outcomes were used to investigate the hypothesis that low body condition score (BCS) is associated with an increased risk of lameness in dairy cows. Mixed effect multinomial logistic regression models were used to investigate the association between prior BCS and repeat lameness events during the longitudinal period of the study. Discrete time survival models were used to explore the relationship between prior BCS and first lifetime lameness events. In total, 79,565 cow weeks at risk were obtained for 724 cows. The number of lameness events was 17,114, of which 8,799 were categorized as mildly lame and 8,315 as severely lame. The median BCS was 2.25 (range, 0.75 to 4.25) and the mean body weight (BW) and age at first calving were 619.5 kg (range, 355.6 to 956.4 kg) and 25.8 mo (range, 20.5 to 37.8 mo), respectively. Subsets of the data were used in the discrete time survival models: 333 mild and 211 severe first lifetime lameness events in heifers (first lactation cows), and 81 mild and 49 severe first lifetime lameness events in cows second lactation or greater. Low BCS 3 wk before a repeated lameness event was associated with a significantly increased risk of lameness. Cows with BCS <2 were at greatest risk of mild or severe lameness, and an increased BCS above 2 was associated with a reduced risk of mild or severe lameness. Low BCS 16 or 8 wk before a first mild or severe lifetime lameness event, respectively, also had a positive association with risk of lameness in cows second lactation or greater. This provides evidence to support targeting management toward maintaining BCS to minimize the risk of lameness. Low BW (independent of BCS) and increased age at first calving above 24 mo were also associated with increased long-term risk of repeated lameness events. Overall, the model explained 62 and 60% of the variability for mild and severe lameness, respectively, highlighting the importance of these variables as risk factors and hence where management could be targeted to significantly affect reducing the risk of lameness

Lameness in dairy heifers; impacts of hoof lesions present around first calving on future lameness, milk yield and culling risk

The importance of lameness in primiparous dairy heifers is increasingly recognised. Although it is accepted that clinical lameness in any lactation increases the risk of future lameness, the impact of foot lesions during the first lactation on long-term lameness risk is less clear. This retrospective cohort study aimed to investigate the impacts of foot lesions occurring around the time of first calving in heifers on future lameness risk, daily milk yield and survival within a dairy herd. Records were obtained for 158 heifers from one UK dairy herd. Heifers were examined in 2 month blocks from 2 months pre-calving through to 4 months post-calving. Sole lesions and white line lesions were scored on a zero to 10 scale and digital dermatitis on a zero to 3 scale. Outcomes investigated were; lameness risk based on weekly locomotion scores, average daily milk yield and culling risk. Mixed effect models were used to investigate associations between maximum lesion scores and outcomes. Lesion scores in the highest score categories for claw horn lesions (sole lesions and white line lesions) in the 2 to 4 month post-calving period were associated with an increased risk of future lameness; heifers with white line lesion scores ≥3 compared with scores zero to 1 and heifers with sole lesion scores ≥4 compared with score 2, at this time point, had a predicted increased risk of future lameness of 1.6 and 2.6 respectively. Sole lesions ≥4 were also associated with a reduction in average daily milk yield of 2.68 kg. Managing heifers to reduce claw horn lesions during this time period post-calving may provide health, welfare and production benefits for the long-term future of those animals. A novel finding from the study was that mild lesion scores compared with scores zero to 1, were associated with a reduced risk of future lameness for white line lesions and sole lesions occurring in the pre-calving or 2 to 4 months post-calving periods respectively. Mild sole lesions in the pre-calving period were also associated with a reduced risk of premature culling. One hypothesis for this result is that a mild insult may result in adaptive changes to the foot leading to greater biomechanical resilience and so increased longevity