Long- and short-time asymptotics of the first-passage time of the Ornstein-Uhlenbeck and other mean-reverting processes

The first-passage problem of the Ornstein-Uhlenbeck process to a boundary is a long-standing problem with no known closed-form solution except in specific cases. Taking this as a starting-point, and extending to a general mean-reverting process, we investigate the long- and short-time asymptotics using a combination of Hopf-Cole and Laplace transform techniques. As a result we are able to give a single formula that is correct in both limits, as well as being exact in certain special cases. We demonstrate the results using a variety of other models

Analytical approximation to the multidimensional Fokker--Planck equation with steady state

The Fokker--Planck equation is a key ingredient of many models in physics, and related subjects, and arises in a diverse array of settings. Analytical solutions are limited to special cases, and resorting to numerical simulation is often the only route available; in high dimensions, or for parametric studies, this can become unwieldy. Using asymptotic techniques, that draw upon the known Ornstein--Uhlenbeck (OU) case, we consider a mean-reverting system and obtain its representation as a product of terms, representing short-term, long-term, and medium-term behaviour. A further reduction yields a simple explicit formula, both intuitive in terms of its physical origin and fast to evaluate. We illustrate a breadth of cases, some of which are `far' from the OU model, such as double-well potentials, and even then, perhaps surprisingly, the approximation still gives very good results when compared with numerical simulations. Both one- and two-dimensional examples are considered.Comment: Updated version as publishe

Patient experiences of an ankle fracture and the most important factors in their recovery : a qualitative interview study

Objective The objective of this qualitative research study is to explore patient experiences of ankle fracture and the factors most important to them in recovery. Design Semistructured interviews exploring patient experiences of ankle fracture recovery at 16–23 weeks following injury. Interviews followed a topic guide and were recorded with an encrypted audio recorder and then transcribed verbatim. Thematic content analysis was used to identify themes in the data. Setting Individuals were recruited from a sample of participants of a UK-based clinical trial of immobilisation methods for ankle fracture (ISRCTN15537280 at the pre-results stage at time of writing). Interviews were conducted at the participants’ own homes or on a university campus setting. Participants A purposive sample was used to account for key variables of age, gender and fracture management. Participants recruited from the clinical trial sample were adults aged 18 years or over with a closed ankle fracture. Results Ten participants were interviewed, five of whom were female and six of whom needed an operation to fix their ankle fracture. The age range of participants was 21–75 years with a mean of 51.6 years. Eight themes emerged from the data during analysis; mobility, loss of independence, healthcare, psychological effects, social and family life, ankle symptoms, sleep disturbance and fatigue, and activities of daily living. Factors of importance to participants included regaining their independence, sleep quality and quantity, ability to drive, ability to walk without walking aids or weight-bearing restrictions, and radiological union. Conclusions The results of this research demonstrates the extensive impact of ankle fracture on individuals’ lives, including social and family life, sleep, their sense of independence and psychological well-being. The results of this study will enable an increased understanding of the factors of relevance to individuals with ankle fracture, allowing collection of appropriate outcomes in clinical studies for this condition. Ultimately these results will help formulate appropriate patient-centred rehabilitation plans for these patients

An exactly solvable self-convolutive recurrence

We consider a self-convolutive recurrence whose solution is the sequence of coefficients in the asymptotic expansion of the logarithmic derivative of the confluent hypergeometic function $U(a,b,z)$. By application of the Hilbert transform we convert this expression into an explicit, non-recursive solution in which the $n$th coefficient is expressed as the $(n-1)$th moment of a measure, and also as the trace of the $(n-1)$th iterate of a linear operator. Applications of these sequences, and hence of the explicit solution provided, are found in quantum field theory as the number of Feynman diagrams of a certain type and order, in Brownian motion theory, and in combinatorics

Effect of carrageenan on the non-specific resistance of mice to injected syngeneic tumour cells, alone or in mixtures.

The mechanisms of non-specific resistance to syngeneic methylcholanthrene-induced fibrosarcomas of mice were investigated. Results showed that a small tumour graft of 0.05 X 10(5) cells is greatly enhanced in growth when admixed with large numbers of cell fragments, killed cells or viable non-replicating cells. The enhancement of small tumour grafts in cell mixtures was found to be non-specific. Carrageenan, a known anti-macrophage agent, significantly increased tumour growth in normal mice. However, it did not enhance the increased tumour growth of 0.05 X 10(5) cells mixed with 10(6) viable, non-replicating mitomycin C-treated tumour cells. The latter observation indicates that carrageenan and admixed cells interfere with the same tumour-inhibitory mechanism and therefore cannot produce additive effects. The results give support to the concept of a non-specific macrophage "surveillance" system which appears crucial in controlling tumour growth, since it determines the establishment of small numbers of tumour cells while they can still be easily destroyed

The effect of free-stream turbulence on heat transfer to a strongly accelerated turbulent boundary layer

Free-stream turbulence effects on heat transfer to strongly accelerated turbulent boundary laye

Genome-wide association study for calving performance using high-density genotypes in dairy and beef cattle

peer-reviewedBackground Calving difficulty and perinatal mortality are prevalent in modern-day cattle production systems. It is well-established that there is a genetic component to both traits, yet little is known about their underlying genomic architecture, particularly in beef breeds. Therefore, we performed a genome-wide association study using high-density genotypes to elucidate the genomic architecture of these traits and to identify regions of the bovine genome associated with them. Results Genomic regions associated with calving difficulty (direct and maternal) and perinatal mortality were detected using two statistical approaches: (1) single-SNP (single nucleotide polymorphism) regression and (2) a Bayesian approach. Data included high-density genotypes on 770 Holstein-Friesian, 927 Charolais and 963 Limousin bulls. Several novel or previously identified genomic regions were detected but associations differed by breed. For example, two genomic associations, one each on chromosomes 18 and 2 explained 2.49 % and 3.13 % of the genetic variance in direct calving difficulty in the Holstein-Friesian and Charolais populations, respectively. Imputed Holstein-Friesian sequence data was used to refine the genomic regions responsible for significant associations. Several candidate genes on chromosome 18 were identified and four highly significant missense variants were detected within three of these genes (SIGLEC12, CTU1, and ZNF615). Nevertheless, only CTU1 contained a missense variant with a putative impact on direct calving difficulty based on SIFT (0.06) and Polyphen (0.95) scores. Using imputed sequence data, we refined a genomic region on chromosome 4 associated with maternal calving difficulty in the Holstein-Friesian population and found the strongest association with an intronic variant in the PCLO gene. A meta-analysis was performed across the three breeds for each calving performance trait to identify common variants associated with these traits in the three breeds. Our results suggest that a portion of the genetic variation in calving performance is common to all three breeds. Conclusion The genomic architecture of calving performance is complex and mainly influenced by many polymorphisms of small effect. We identified several associations of moderate effect size but the majority were breed-specific, indicating that breed-specific alleles exist for calving performance or that the linkage phase between genotyped allele and causal mutation varies between breeds

Potentiation of tumour growth by endotoxin in serum from syngeneic tumour-bearing mice.

The subcutaneous growth of 2 antigenically distinct syngeneic methylcholanthrene-induced murine fibrosarcomas, designated H1 and H7, were significantly augmented by the concomitant administration of E. coli endotoxin (LPS). Amounts as little as 0.2 micrograms i.p. potentiated tumour growth. The weakly antigenic tumour, H1, was more susceptible to provocation by LPS than the more strongly antigenic H7. Maximum provocation of H1 tumour growth occurred when LPS was injected 1 day before the administration of 5000 tumour cells. In contrast, significant anti-tumour resistance resulted if LPS was administered 6 days before the inoculation of tumour cells. Preliminary evidence indicates that low doses of LPS can facilitate the "sneaking through" phenomenon. Enhancement of tumour growth could not be demonstrated with sera or plasma from tumour-bearing mice, unless the samples were contaminated with endotoxin. The results illustrate the importance of excluding endotoxin from solutions used in studies of experimental tumours

Testing metabolic theories

Metabolism is the process by which individual organisms acquire energy and materials from their environment and use them for maintenance, differentiation, growth, and reproduction. There has been a recent push to build an individual-based metabolic underpinning into ecological theory—that is, a metabolic theory of ecology. However, the two main theories of individual metabolism that have been applied in ecology—Kooijman’s dynamic energy budget (DEB) theory and theWest, Brown, and Enquist (WBE) theory— have fundamentally different assumptions. Surprisingly, the core assumptions of these two theories have not been rigorously compared from an empirical perspective. Before we can build an understanding of ecology on the basis of individual metabolism, we must resolve the differences between these theories and thus set the appropriate foundation. Here we compare the DEB and WBE theories in detail as applied to ontogenetic growth and metabolic scaling, from which we identify circumstances where their predictions diverge most strongly. Promising experimental areas include manipulative studies of tissue regeneration, body shape, body condition, temperature, and oxygen. Much empirical work designed specifically with DEB and WBE theory in mind is required before any consensus can be reached on the appropriate theoretical basis for a metabolic theory of ecology