This paper is the sequel to another with the same name (Buttigieg et al., Comput. Methods Funct. Theory, 2023), and is concerned with results of the same type. We deduce a result on the moments of the exit time of Brownian motion from domains whose boundary curve is replaced by a dashed line, and from domains arising from a periodic tiling of the plane. We also give a construction of a type of domain which is similar to a wedge domain, but the behaviour of whose exit time moments answer several questions that had been speculated upon.

In this note, by an elementary use of Girsanov's transform, we show that the exit time for either a biased random walk or a drifted Brownian motion on a symmetric interval is stochastically monotone with respect to the drift parameter. In the random walk case, this gives an alternative proof of a recent result of E. Pek & ouml;z and R. Righter in 2024, while the Brownian motion case is the continuous analogue as discussed in the same paper. Our arguments in both discrete and continuous cases are parallel to each other. We also outline a simple SDE proof for the Brownian case based on a standard comparison theorem.

In this paper, we classify non-geometric distance-regular graphs of diameter at least 3 with smallest eigenvalue at least -3. This is progress towards what is hoped to be an eventual complete classification of distance-regular graphs with smallest eigenvalue at least -3, analogous to existing classification results available in the case that the smallest eigenvalue is at least -2.

In 1979, Neumaier gave a bound on λ in terms of m and μ, where -m is the smallest eigenvalue of a primitive strongly regular graph, unless the graph in question belongs to one of the two infinite families of strongly regular graphs. We improve this result. We also indicate how our methods can be used to give an alternate derivation of Bruck's Completion Theorem for orthogonal arrays.

We show that the derivative of the intersection and self-intersection local times of alpha-stable processes are exponentially integrable for certain parameter values. This includes the Brownian motion case. We also discuss related results present in the literature for fractional Brownian motion, and in particular give a counter-example to a result in Guo et al. (2019) related to this question.

We give the correct condition for existence of the k-th derivative of the intersection local time for fractional Brownian motion, which was originally discussed in [Guo, J., Hu, Y., and Xiao, Y., Higher-order derivative of intersection local time for two independent fractional Brownian motions, Journal of Theoretical Probability 32, (2019), pp. 1190-1201]. We also show that the k-th derivative of the intersection and self-intersection local times of fractional Brownian motion are exponentially integrable for certain parameter values. In addition, we show convergence in distribution when the existence condition is violated for the k-th derivative of self-intersection local time of fractional Brownian motion under scaling.

Subtropical Asia has a rich diversity of reptiles and ticks, though the role of reptiles in the sylvatic cycles of medically important ticks in the region is poorly known. Habu vipers (Protobothrops flavoviridis) are widespread and common in the Japanese subtropics but their role as hosts for ticks has not been carefully explored. For 15 months in 2023/24, habu vipers were screened for ticks and were found to be important hosts for immature stages of the tick Amblyomma testudinarium, with a 22% infestation rate. Amblyomma testudinarium was found to have weak attachment site preferences on P. flavoviridis and host body length was found to have no relationship with either the risk of infestation or the tick load in infested snakes. The phenological profile of A. testudinarium was mapped for the first time in the subtropics based on mean tick loads on P. flavoviridis and historical flagging data. March-April were identified as the period of highest activity, May to July was a period of declining activity, August-September was a period of almost complete inactivity, and October to February was a periods of increasing activity. The role of habu vipers in the sylvatic cycle of A. testudinarium in subtropical Asia is discussion. Additionally, the potential of habu vipers to serve as reservoirs of pathogens or as dilution hosts is also explored.

In this paper, we show that if G is strongly regular then the Gallai graph Gamma(G) and the anti-Gallai graph Delta(G) of G are edge-regular. We also identify conditions under which the Gallai and anti-Gallai graphs are themselves strongly regular, as well as conditions under which they are 2-connected. We include also a number of concrete examples and a discussion of spectral properties of the Gallai and anti-Gallai graphs.

Bats and ticks are important sources of zoonotic pathogens. Therefore, understanding the diversity, distribution, and ecology of both groups is crucial for public health preparedness. Soft ticks (Argasidae) are a major group of ectoparasites commonly associated with bats. The multi-host life cycle of many argasids make them important vectors of pathogens. Over nine years (2011-2020), surveillance was undertaken to identify the ticks associated with common bats in Singapore. During this period, the bat tick Ornithodoros batuensis was detected within populations of two cave roosting bat species: Eonycteris spelaea and Penthetor lucasi. We examined the relationship between bat species, roosting behaviour, and probability of O. batuensis infestation. We also estimated the relationship between bat life history variables (body condition index, sex, and age) on the probability of infestation and tick count. This represents the first detection of O. batuensis and the genus Ornithodoros within Singapore. We also provide evidence of the continued persistence of Argas pusillus in Singapore with the second local record.

India’s currency-to-GDP ratio indicates a strong and persistent demand for cash. Despite the withdrawal of two high-value banknotes in 2016 and the recent economic contraction due to the COVID-19 pandemic, cash continues to rule in India. Although there is extensive work on the demand for cash in India, relatively little is known about the quality and supply of banknotes. This paper provides an overview of currency management policies in India using publicly available annual aggregate issuances and disposals data from the Reserve Bank of India (RBI). We model the life of a banknote using survival analysis models, finding that low-value banknotes (INR 10 and 20) have a median life of 4–5 years. The estimates of longevity of a banknote are significantly associated with velocity of circulation as well as exogenous shocks. We demonstrate the value of survival analysis methods in informing currency management policies in India and providing avenues for future work in this domain.

SummaryThe trigonometric angle-sum formulas are given a new interpretation as statements about conformal maps. In particular, we show how the angle-sum formula for tangent can be realized by equating two different conformal maps from an infinite strip to a disk, and the formulas for sine and cosine by similarly equating conformal maps from an infinite strip to a certain slit domain.

In this paper, we classify non-geometric distance-regular graphs of diameter at least $3$ with smallest eigenvalue at least $-3$. This is progress towards what is hoped to be an eventual complete classification of distance-regular graphs with smallest eigenvalue at least $-3$, analogous to existing classification results available in the case that the smallest eigenvalue is at least $-2$.

The Asian rodent tick (Ixodes granulatus) occurs throughout much of Asia, it frequently bites humans, and zoonotic pathogens, such as Borrelia burgdorferi (sensu lato) and Rickettsia honei, have been detected within it. Unfortunately, the ecology of I. granulatus remains poorly known, including drivers of its abundance and the interaction ecology with its sylvatic hosts. To elucidate the ecology of this medically important species, the habitat preferences of I. granulatus were assessed in Singapore and Malaysia. Ixodes granulatus showed strong associations with old forest habitats, though across different age classes of old forest there was limited variation in abundance. Ixodes granulatus was absent from other habitats including young forest, scrubland, and parks/gardens. Within its sylvatic rodent hosts, a range of factors were found to be statistically significant predictors of I. granulatus load and/or infestation risk, including sex and body condition index. Male rodents were significantly more likely to be infested and to have higher loads than females, similarly, animals with a lower body condition index were significantly more likely to be infested. Proactive public health efforts targeted at preventing bites by this tick should carefully consider its ecology to minimise ecological overlap between humans and I. granulatus.

In an elegant recent paper \cite{geng2022conway}, Geng and Xia settled the question of the infinite divisibility of the Conway--Maxwell--Poisson distribution, using in large part several results from complex analysis. In this note we show how these complex analytic methods can be circumvented, thereby giving a proof of their result which is completely elementary.

We prove a number of results relating exit times of planar Brownian motion with the geometric properties of the domains in question. Included are proofs of the conformal invariance of moduli of rectangles and annuli using Brownian motion; similarly probabilistic proofs of some recent results of Karafyllia on harmonic measure on starlike domains; examples of domains and their complements which are simultaneously large when measured by the moments of exit time of Brownian motion, and examples of domains and their complements which are simultaneously small; and proofs of several identities involving the Cauchy distribution using the optional stopping theorem.

Urbanisation has a wide range of impacts on biodiversity, but its effects on parasitic arthropods, particularly those of bats, remain poorly studied. Ectoparasites of the large-footed myotis (Myotis macropus) in eastern Australia were sampled from 10 roost sites across an urban gradient. In total, 265 bats were examined and 447 ectoparasites were collected, comprising three species of Hippoboscoidea: Basilia hamsmithi (Nycteribiidae), Penicillidia setosala (Nycteribiidae), Brachytarsina amboinensis (Streblidae), and an acarine, Spinturnix novaehollandiae (Mesostigmata, Spinturnicidae). Degree of urbanisation was found to have a significant effect on the abundance of the batfly B. hamsmithi but had no significant effect on the abundance of the wing mite S. novaehollandiae. We hypothesise that this is due to differences in the life history of these two species and the advantage components of these differences confer in exploiting variations in host roost habits. The prevalence of the batfly B. hamsmithi was high in urban sites but comparatively low in suburban and non-urban sites. Mass, sex, and body condition were found to have no significant impact on either the parasite load or the chance of infestation. Both P. setosala and B. amboinensis were recorded from M. macropus for the first time, though only in small numbers. They were associated with mixed-species roosts in a suburban site and are evidence of parasite spillover between sympatric bat species.

SummaryEach of the Pythagorean means corresponds to the centroid of a region in the Cartesian plane. We show how this insight leads to a short proof of a result that generalizes the HM-GM-AM inequality.

Let $T_D$ denote the first exit time of a Brownian motion from a domain $D$ in ${\mathbb R}^n$. Given domains $U,W \subseteq {\mathbb R}^n$ containing the origin, we investigate the cases in which we are more likely to have fast exits from $U$ than $W$, meaning ${\bf P}(T_U {\bf P}(T_Wt) > {\bf P}(T_W>t)$ for $t$ large. This result, which applies only in two dimensions, shows that the unit disk has the lowest probability of long stays amongst all Schlicht domains.

In a recent paper by Yu (arXiv:2008.05633, 2020), higher order derivatives of self-intersection local time of fractional Brownian motion were defined, and existence over certain regions of the Hurst parameter H was proved. Utilizing the Wiener chaos expansion, we provide new proofs of Yu’s results and show how a Varadhan-type renormalization can be used to extend the range of convergence for the even derivatives.

Calliphora augur (Diptera: Calliphoridae) is a common carrion-breeding blowfly of forensic, medical and agricultural importance in eastern Australia. Despite this, detailed information on the developmental biology of C. augur is lacking. Here, we present the first comprehensive study on the development of all three larval instars of C. augur , fed on sheep’s liver, at constant temperatures of 15, 20, 25, 30 and 35°C. We provide thermal summation models describing instar duration, as well as 95% prediction intervals for larval length at each constant temperature, enabling the age of larvae of C. augur to be estimated from their developmental stage and their average length. These data provide a basis for the application of this species to forensic investigations.