Fix integers $r \ge 2$ and $1\le s_1\le \cdots \le s_{r-1}\le t$ and set $s=\prod _{i=1}^{r-1}s_i$. Let $K=K(s_1, \ldots , s_{r-1}, t)$ denote the complete $r$-partite $r$-uniform hypergraph with parts of size $s_1, \ldots , s_{r-1}, t$. We prove that the Zarankiewicz number $z(n, K)= n^{r-1/s-o(1)}$ provided $t\gt 3^{s+o(s)}$. Previously this was known only for $t \gt ((r-1)(s-1))!$ due to Pohoata and Zakharov. Our novel approach, which uses Behrend’s construction of sets with no 3-term arithmetic progression, also applies for small values of $s_i$, for example, it gives $z(n, K(2,2,7))=n^{11/4-o(1)}$ where the exponent 11/4 is optimal, whereas previously this was only known with 7 replaced by 721.

We study stationary distributions in the context of stochastic reaction networks. In particular, we are interested in complex balanced reaction networks and the reduction of such networks by assuming that a set of species (called non-interacting species) are degraded fast (and therefore essentially absent from the network), implying that some reaction rates are large relative to others. Technically, we assume that these reaction rates are scaled by a common parameter N and let $N\to\infty$. The limiting stationary distribution as $N\to\infty$ is compared with the stationary distribution of the reduced reaction network obtained by elimination of the non-interacting species. In general, the limiting stationary distribution could differ from the stationary distribution of the reduced reaction network. We identify various sufficient conditions under which these two distributions are the same, including when the reaction network is detailed balanced and when the set of non-interacting species consists of intermediate species. In the latter case, the limiting stationary distribution essentially retains the form of the complex balanced distribution. This finding is particularly surprising given that the reduced reaction network could be non-weakly reversible and might exhibit unconventional kinetics.

The money exchange model is a type of agent-based model used to study how wealth distribution and inequality evolve through monetary exchanges between individuals. The primary focus of this model is to identify the limiting wealth distributions that emerge at the macroscopic level, given the microscopic rules governing the exchanges among agents. In this paper, we formulate generalized versions of the immediate exchange model, the uniform reshuffling model, and the uniform saving model, all of which are types of money exchange model, as discrete-time interacting particle systems and characterize their stationary distributions. Furthermore, we prove that, under appropriate scaling, the asymptotic wealth distribution converges to an exponential distribution for the uniform reshuffling model, and to either an exponential distribution or a gamma distribution depending on the tail behavior of the number of coins given/saved in the immediate exchange model and the random saving model, which generalizes the uniform saving model. In particular, our results provide a mathematically rigorous formulation and generalization of the assertions previously predicted in studies based on numerical simulations and heuristic arguments.

This paper investigates a continuous-time multidimensional risk model with stochastic returns driven by a geometric Lévy process, where each main claim is accompanied by a random number of delayed claims. By employing a framework of multivariate regular variation for claim sizes and allowing for arbitrarily dependent claim-number processes, we conduct asymptotic analyses for two types of ruin probabilities. Numerical examples are used to demonstrate the accuracy of our asymptotic estimates.

A novel family of statistical distributions, called enriched truncated exponentiated generalized family, is theoretically developed to model heavy-tailed data. One of the three-parameter sub-models of this family derived from log-logistic distribution is comprehensively studied. The statistical properties are explored, including moments and Fisher information matrix. In addition, tail-heaviness is studied using the tail-index approach. The method of maximum likelihood is used for parameter estimation, and existence and uniqueness of these estimators are shown. The flexibility of the new family is further validated by applying to the Norwegian fire insurance claim dataset. The goodness-of-fit measures are used to illustrate the adequacy of the proposed family of distributions. Furthermore, a backtesting procedure is conducted for well-known risk measures to assess the accuracy of the right tail fit.

The famous Sidorenko’s conjecture asserts that for every bipartite graph $H$, the number of homomorphisms from $H$ to a graph $G$ with given edge density is minimised when $G$ is pseudorandom. We prove that for any graph $H$, a graph obtained from replacing edges of $H$ by generalised theta graphs consisting of even paths satisfies Sidorenko’s conjecture, provided a certain divisibility condition on the number of paths. To achieve this, we prove unconditionally that bipartite graphs obtained from replacing each edge of a complete graph with a generalised theta graph satisfy Sidorenko’s conjecture, which extends a result of Conlon, Kim, Lee and Lee [J. Lond. Math. Soc., 2018].

This cross-sectional study investigated how care home size influences COVID-19 transmission dynamics, focusing on outbreaks in England during the second wave of COVID-19 (Wave 2; December 2020 to March 2021) and the Omicron wave (December 2021 to February 2022). Using data from the UK Health Security Agency and the Care Quality Commission, positive SARS-CoV-2 test results were matched to care home registration and occupancy data, examining outbreak trajectories in homes of varying sizes and resident age groups. The study included over 90,000 positive cases across the two waves. Small care homes (SCHs, with 10 or fewer beds), predominantly housing younger adults, showed significantly higher early positivity rates: 42% of residents were positive at outbreak detection, rising to 61% by day 7. In contrast, larger homes had early positivity rates of only 3–6%. These findings suggest that SCHs, often designed for communal living, facilitate rapid within-home transmission similar to household settings. The study concludes that outbreak control strategies in SCHs should differ from those in larger care homes, emphasizing proportionate, individualized approaches that consider resident vulnerability and minimize disruption to social support systems. These results have broader implications for managing future infectious disease outbreaks and support the development of tailored guidance based on care home size and resident demographics.

In this paper, we consider a bidimensional risk model with stochastic returns and dependent subexponential claims, in which every main claim may be accompanied by a delayed claim, occurring after an uncertain period of time. The surplus of each business line is allowed to be invested in a portfolio of risk-free assets, and the price process of the investment is modeled by a geometric Lévy process. Meanwhile, we employ a time-claim-dependent structure to describe the dependence among claims and the interarrival times. Some uniform asymptotic formulas for the finite-time ruin probabilities are derived under this structure. Finally, a simulation study is conducted to evaluate the accuracy of the derived results.

We study Langevin-type algorithms for sampling from Gibbs distributions such that the potentials are dissipative and their weak gradients have finite moduli of continuity not necessarily convergent to zero. Our main result is a non-asymptotic upper bound on the 2-Wasserstein distance between a Gibbs distribution and the law of general Langevin-type algorithms based on a Liptser–Shiryaev-type condition for change of measures and Poincaré inequalities. We apply this bound to show that the Langevin Monte Carlo algorithm can approximate Gibbs distributions with arbitrary accuracy if the potentials are dissipative and their gradients are uniformly continuous. We also propose Langevin-type algorithms with spherical smoothing for distributions whose potentials are not convex or continuously differentiable and show their polynomial complexities.

Effectiveness of nirsevimab against respiratory syncytial virus (RSV) hospitalization during the 2024/2025 season in Spain was estimated using a test-negative design (TND) and hospital-based respiratory infections surveillance data. Children born between 1 April 2024 and 31 March 2025 and hospitalized with severe respiratory infection between the start of the 2024 immunization campaign (regionally variable, between 16 September and 1 October 2024) and 31 March 2025 were systematically RT-PCR RSV-tested within 10 days of symptom onset and classified as cases if positive or controls if negative. Nirsevimab effectiveness ((1 − odds ratio) × 100) was estimated using logistic regression, adjusted for admission week, age, sex, high-risk factors, and regional RSV hospitalization rate. We included 199 cases (68.8% immunized) and 360 controls (86.4% immunized). Overall effectiveness was 65.5% (95% confidence interval: 45.2 to 78.3). Effectiveness was similar among infants born before and after the campaign start (63.6% vs. 70.4%, respectively). We found an unexpected early decrease in effectiveness with increasing time since immunization and age, albeit with wide confidence intervals for some groups. Strong age–period–cohort effects and potential sources of bias were identified, highlighting the need to further explore methodological challenges of implementing the TND in the dynamic population of newborns.

General additive functionals of patricia tries are studied asymptotically in a probabilistic model with independent, identically distributed letters from a finite alphabet. Asymptotic normality is shown after normalization together with asymptotic expansions of the moments. There are two regimes depending on the algebraic structure of the letter probabilities, with and without oscillations in the expansion of moments. As applications firstly the proportion of fringe trees of patricia tries with k keys is studied, which is oscillating around $(1-\rho(k))/(2H)k(k-1)$, where H denotes the source entropy and $\rho(k)$ is exponentially decreasing. The oscillations are identified explicitly. Secondly, the independence number of patricia tries and of tries is considered. The general results for additive functions also apply, where a leading constant is numerically approximated. The results extend work of Janson on tries by relating additive functionals on patricia tries to additive functionals on tries.

We show that for any integer $k\ge 1$ there exists an integer $t_0(k)$ such that, for integers $t, k_1, \ldots , k_{t+1}, n$ with $t\gt t_0(k)$, $\max \{k_1, \ldots , k_{t+1}\}\le k$, and $n \gt 2k(t+1)$, the following holds: If $F_i$ is a $k_i$-uniform hypergraph with vertex set $[n]$ and more than $ \binom{n}{k_i}-\binom{n-t}{k_i} - \binom{n-t-k}{k_i-1} + 1$ edges for all $i \in [t+1]$, then either $\{F_1,\ldots , F_{t+1}\}$ admits a rainbow matching of size $t+1$ or there exists $W\in \binom{[n]}{t}$ such that $W$ intersects $F_i$ for all $i\in [t+1]$. This may be viewed as a rainbow non-uniform extension of the classical Hilton-Milner theorem. We also show that the same holds for every $t$ and $n \gt 2k^3t$, generalizing a recent stability result of Frankl and Kupavskii on matchings to rainbow matchings.

We establish large deviations for dynamical Schrödinger problems driven by perturbed Brownian motions when the noise parameter tends to zero. Our results show that Schrödinger bridges charge exponentially small masses outside the support of the limiting law that agrees with the optimal solution to the dynamical Monge–Kantorovich optimal transport problem. Our proofs build on mixture representations of Schrödinger bridges and establishing exponential continuity of Brownian bridges with respect to the initial and terminal points.

Respiratory infections trigger asthma exacerbations. Despite being less severely affected by COVID-19 than adults, the subsequent lockdowns had a great impact on children. Previous studies showed a decrease in asthma exacerbations during the COVID-19 lockdowns, but findings from secondary care settings are scarce. We aimed to elucidate the trends in frequency and characteristics of asthma exacerbations in children presenting on an emergency department (ED) of a secondary care setting before, during, and after the COVID-19 pandemic. A retrospective analysis was conducted using data from ED visits between January 2018 and November 2022 for asthma exacerbations in children. The incidence of ED visits, hospital admissions, paediatric intensive care unit (PICU) admissions, administered medication, and demographic information were compared. A total of 1121 exacerbations were reported in 670 children, of whom 476 (42%) were admitted to hospital and 44 (3.9%) required PICU admission. We observed a decrease in ED visits for asthma exacerbations during the pandemic but an increased risk in hospital admissions and PICU transfers for exacerbations. This suggests a more severe course of exacerbations. Barriers to health care and lower viral exposure may contribute to this.

Inequality is an inherent quality of society. This paper provides actuarial insights into the recognition, measurement, and consequences of inequality. Key underlying concepts are discussed, with an emphasis on the distinction between inequality of opportunity and inequality of outcome. To better design and maintain approaches and programmes that mitigate its adverse effects, it is important to understand its contributing causes. The paper outlines strategies for reflecting on and addressing inequality in actuarial practice. Actuaries are encouraged to work with policymakers, employers, providers, regulators, and individuals in the design and management of sustainable programmes to address some of the critical issues associated with inequality. These programmes can encourage more equal opportunities and protect against the adverse financial effects of outcomes.

Negative dependence in tournaments has received attention in the literature. The property of negative orthant dependence (NOD) was proved for different tournament models with a special proof for each model. For general round-robin tournaments and knockout tournaments with random draws, Malinovsky and Rinott (2023) unified and simplified many existing results in the literature by proving a stronger property, negative association (NA). For a knockout tournament with a non-random draw, they presented an example to illustrate that ${\boldsymbol{S}}$ is NOD but not NA. However, their proof is not correct. In this paper, we establish the properties of negative regression dependence (NRD), negative left-tail dependence (NLTD), and negative right-tail dependence (NRTD) for a knockout tournament with a random draw and with players being of equal strength. For a knockout tournament with a non-random draw and with equal strength, we prove that ${\boldsymbol{S}}$ is NA and NRTD, while ${\boldsymbol{S}}$ is, in general, not NRD or NLTD.