Let $b \geqslant 3$ be an integer and C(b, D) be the set of real numbers in [0,1] whose base b expansion only consists of digits in a set $D {\subseteq} \{0,...,b-1\}$. We study how close can numbers in C(b, D) be approximated by rational numbers with denominators being powers of some integer t and obtain a zero-full law for its Hausdorff measure in several circumstances. When b and t are multiplicatively dependent, our results correct an error of Levesley, Salp and Velani (Math. Ann. 338 (2007), 97–118) and generalise their theorem. When b and t are multiplicatively independent but have the same prime divisors, we obtain a partial result on the Hausdorff measure and bounds for the Hausdorff dimension, which are close to the multiplicatively dependent case. Based on these results, several conjectures are proposed.

This research employs an enhanced Polar Operation Limit Assessment Risk Indexing System (POLARIS) and multi-scale empirical analysis methods to quantitatively evaluate the risks in icy region navigation. It emphasises the significant influence of spatial effects and external environmental factors on maritime accidents. Findings reveal that geographical location, environmental and ice conditions are crucial contributors to accidents. The models indicate that an increase in ports, traffic volume and sea ice density directly correlates with higher accident rates. Additionally, a novel risk estimation model is introduced, offering a more accurate and conservative assessment than current standards. This research enriches the understanding of maritime accidents in icy regions, and provides a robust framework for different navigation stages and conditions. The proposed strategies and model can effectively assist shipping companies in route planning and risk management to enhance maritime safety in icy regions.

With increased global navigation satellite system (GNSS) signals and degraded observation environments, the correctness of ambiguity resolution is disturbed, causing unexpected real-time kinematic (RTK) positioning solutions. This paper presents an improved fault detection and exclusion (FDE) method based on the generalized least squares (GLS) model. The correlated GLS model is constructed by regarding double-differencing (DD) integer ambiguities as the known parameters. Meanwhile, the validity of residuals as crucial components of fault detection could be enhanced by the iterative re-weighted least squares (IRLS) method rather than the least squares (LS) without robustness. A static test with artificial faults and a dynamic test with natural faults were carried out, respectively. By analyzing test statistics of the enhanced FDE algorithm and comparing its positioning errors with those from the classical LS, it is shown that our method can provide high-precision and high-reliability RTK solutions facing wrong DD fixed ambiguities due to observation faults.

The q-colour Ramsey number of a k-uniform hypergraph H is the minimum integer N such that any q-colouring of the complete k-uniform hypergraph on N vertices contains a monochromatic copy of H. The study of these numbers is one of the central topics in Combinatorics. In 1973, Erdős and Graham asked to maximise the Ramsey number of a graph as a function of the number of its edges. Motivated by this problem, we study the analogous question for hypergaphs. For fixed $k \ge 3$ and $q \ge 2$ we prove that the largest possible q-colour Ramsey number of a k-uniform hypergraph with m edges is at most $\mathrm{tw}_k(O(\sqrt{m})),$ where tw denotes the tower function. We also present a construction showing that this bound is tight for $q \ge 4$. This resolves a problem by Conlon, Fox and Sudakov. They previously proved the upper bound for $k \geq 4$ and the lower bound for $k=3$. Although in the graph case the tightness follows simply by considering a clique of appropriate size, for higher uniformities the construction is rather involved and is obtained by using paths in expander graphs.

Given a surface $\Sigma$ equipped with a set P of marked points, we consider the triangulations of $\Sigma$ with vertex set P. The flip-graph of $\Sigma$ is the graph whose vertices are these triangulations, and whose edges correspond to flipping arcs in these triangulations. The flip-graph of a surface appears in the study of moduli spaces and mapping class groups. We consider the number of geodesics in the flip-graph of $\Sigma$ between two triangulations as a function of their distance. We show that this number grows exponentially provided the surface has enough topology, and that in the remaining cases the growth is polynomial.

Vessel collision risk estimation is crucial in navigation manoeuvres, route planning, risk control, safety management and forewarning issues. The interaction possibility is a good method to quantify the near-miss collision risks of multi-ships. Current models, however, are mostly concerned about the movements in an unrestricted isotropic travel environment or network environment. This article simultaneously addresses these issues by developing a novel environment–kinetic compound space–time prism to capture potential spatial–temporal interactions of multi-ships in constrained dynamic environments. The approach could significantly reduce the overestimation of the individual vessel’s potential travel area and the interaction possibility of encountering vessels in restricted water. The proposed environmental–kinetical compound space–time prism (EKC-STP)-based method enables identifying where and when multi-ships possibly interacted in the constraint water area, as well as how the interaction possibility pattern changed from day to day. The collision risk evaluation results were validated through comparison with other methods. The full picture of hierarchical collision risk distribution in port areas is determined and could be employed to provide quantifiable references for efficient and practical anti-collision measures establishment.

The Plane or Plain Scale is a navigational device that dates back to the early 1600s but has long since ceased to be used in practice. It could perform the function of a protractor and be used to solve problems in plane trigonometry. In addition, coupled with a suite of remarkable geometric constructions based on stereographic projection, the Plane Scale could efficiently solve problems in spherical trigonometry and hence navigation on a sphere. The methods used seem today to be largely unknown. This paper describes the Plane Scale and how it was used.

The optimisation of inter-island transportation systems constitutes a critical determinant of regional economic development and the efficacy of mobility infrastructure. This study presents a comparative analysis of passenger mode selection between short-sea shipping (SSS) and road transport alternatives through stated preference surveys conducted via anonymised questionnaires. Employing advanced discrete choice modelling techniques – specifically the multinomial logit (MNL), random parameter logit (RPL) and latent class (LC) frameworks – we quantitatively disentangle the complex determinants influencing modal preferences. Our systematic sensitivity analysis reveals distinct behavioural patterns: passengers opting for SSS prioritise journey convenience, whereas road transport users exhibit stronger cost sensitivity. These findings provide actionable insights for formulating evidence-based policies to enhance intermodal transportation networks in the Zhoushan Archipelago of China. Beyond its immediate geographical focus, this research contributes methodological innovations by applying finite mixture models to capture unobserved heterogeneity in maritime transport decisions. The framework demonstrates significant transferability potential for island territories globally and urban freight corridor optimisation challenges, particularly in contexts requiring trade-off analyses between maritime efficiency and terrestrial logistics constraints.