We prove that every homeomorphism of a compact manifold with dimension one has zero topological emergence, whereas in dimension greater than one the topological emergence of a $C^0-$generic homeomorphism is maximal, equal to the dimension of the manifold. We also show that the metric emergence of a continuous self-map on compact metric space has the intermediate value property.

In this paper, we give an explicit formula as well as a practical algorithm for computing the Cassels–Tate pairing on $\text{Sel}^{2}(J) \times \text{Sel}^{2}(J)$ where J is the Jacobian variety of a genus two curve under the assumption that all points in J[2] are K-rational. We also give an explicit formula for the Obstruction map $\text{Ob}: H^1(G_K, J[2]) \rightarrow \text{Br}(K)$ under the same assumption. Finally, we include a worked example demonstrating that we can improve the rank bound given by a 2-descent via computing the Cassels–Tate pairing.

We introduce a number field analogue of the Mertens conjecture and demonstrate its falsity for all but finitely many number fields of any given degree. We establish the existence of a logarithmic limiting distribution for the analogous Mertens function, expanding upon work of Ng. Finally, we explore properties of the generalised Mertens function of certain dicyclic number fields as consequences of Artin factorisation.

We study the counts of smooth permutations and smooth polynomials over finite fields. For both counts we prove an estimate with an error term that matches the error term found in the integer setting by de Bruijn more than 70 years ago. The main term is the usual Dickman $\rho$ function, but with its argument shifted.

We determine the order of magnitude of $\log(p_{n,m}/\rho(n/m))$ where $p_{n,m}$ is the probability that a permutation on n elements, chosen uniformly at random, is m-smooth.

We uncover a phase transition in the polynomial setting: the probability that a polynomial of degree n in $\mathbb{F}_q$ is m-smooth changes its behaviour at $m\approx (3/2)\log_q n$.

We define a knot to be half ribbon if it is the cross-section of a ribbon 2-knot, and observe that ribbon implies half ribbon implies slice. We introduce the half ribbon genus of a knot K, the minimum genus of a ribbon knotted surface of which K is a cross-section. We compute this genus for all prime knots up to 12 crossings, and many 13-crossing knots. The same approach yields new computations of the double slice genus. We also introduce the half fusion number of a knot K, that measures the complexity of ribbon 2-knots of which K is a cross-section. We show that it is bounded below by the Levine–Tristram signatures, and differs from the standard fusion number by an arbitrarily large amount.

Let $r_5(N)$ be the largest cardinality of a set in $\{1,\ldots,N\}$ which does not contain 5 elements in arithmetic progression. Then there exists a constant $c\in (0,1)$ such that

\[r_5(N)\ll \frac{N}{\exp\!((\!\log\log N)^{c})}.\]

$U^4$

Among maritime accidents, fishing vessel collisions are particularly prone to both high frequency and severity. This study aims to identify the correlation between effective collision speed (Delta-V) and the severity of hull damage in fishing vessel collisions. Using data from collisions in South Korea, the study examines the influence of collision-related factors including Delta-V, collision location, collision subject, collision angle and the hull material of the impacted vessel on the extent of vessel damage. Statistical analyses and binary logistic regression were employed to assess trends and relationships between these variables. The findings confirm direct associations between hull damage severity and factors such as tonnage, collision location, the striking vessel and the extent of hull damage.

Augmented reality (AR) is a technology designed to display three-dimensional virtual elements in a real environment. This technology could reduce the cognitive load of marine operators by simplifying information interpretation. However, field tests often reveal qualitative reports of inaccurately projected virtual elements. To address this issue, we present a theoretical model to quantify the error between virtual projections and their observed positions. Numerical simulations, using normal random variables, indicate agreement between the predicted model variance and the error’s standard deviation. Furthermore, a real navigation experiment is conducted where observed errors are inferior to corresponding estimates for error bounds, further indicating the model’s adequacy. The proposed model enables real-time error estimation, system performance prediction and the specification of accuracy requirements. Overall, this study aims to contribute to the systematic definition of accuracy standards for AR-based maritime navigational assistance.

The question arises when developing and testing Unmanned Surface Vessel (USV) Manoeuvring Autonomy (MA): ‘is the performance we are seeing in our current on-water tests better than that of the last autonomy software version we deployed?’ An approach to answer this question is inspired by educators’ rubrics, in which a teacher grades a student’s work to objective criteria and then sums the individual criteria to determine the student’s overall grade. Here, individual metrics are used to evaluate a USV manoeuvring within range of another vessel. A weighted average is then applied to determine the overall score. With that objective performance value now obtained, similar manoeuvring tests can be compared between autonomy software versions to determine if the autonomy under development is progressively improving. This paper does not determine the threshold score needed to establish that a USV is safe to operate; thresholding of sufficient performance is recommended for future study.

Automatic Identification System (AIS) provides estimated position time along with reception time and a time stamp at the receiving station; however, the exact position estimation time remains unidentified. Therefore, this study examines the extent of positional error when using current AIS reception time. As a result, a maximum positional error of 116.9 m was observed between AIS and RTK-GPS (Real-Time Kinematic GPS). Subsequent time correction reduced this error to less than 10 m, with the product of ship speed and correction time nearly matching the error pre-correction. Consequently, it was concluded that transmitting position estimation time is essential for maintaining the reliability of Position Accuracy transmitted by AIS or VHF Data Exchange System (VDES). Furthermore, VDES may possess the communication capacity to transmit and receive vessel attitude data. Therefore, to assess the required transmission frequency, the data transmission period of roll and pitch attitude data was analysed through the mutual correlation of acceleration and angular velocity. The results indicated that the correlation coefficient for each axis exceeded 0.65 at frequencies of 0.5 Hz or higher.

Global Navigation Satellite Systems (GNSS) positioning and integrity monitoring models and algorithms currently generically assume that measurement errors follow a Gaussian distribution. As this is not always the case, there is a trade-off affecting system safety and availability, emphasising the need for better error characterisation in mission-critical applications. Research to date has shown advantages of Generalised Extreme Value (GEV) distribution for mapping extreme events. However, it is more complex than the Gaussian distribution, especially in the error convolution process. This paper derives a distribution, referred to as the GEV-based Gaussian distribution, that benefits from the advantages of both the GEV and Gaussian distributions in mapping extreme events and simplicity, respectively. The proposed distribution is tested against Gaussian, GEV and Generalised t distribution. The results show that the proposed distribution can provide a better bound for extreme events than the tested distribution both for pseudorange and carrier phase errors.

In this paper, we investigate extensions between graded Verma modules in the Bernstein–Gelfand–Gelfand category $\mathcal{O}$. In particular, we determine exactly which information about extensions between graded Verma modules is given by the coefficients of the R-polynomials. We also give some upper bounds for the dimensions of graded extensions between Verma modules in terms of Kazhdan–Lusztig combinatorics. We completely determine all extensions between Verma module in the regular block of category $\mathcal{O}$ for $\mathfrak{sl}_4$ and construct various “unexpected” higher extensions between Verma modules.

We classify hyperbolic polynomials in two real variables that admit a transitive action on some component of their hyperbolic level sets. Such surfaces are called special homogeneous surfaces, and they are equipped with a natural Riemannian metric obtained by restricting the negative Hessian of their defining polynomial. Independent of the degree of the polynomials, there exist a finite number of special homogeneous surfaces. They are either flat, or have constant negative curvature.