For odd n we construct a path $\rho\;:\;\thinspace \Pi_1(S) \to SL(n\mathbb{R})$ of discrete, faithful, and Zariski dense representations of a surface group such that $\rho_t(\Pi_1(S)) \subset SL(n,\mathbb{Q})$ for every $t\in \mathbb{Q}$.

We prove that the Hodge–Tate spectral sequence of a proper smooth rigid analytic variety can be reconstructed from its infinitesimal $\mathbb{B}_{\text{dR}}^+$-cohomology through the Bialynicki–Birula map. We also give a new proof of the torsion-freeness of the infinitesimal $\mathbb{B}_{\text{dR}}^+$-cohomology independent of Conrad–Gabber spreading theorem, and a conceptual explanation that the degeneration of Hodge–Tate spectral sequences is equivalent to that of Hodge–de Rham spectral sequences.

In this paper we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation $\pi$ we consider a special small $\mathrm{GL}_2(\mathbb{Z}_p)$-type V in $\pi$ and prove global sup-norm bounds for an average over an orthonormal basis of V. We achieve a non-trivial saving when the dimension of V grows.

We study a pair consisting of a smooth 3-fold defined over an algebraically closed field and a “general” ${\Bbb R}$-ideal. We show that the minimal log discrepancy (“mld” for short) of every such a pair is computed by a prime divisor obtained by at most two weighted blow-ups. This bound is regarded as a weighted blow-up version of Mustaţă–Nakamura’s conjecture. We also show that if the mld of such a pair is not less than 1, then it is computed by at most one weighted blow-up. As a consequence, ACC of mld holds for such pairs.

Given any polynomial in two variables of degree at most three with rational integer coefficients, we obtain a new search bound to decide effectively if it has a zero with rational integer coefficients. On the way we encounter a natural problem of estimating singular points. We solve it using elementary invariant theory but an optimal solution would seem to be far from easy even using the full power of the standard Height Machine.

We prove an equality, predicted in the physical literature, between the Jeffrey–Kirwan residues of certain explicit meromorphic forms attached to a quiver without loops or oriented cycles and its Donaldson–Thomas type invariants.

In the special case of complete bipartite quivers we also show independently, using scattering diagrams and theta functions, that the same Jeffrey–Kirwan residues are determined by the the Gross–Hacking–Keel mirror family to a log Calabi–Yau surface.

Missed detection probability is a critical metric for the integrity performance of receiver autonomous integrity monitoring (RAIM) in the presence of faults. The traditional missed detection probability evaluation method for RAIM is limited by impractical time consumption because of the absence of accurate searching interval for the magnitude of a worst-case fault. To address this issue, the searching interval for the magnitude of a worst-case fault is constructed by the combination of minimum detectable magnitude and minimum hazardous magnitude, and the searching interval adjustment is designed to avoid the absence of worst-case fault magnitude so that the maximum missed detection probability can be accurately evaluated. The simulation result indicates that the proposed method can achieve higher accuracy for the worst-case fault magnitude searching. Furthermore, the accuracy of worldwide evaluated missed detection rate can achieve an improvement of 57·66% at most by the proposed method for the different classical RAIM algorithms.

By simultaneously estimating satellite clock drifts (SCDs) as either constant parameters or piece-wise parameters, we present an improved integrated orbit determination and time synchronization approach for BDS-3 satellites with raw inter-satellite link (ISL) observations. Experiments with L-band data from seven monitoring stations in China and ISL data from eight satellites of the third-generation Beidou Navigation Satellite System (BDS-3) were carried out and the two SCD estimation strategies are validated. It is demonstrated that, with SCDs estimated, the quality of orbits and clock offsets is comparable to those with SCDs corrected using predicted values. The accuracy of the estimated orbits and clocks are up to 0.019 m (radial) and 0.095 ns, respectively, with improvements of 95% and 90%, when compared with the results using the L-band data alone. It is also demonstrated that estimating SCDs time slice by time slice is slightly worse in accuracy but superior in coping with possible frequency jump of satellite clocks.

On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros $1/2+i\gamma$ of the Riemann zeta function, we show that the sequence

\begin{equation*}\Gamma_{[a, b]} =\Bigg\{ \gamma : \gamma>0 \quad \mbox{and} \quad \frac{ \log\big(| \zeta^{(m_{\gamma })} (\frac12+ i{\gamma }) | / (\!\log{{\gamma }} )^{m_{\gamma }}\big)}{\sqrt{\frac12\log\log {\gamma }}} \in [a, b] \Bigg\},\end{equation*}

${\gamma }$

$a<b$

$m_\gamma$

$1/2+i{\gamma }$

${\gamma }$

$\gamma (\!\log T)/2\pi$

${\gamma }\in \Gamma_{[a, b]}$

$0<{\gamma }\leq T$

The development of Maritime Autonomous Surface Ship (MASS) is progressing rapidly within the maritime industry. Degree Two of MASS (MASS-DoA2), balancing human oversight and autonomous efficiency, will likely gain regulatory approval and industry acceptance. MASS-DoA2 possesses different control modes to adapt to various scenarios. However, the control-switching mechanisms among operators at shore control centres, autonomous navigation systems and number of seafarers onboard remain ambiguous, which poses a new risk that may significantly influence navigation safety. This study focuses on MASS-DoA2 and carries out a systematic review of autonomous ship guidelines. A questionnaire was designed based on the review findings, and a survey was carried out among captains and researchers in related fields. The review identified 11 control-switching scenarios with suggested takeover agents and the switching process and outlined the priority relationship between various takeover agents. Finally, a control-switching framework for MASS – DoA2 is proposed. It can serve as a theoretical framework for research on MASS's dynamic degree of autonomy and provide a reference for maritime regulatory authorities in establishing MASS – DoA2 control-switching mechanisms.

The geostationary orbit (GEO) belt hosts a substantial number of high-value satellites, making the study of autonomous navigation within this area significant. Features of autonomous operations such as patrolling the GEO belt and frequent manoeuvres at a certain location make real-time positioning using the Global Navigation Satellite System (GNSS) valuable. This paper studies the performance of positioning with GNSS considering main lobe and side lobe signals at different longitudes in the GEO belt. The research delves into the visibility and Position Dilution of Precision (PDOP) across the GEO belt, analysing the performance of the Global Positioning System (GPS), GLObalnaya NAvigatsionnaya Sputnikovaya Sistema (GLONASS) in Russian, BeiDou Navigation Satellite System (BDS), Galileo Navigation Satellite System (Galileo) and multi-systems. In particular, this paper investigates the impact of asymmetric constellations of mixed GEO, Inclined Geosynchronous Orbit (IGSO) and Medium Earth Orbit (MEO) satellites. The study reveals that BDS hybrid constellation provides long-term stable signal coverage over the GEO space above North America and Atlantic Ocean, where GEO signals are more sustainable while IGSO signals have wider coverage. This advantage positions BDS favourably in terms of performance in these regions.

In previous research, several computational methods have been proposed to analyse the navigation, transportation safety and collision risks of maritime vessels. The objective of this study is to use Automatic Identification System (AIS) data to assess the collision risk between two vessels before an actual collision occurs. We introduce the concept of an angle interval in the model to enable real-time response to vessel collision risks. When predicting collision risks, we consider factors such as relative distance, relative velocity and phase between the vessels. Lastly, the collision risk is divided into different regions and represented by different colours. The green region represents a low-risk area, the yellow region serves as a cautionary zone and the red region indicates a high-alert zone. If a signal enters the red region, the vessel's control system will automatically intervene and initiate evasive manoeuvres. This reactive mechanism enhances the safety of vessel operations, ensuring the implementation of effective collision avoidance measures.

Indoor navigation for micro aerial vehicles (MAVs) is challenging in GPS signal-obstructed indoor corridor environments. Position and heading estimation for a MAV is required to navigate without colliding with obstacles. The connected components algorithm and k-means clustering algorithm have been integrated for line and vanishing point detection in the corridor image frames to estimate the position and heading of the MAV. The position of the vanishing point indicates the position of the MAV (centre, left or right) in the corridor. Furthermore, the Euclidean distance between the image centre and mid-pixel coordinates at the last row of the image and the detected vanishing point pixel coordinates in the successive corridor image frames are used to compute the heading of the MAV. When the MAV deviates from the corridor centre, the position and heading measurement can send a suitable control signal to the MAV and align the MAV at the centre of the corridor. When compared with a grid-based vanishing point detection method heading accuracy of ±1⋅5°, the k-means clustering-based vanishing point detection is suitable for real-time heading measurement for indoor MAVs with an accuracy of ±0⋅5°.

Official interpretations of Doppler shifts from the final satellite communications of missing Malaysian Airlines MH370 were based on a motion-decoupled ‘Up-Down model’. That model predicted an uncontrolled high-speed gravitationally accelerated dive following fuel starvation. Here, I challenge that model using a more-realistic motion-coupled ‘Declination model’. Aerial, satellite and underwater searches failed to find the predicted official violent crash-site near the 7th arc. Meticulous re-examination of debris damage by air-crash investigator Larry Vance concluded that the aircraft glide-landed under power with extended wing-flaps. The trailing-edges were then damaged, broke off their mountings, flailing about and retracted along the guides to cause the observed wing-flap damage. Larry's conclusions complement interpretations from the ‘Declination model’ which we demonstrate here with three example flight tracks. Our revised Doppler-shift analyses support the hypothesis of a controlled eastward descent. We conclude that the official theory of fuel starvation and a high-speed dive are fundamentally flawed.

Vessel traffic services (VTS) is a marine information exchange system vital for the safety and efficiency of ship traffic within designated regions. The harmonisation, integration and exchange of marine information have emerged as significant components in promoting maritime safety, in line with the concept of e-navigation. This study aimed to analyse the flow of information between VTS areas employing social network analysis to ensure seamless marine information exchange across VTS areas. Information flow was analysed based on data obtained from ships navigating through coastal waters and ports in Korea, revealing that the sea area near Busan New Port exhibited the highest concentration of information flow, while the Tongyoung Coast VTS area represented a critical link in the flow of information. Given its history of marine accidents, the current Masan (Opko) VTS region emerged as a susceptible area. The study provides valuable foundational data for a comprehensive coastal surveillance system.