In a 1968 issue of the Proceedings, P. M. Cohn famously claimed that a commutative domain is atomic if and only if it satisfies the ascending chain condition on principal ideals (ACCP). Some years later, a counterexample was however provided by A. Grams, who showed that every commutative domain with the ACCP is atomic, but not vice versa. This has led to the problem of finding a sensible (ideal-theoretic) characterisation of atomicity.

The question (explicitly stated on p. 3 of A. Geroldinger and F. Halter–Koch’s 2006 monograph on factorisation) is still open. We settle it here by using the language of monoids and preorders.

We show that a group that is hyperbolic relative to strongly shortcut groups is itself strongly shortcut, thus obtaining new examples of strongly shortcut groups. The proof relies on a result of independent interest: we show that every relatively hyperbolic group acts properly and cocompactly on a graph in which the parabolic subgroups act properly and cocompactly on convex subgraphs.

Let F be a finite extension of ${\mathbb Q}_p$. Let $\Omega$ be the Drinfeld upper half plane, and $\Sigma^1$ the first Drinfeld covering of $\Omega$. We study the affinoid open subset $\Sigma^1_v$ of $\Sigma^1$ above a vertex of the Bruhat–Tits tree for $\text{GL}_2(F)$. Our main result is that $\text{Pic}\!\left(\Sigma^1_v\right)[p] = 0$, which we establish by showing that $\text{Pic}({\mathbf Y})[p] = 0$ for ${\mathbf Y}$ the Deligne–Lusztig variety of $\text{SL}_2\!\left({\mathbb F}_q\right)$. One formal consequence is a description of the representation $H^1_{{\acute{\text{e}}\text{t}}}\!\left(\Sigma^1_v, {\mathbb Z}_p(1)\right)$ of $\text{GL}_2(\mathcal{O}_F)$ as the p-adic completion of $\mathcal{O}\!\left(\Sigma^1_v\right)^\times$.

What is the probability that a random UHF algebra is of infinite type? What is the probability that a random simple AI algebra has at most k extremal traces? What is the expected value of the radius of comparison of a random Villadsen-type AH algebra? What is the probability that such an algebra is $\mathcal{Z}$-stable? What is the probability that a random Cuntz–Krieger algebra is purely infinite and simple, and what can be said about the distribution of its K-theory? By constructing $\mathrm{C}^*$-algebras associated with suitable random (walks on) graphs, we provide context in which these are meaningful questions with computable answers.

Gross and Siebert developed a program for constructing in arbitrary dimension a mirror family to a log Calabi–Yau pair (X, D), consisting of a smooth projective variety X with a normal-crossing anti-canonical divisor D in X. In this paper, we provide an algorithm to practically compute explicit equations of the mirror family in the case when X is obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary, and D is the strict transform of the toric boundary. The main ingredient is the heart of the canonical wall structure associated to such pairs (X, D), which is constructed purely combinatorially, following our previous work with Mark Gross. In the case when we blow up a single hypersurface we show that our results agree with previous results computed symplectically by Aroux–Abouzaid–Katzarkov. In the situation when the locus of blow-up is formed by more than a single hypersurface, due to infinitely many walls interacting, writing the equations becomes significantly more challenging. We provide the first examples of explicit equations for mirror families in such situations.

We prove that there exist finitely generated, stably finite algebras which are not linear sofic. This was left open by Arzhantseva and Păunescu in 2017.

We study intermediate-scale statistics for the fractional parts of the sequence $\left(\alpha a_{n}\right)_{n=1}^{\infty}$, where $\left(a_{n}\right)_{n=1}^{\infty}$ is a positive, real-valued lacunary sequence, and $\alpha\in\mathbb{R}$. In particular, we consider the number of elements $S_{N}\!\left(L,\alpha\right)$ in a random interval of length $L/N$, where $L=O\!\left(N^{1-\epsilon}\right)$, and show that its variance (the number variance) is asymptotic to L with high probability w.r.t. $\alpha$, which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotic holds almost surely in $\alpha\in\mathbb{R}$ when $L=O\!\left(N^{1/2-\epsilon}\right)$. For slowly growing L, we further prove a central limit theorem for $S_{N}\!\left(L,\alpha\right)$ which holds for almost all $\alpha\in\mathbb{R}$.

We prove a formula expressing the K-theoretic log Gromov-Witten invariants of a product of log smooth varieties $V \times W$ in terms of the invariants of V and W. The proof requires introducing log virtual fundamental classes in K-theory and verifying their various functorial properties. We introduce a log version of K-theory and prove the formula there as well.

When $k\geqslant 4$ and $0\leqslant d\leqslant (k-2)/4$, we consider the system of Diophantine equations

\begin{align*}x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\leqslant j\leqslant k,\, j\ne k-d).\end{align*}

$d=o\!\left(k^{1/4}\right)$

In this paper we prove that the set $\{|x^1-x^2|,\dots,|x^k-x^{k+1}|\,{:}\,x^i\in E\}$ has non-empty interior in $\mathbb{R}^k$ when $E\subset \mathbb{R}^2$ is a Cartesian product of thick Cantor sets $K_1,K_2\subset\mathbb{R}$. We also prove more general results where the distance map $|x-y|$ is replaced by a function $\phi(x,y)$ satisfying mild assumptions on its partial derivatives. In the process, we establish a nonlinear version of the classic Newhouse Gap Lemma, and show that if $K_1,K_2, \phi$ are as above then there exists an open set S so that $\bigcap_{x \in S} \phi(x,K_1\times K_2)$ has non-empty interior.

Let $\textsf{T}$ be a triangulated category with shift functor $\Sigma \colon \textsf{T} \to \textsf{T}$. Suppose $(\textsf{A},\textsf{B})$ is a co-t-structure with coheart $\textsf{S} = \Sigma \textsf{A} \cap \textsf{B}$ and extended coheart $\textsf{C} = \Sigma^2 \textsf{A} \cap \textsf{B} = \textsf{S}* \Sigma \textsf{S}$, which is an extriangulated category. We show that there is a bijection between co-t-structures $(\textsf{A}^{\prime},\textsf{B}^{\prime})$ in $\textsf{T}$ such that $\textsf{A} \subseteq \textsf{A}^{\prime} \subseteq \Sigma \textsf{A}$ and complete cotorsion pairs in the extended coheart $\textsf{C}$. In the case that $\textsf{T}$ is Hom-finite, $\textbf{k}$-linear and Krull–Schmidt, we show further that there is a bijection between complete cotorsion pairs in $\textsf{C}$ and functorially finite torsion classes in $\textsf{mod}\, \textsf{S}$.

For each central essential hyperplane arrangement $\mathcal{A}$ over an algebraically closed field, let $Z_\mathcal{A}^{\hat\mu}(T)$ denote the Denef–Loeser motivic zeta function of $\mathcal{A}$. We prove a formula expressing $Z_\mathcal{A}^{\hat\mu}(T)$ in terms of the Milnor fibers of related hyperplane arrangements. This formula shows that, in a precise sense, the degree to which $Z_{\mathcal{A}}^{\hat\mu}(T)$ fails to be a combinatorial invariant is completely controlled by these Milnor fibers. As one application, we use this formula to show that the map taking each complex arrangement $\mathcal{A}$ to the Hodge–Deligne specialization of $Z_{\mathcal{A}}^{\hat\mu}(T)$ is locally constant on the realization space of any loop-free matroid. We also prove a combinatorial formula expressing the motivic Igusa zeta function of $\mathcal{A}$ in terms of the characteristic polynomials of related arrangements.

Autonomous vehicles rely on a combination of sensors for safe navigation around the world. For precise localisation, high-definition (HD) maps are used. These maps are a representation of the world containing information about objects on the road infrastructure. Currently, there are tens of HD map makers, however, no rigorous description of the requirements for the accuracy of HD maps has been published yet. This study fills the gap and offers a mathematical description of the minimum required accuracy for HD maps. In the first part, we identify factors that influence the quality of a map. Based on that, we proceed to present our solution for determining the minimum required accuracy for HD maps, both for static and dynamic models, and present a new formula for the minimum necessary accuracy for HD maps.

In missile test ranges, complex missions demand precise trajectory generated by radar. Both the radar and Global Navigation Satellite System (GNSS) signals are affected by atmospheric effects, degrading their accuracy and performance. The Indian Regional Navigation Satellite System/Navigation with Indian Constellation (IRNSS/NavIC) transmits signals in the S-band together with the L-band. This paper presents a novel experimental technique to improve the tracking accuracy of S-band radars using the concurrent NavIC S-band signal. The ionospheric delay using the NavIC S-band signal is calculated first, and the results are used to improve the trajectory data of simultaneously operating S-band radars. This is a unique application of the NavIC S-band signals apart from its conventional usage. During a launch mission, for low elevation angles, the ionospheric error is found to be ~130 m while at higher elevation angles the error values are found to be ~1–3 m. The concept is validated using data from a missile test mission. This report on the use of S-band GNSS signals for the correction of S-band radar range data offers a clear advantage of simplicity and accuracy.

It has been acknowledged that the Doppler is beneficial to the GNSS positioning of smartphones. However, analysis of Doppler precision on smartphones is insufficient. In this paper, we focus on the characteristic analysis of the raw Doppler measurement from Android smartphones. A comprehensive investigation of the Doppler was conducted. The results illustrate that the availability of Doppler is stable and higher than that of carrier measurements, which means that the Doppler-smoothed code (DSC) method is more effective. However, there is a constant bias between the Doppler and the code rate in Xiaomi MI8, which indicates that extra processing of the DSC method is necessary for this phone. Additionally, it is demonstrated that the relationship between the Doppler and C/N0 can be expressed as an exponential function, and the fitting parameters are provided. The numerical experiment in car-borne and hand-held scenes was conducted for evaluating the performance of the Doppler-aided positioning algorithm. For positioning, the improvement reaches 37 ⋅ 69%/37 ⋅ 14%/26 ⋅ 61% in the east, north and up components, respectively, after applying the Doppler aiding. For velocity estimation, the improvement reaches 29 ⋅ 62%/39 ⋅ 63%/29 ⋅ 37% in the three components, respectively.

Ship collision avoidance has always been one of the classic topics in the field of marine research. In traditional encounter situations, officers on watch (OOWs) usually use a very high frequency (VHF) radio to coordinate each other. In recent years, with the continuous development of autonomous ships, there will be a mixed situation where ships of different levels of autonomy coexist at the same time. Under such a scenario, different decision makers have different perceptions of the current scene and different decision-making logic, so conventional collision avoidance methods may not be applicable. Therefore, this paper proposes a collaborative collision avoidance strategy for multi-ship collision avoidance under mixed scenarios. It builds a multi-ship cooperative network to determine cooperative objects and timing, at the same time. Based on a cooperative game model, a global collision avoidance responsibility distribution that satisfies group rationality and individual rationality is realised, and finally achieves a collaborative strategy according to the generalised reciprocal velocity obstacle (GRVO) algorithm. Case studies show that the strategy proposed in this paper can make all ships pass each other clearly and safely.

Complex domestic airspace requires collision risk models and monitoring tools suitable for arbitrary aircraft trajectories. This paper presents a new mathematically based collision risk approach that extends the International Civil Aviation Organisation (ICAO) models to full aircraft encounters based on real trajectory data. A new continuous time intervention model is presented, along with a position uncertainty propagation model that better reflects aircraft behaviour and allows generalisation to all trajectories to eliminate degenerate cases. The proposed risk model is computationally efficient compared to the models it is based on and can be applied to large-scale trajectory data. The utility of the model is demonstrated through a series of case studies using real aircraft trajectories.