New bounds for odd colourings of graphs

Given a graph $G$, a vertex-colouring $\sigma$ of $G$, and a subset $X\subseteq V(G)$, a colour $x \in \sigma(X)$ is said to be \emph{odd} for $X$ in $\sigma$ if it has an odd number of occurrences in $X$. We say that $\sigma$ is an \emph{odd colouring} of $G$ if it is proper and every (open) neighbourhood has an odd colour in $\sigma$. The odd chromatic number of a graph $G$, denoted by $\chi_o(G)$, is the minimum $k\in\mathbb{N}$ such that an odd colouring $\sigma \colon V(G)\to [k]$ exists. In a recent paper, Caro, Petru\v sevski and \v Skrekovski conjectured that every connected graph of maximum degree $\Delta\ge 3$ has odd-chromatic number at most $\Delta+1$. We prove that this conjecture holds asymptotically: for every connected graph $G$ with maximum degree $\Delta$, $\chi_o(G)\le\Delta+O(\ln\Delta)$ as $\Delta \to \infty$. We also prove that $\chi_o(G)\le\lfloor3\Delta/2\rfloor+2$ for every $\Delta$. If moreover the minimum degree $\delta$ of $G$ is sufficiently large, we have $\chi_o(G) \le \chi(G) + O(\Delta \ln \Delta/\delta)$ and $\chi_o(G) = O(\chi(G)\ln \Delta)$. Finally, given an integer $h\ge 1$, we study the generalisation of these results to $h$-odd colourings, where every vertex $v$ must have at least $\min \{\deg(v),h\}$ odd colours in its neighbourhood. Many of our results are tight up to some multiplicative constant

Dynamic design and analysis of subsea CO2 discharging flowline for cargo submarines used for CCS in low-carbon and renewable energy value chains

Developing offshore low carbon and renewable energy value chains to realize a net-zero energy future requires combining offshore renewable energy and carbon capture storage (CCS) solutions. The subsea shuttle tanker (SST) was presented in recently published works to accelerate the adoption of offshore CCS systems. The SST is a novel underwater vessel designed to transport CO2 autonomously from offshore facilities to subsea wells for direct injection at marginal fields using a flowline connected. The SST will be subjected to stochastic currents and experience dynamic responses during this offloading process. The offloading flowline must be designed to handle this dynamic response. As such, this paper establishes the baseline design for this flowline. The cross-section and global configuration designs drive the flowline design. For the cross-section design, the pressure containment, collapse and local buckling criteria defined in DNV-OS-F101 are applied to validate the required structural capacity at specified water depths. For the configuration design, the principle factors concerning the water depth, internal flow rate, and current speed are investigated to further validate the stress capacity according to the allowed von Mises stress level for a more robust baseline design. Finally, the flowline connecting and disassembly methodology is proposed, and the critical factor of well-coordinated speed between flowline and SST is investigated to avoid overbending during the lifting and lowering phases.publishedVersio

幼若動物組織浸出液(血液)の微生物發育に及ぼす影響

Embedded adaptive interventions in the SMART design of Figs. 1 and 2. Table S1. Eight embedded adaptive interventions in the SMART design of Fig. 1. Table S2. Four embedded adaptive interventions in the SMART design of Fig. 2, word document. (DOCX 14 kb

Image Clustering via the Principle of Rate Reduction in the Age of Pretrained Models

The advent of large pre-trained models has brought about a paradigm shift in both visual representation learning and natural language processing. However, clustering unlabeled images, as a fundamental and classic machine learning problem, still lacks effective solution, particularly for large-scale datasets. In this paper, we propose a novel image clustering pipeline that leverages the powerful feature representation of large pre-trained models such as CLIP and cluster images effectively and efficiently at scale. We show that the pre-trained features are significantly more structured by further optimizing the rate reduction objective. The resulting features may significantly improve the clustering accuracy, e.g., from 57\% to 66\% on ImageNet-1k. Furthermore, by leveraging CLIP's image-text binding, we show how the new clustering method leads to a simple yet effective self-labeling algorithm that successfully works on unlabeled large datasets such as MS-COCO and LAION-Aesthetics. We will release the code in https://github.com/LeslieTrue/CPP.Comment: 21 pages, 13 figure

Unsupervised Manifold Linearizing and Clustering

We consider the problem of simultaneously clustering and learning a linear representation of data lying close to a union of low-dimensional manifolds, a fundamental task in machine learning and computer vision. When the manifolds are assumed to be linear subspaces, this reduces to the classical problem of subspace clustering, which has been studied extensively over the past two decades. Unfortunately, many real-world datasets such as natural images can not be well approximated by linear subspaces. On the other hand, numerous works have attempted to learn an appropriate transformation of the data, such that data is mapped from a union of general non-linear manifolds to a union of linear subspaces (with points from the same manifold being mapped to the same subspace). However, many existing works have limitations such as assuming knowledge of the membership of samples to clusters, requiring high sampling density, or being shown theoretically to learn trivial representations. In this paper, we propose to optimize the Maximal Coding Rate Reduction metric with respect to both the data representation and a novel doubly stochastic cluster membership, inspired by state-of-the-art subspace clustering results. We give a parameterization of such a representation and membership, allowing efficient mini-batching and one-shot initialization. Experiments on CIFAR-10, -20, -100, and TinyImageNet-200 datasets show that the proposed method is much more accurate and scalable than state-of-the-art deep clustering methods, and further learns a latent linear representation of the data

Dynamics of Sediment Microbial Functional Capacity and Community Interaction Networks in an Urbanized Coastal Estuary

Coastal estuaries and bays are exposed to both natural and anthropogenic environmental changes, inflicting intensive stress on the microbial communities inhabiting these areas. However, it remains unclear how microbial community diversity and their eco-functions are affected by anthropogenic disturbances rather than natural environmental changes. Here, we explored sediment microbial functional genes dynamics and community interaction networks in Hangzhou Bay (HZB), one of the most severely polluted bays on China’s eastern coast. The results indicated key microbial functional gene categories, including N, P, S, and aromatic compound metabolism, and stress response, displayed significant spatial dynamics along environmental gradients. Sensitive feedbacks of key functional gene categories to N and P pollutants demonstrated potential impacts of human-induced seawater pollutants to microbial functional capacity. Seawater ammonia and dissolved inorganic nitrogen (DIN) was identified as primary drivers in selecting adaptive populations and varying community composition. Network analysis revealed distinct modules that were stimulated in inner or outer bay. Importantly, the network keystone species, which played a fundamental role in community interactions, were strongly affected by N-pollutants. Our results provide a systematic understanding of the microbial compositional and functional dynamics in an urbanized coastal estuary, and highlighted the impact of human activities on these communities

Certains problèmes de coloration des sommets et une généralisation de la Hamilton-connectivité dans des graphes

The decomposition of graphs refers to the process of breaking down a complex graph into simpler, smaller components, often with the goal of analysing or solving problems related to the graph. It is an important tool to display the global structure and properties in a more fine-grained manner, and also useful in solving problems that involve finding specific structures in a graph. There are several common types of graph decomposition techniques that are widely used in graph theory and related fields, including tree decomposition, block decomposition, modular decomposition, hierarchical decomposition, etc. This thesis studies two kinds of vertex decomposition of a graph: proper colourings (decomposition into independent sets) and Hamilton-connectivity (decomposition into internally-disjoint paths between two sets where the paths cover all the vertices of graphs).La décomposition des graphes fait référence au processus de décomposer un graphe complexe en composantes plus simples et plus petites, souvent dans le but d'analyser ou de résoudre des problèmes liés au graphe. Il s'agit d'un outil important pour représenter la structure globale et les propriétés d'une manière plus détaillée. Il est aussi également utile pour résoudre des problèmes impliquant la recherche de structures spécifiques dans un graphe. Il existe plusieurs types courants de techniques de décomposition de graphe largement utilisées en théorie des graphes et dans des domaines connexes, notamment la décomposition en arbres, la décomposition en blocs, la décomposition modulaire, la décomposition hiérarchique, etc. Cette thèse étudie deux types de décomposition de sommets d'un graphe : les colorations propres (décomposition en ensembles indépendants) et la Hamilton-connectivité (décomposition en chemins internement disjoints entre deux ensembles où les chemins couvrent tous les sommets du graphe)

Certains problèmes de coloration des sommets et une généralisation de la Hamilton-connectivité dans des graphes

La décomposition des graphes fait référence au processus de décomposer un graphe complexe en composantes plus simples et plus petites, souvent dans le but d'analyser ou de résoudre des problèmes liés au graphe. Il s'agit d'un outil important pour représenter la structure globale et les propriétés d'une manière plus détaillée. Il est aussi également utile pour résoudre des problèmes impliquant la recherche de structures spécifiques dans un graphe. Il existe plusieurs types courants de techniques de décomposition de graphe largement utilisées en théorie des graphes et dans des domaines connexes, notamment la décomposition en arbres, la décomposition en blocs, la décomposition modulaire, la décomposition hiérarchique, etc. Cette thèse étudie deux types de décomposition de sommets d'un graphe : les colorations propres (décomposition en ensembles indépendants) et la Hamilton-connectivité (décomposition en chemins internement disjoints entre deux ensembles où les chemins couvrent tous les sommets du graphe).The decomposition of graphs refers to the process of breaking down a complex graph into simpler, smaller components, often with the goal of analysing or solving problems related to the graph. It is an important tool to display the global structure and properties in a more fine-grained manner, and also useful in solving problems that involve finding specific structures in a graph. There are several common types of graph decomposition techniques that are widely used in graph theory and related fields, including tree decomposition, block decomposition, modular decomposition, hierarchical decomposition, etc. This thesis studies two kinds of vertex decomposition of a graph: proper colourings (decomposition into independent sets) and Hamilton-connectivity (decomposition into internally-disjoint paths between two sets where the paths cover all the vertices of graphs)