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A graph $H$ is said to be $F$-saturated relative to $G$, if $H$ does not contain any copy of $F$ but the addition of any edge $e$ in $E(G)\backslash E(H)$ would create a copy of $F$. The minimum size of a $F$-saturated graph relative to $G$ is denoted by $sat(G,F)$. We use $sat(n,k,F)$ to denote the partite saturation number $sat(K_{k \times n},F)$, where $K_{k \times n}$ is the complete $k$-partite graph with $n$ vertices in each part. In this paper we prove that $sat(n,k, C_{\ell+1})= kn + O(\ell^2)$ holds for all $\ell \geq k \geq 3$, and $sat(n,k, C_{\ell+1})= kn + O(\ell)$ holds for $\ell \geq 60k+11$. Also we determine the exact value of $sat(n,k, C_{\ell+1})$ for all $4 \geq \ell \geq k \geq 3$., Comment: 20 pages

Floods are among the most common and devastating natural hazards, imposing immense costs on our society and economy due to their disastrous consequences. Recent progress in weather prediction and spaceborne flood mapping demonstrated the feasibility of anticipating extreme events and reliably detecting their catastrophic effects afterwards. However, these efforts are rarely linked to one another and there is a critical lack of datasets and benchmarks to enable the direct forecasting of flood extent. To resolve this issue, we curate a novel dataset enabling a timely prediction of flood extent. Furthermore, we provide a representative evaluation of state-of-the-art methods, structured into two benchmark tracks for forecasting flood inundation maps i) in general and ii) focused on coastal regions. Altogether, our dataset and benchmark provide a comprehensive platform for evaluating flood forecasts, enabling future solutions for this critical challenge. Data, code & models are shared at https://github.com/Multihuntr/GFF under a CC0 license., Comment: Accepted at NeurIPS 2024 Datasets & Benchmarks

A graph $G$ is $H$-saturated if $H$ is not a subgraph of $G$ but $H$ is a subgraph of $G + e$ for any edge $e$ in $\overline{G}$. The saturation number $sat(n,H)$ for a graph $H$ is the minimal number of edges in any $H$-saturated graph of order $n$. The $sat(n, K_{p_1} \cup K_{p_2} \cup K_{p_3})$ with $p_3 \ge p_1 + p_2$ was given in [Discrete Math. 347 (2024) 113868]. In this paper, $sat(n,K_{p_1} \cup K_{p_2} \cup K_{p_3} \cup K_{p_4})$ with $p_{i+1} - p_i \ge p_1$ for $2 \le i\le 3$ and $4\le p_1\le p_2$ is determined., Comment: 17pages

Let $H$ be a fixed graph, a graph G is $H$-saturated if it has no copy of $H$ in $G$, but the addition of any edge in $E(\overline G)$ to $G$ results in an $H$-subgraph. The saturation number sat$(n,H)$ is the minimum number of edges in an $H$-saturated graph on $n$ vertices. In this paper, we determine the saturation number sat$(n,P_7+tP_2)$ for $n\geq \frac {14}{5}t+27$ and characterize the extremal graphs for $n\geq \frac{14}{13}(3t+25)$., Comment: 11 pages

Let $H$ be a fixed graph. A graph $G$ is called {\it $H$-saturated} if $H$ is not a subgraph of $G$ but the addition of any missing edge to $G$ results in an $H$-subgraph. The {\it saturation number} of $H$, denoted $sat(n,H)$, is the minimum number of edges over all $H$-saturated graphs of order $n$, and $Sat(n,H)$ denote the family of $H$-saturated graphs with $sat(n,H)$ edges and $n$ vertices. In this paper, we resolve a conjecture of Chen and Yuan in[Discrete Math. 347(2024)113868] by determining $Sat(n,K_p\cup (t-1)K_q)$ for every $2\le p\le q$ and $t\ge 2$., Comment: 7 pages, 2 figures

Color centers in silicon carbide (SiC) have demonstrated significant promise for quantum information processing. However, the undesirable ionization process that occurs during optical manipulation frequently causes fluctuations in the charge state and performance of these defects, thereby restricting the effectiveness of spin-photon interfaces. Recent predictions indicate that divacancy defects near stacking faults possess the capability to stabilize their neutral charge states, thereby providing robustness against photoionization effects. In this work, we present a comprehensive protocol for the scalable and targeted fabrication of single divacancy arrays in 4H-SiC using a high-resolution focused helium ion beam. Through photoluminescence emission (PLE) experiments, we demonstrate long-term emission stability with minimal linewidth shift ($\sim$ 50 MHz over 3 hours) for the single c-axis divacancies within stacking faults. By measuring the ionization rate for different polytypes of divacancies, we found that the divacancies within stacking faults are more robust against resonant excitation. Additionally, angle-resolved PLE spectra reveal their two resonant-transition lines with mutually orthogonal polarizations. Notably, the PLE linewidths are approximately 7 times narrower and the spin-coherent times are 6 times longer compared to divacancies generated via carbon-ion implantation. These findings highlight the immense potential of SiC divacancies for on-chip quantum photonics and the construction of efficient spin-to-photon interfaces, indicating a significant step forward in the development of quantum technologies., Comment: 11 pages, 4 figures

A graph is $(\mathcal{I}, \mathcal{F})$-colorable if its vertex set can be partitioned into two subsets, one of which is an independent set, and the other induces a forest. In this paper, we prove that every planar graph without $5^{-}$-cycles at distance less than $3$ is $(\mathcal{I}, \mathcal{F})$-colorable., Comment: 11 pages, 3 figures

In this paper, we prove the norm inflation and get the ill-posedness for the modified Camassa-Holm equation in $B_{\infty,1}^0$. Therefore we completed all well-posedness and ill-posedness problem for the modified Camassa-Holm equation in all critical spaces $B_{p,1}^\frac{1}{p}$ with $p\in[1,\infty]$.

In this paper,for a given conservative solution, we introduce a set of auxiliary variables tailored to this particular solution, and prove that these variables satisfy a particular semilinear system having unique solutions. In turn, we get the uniqueness of the conservative solution in the original variables., Comment: arXiv admin note: text overlap with arXiv:1401.0312, arXiv:1509.08569 by other authors

Nowadays, artificial intelligence (AI) technology with large models plays an increasingly important role in both academia and industry. It also brings a rapidly increasing demand for the computing power of the hardware. As the computing demand for AI continues to grow, the growth of hardware computing power has failed to keep up. This has become a significant factor restricting the development of AI. The augmentation of hardware computing power is mainly propelled by the escalation of transistor density and chip area. However, the former is impeded by the termination of the Moore's Law and Dennard scaling, and the latter is significantly restricted by the challenge of disrupting the legacy fabrication equipment and process. In recent years, advanced packaging technologies that have gradually matured are increasingly used to implement bigger chips that integrate multiple chiplets, while still providing interconnections with chip-level density and bandwidth. Compared to conventional high-performance computing paradigms such as multi-accelerator and datacenter-scale computing, Wafer-scale Computing shows remarkable advantages in communication bandwidth, integration density, and programmability potential. Not surprisingly, disruptive Wafer-scale Computing also brings unprecedented design challenges for hardware architecture, design-system-technology co-optimization, power and cooling systems, and compiler tool chain. At present, there are no comprehensive surveys summarizing the current state and design insights of Wafer-scale Computing. This paper aims to take the first step to help academia and industry review existing wafer-scale chips and essential technologies in a one-stop manner. So that people can conveniently grasp the basic knowledge and key points, understand the achievements and shortcomings of existing research, and contribute to this promising research direction.

Recently there have been many algorithms proposed for the classification of very high resolution whole slide images (WSIs). These new algorithms are mostly focused on finding novel ways to combine the information from small local patches extracted from the slide, with an emphasis on effectively aggregating more global information for the final predictor. In this paper we thoroughly explore different key design choices for WSI classification algorithms to investigate what matters most for achieving high accuracy. Surprisingly, we found that capturing global context information does not necessarily mean better performance. A model that captures the most global information consistently performs worse than a model that captures less global information. In addition, a very simple multi-instance learning method that captures no global information performs almost as well as models that capture a lot of global information. These results suggest that the most important features for effective WSI classification are captured at the local small patch level, where cell and tissue micro-environment detail is most pronounced. Another surprising finding was that unsupervised pre-training on a larger set of 33 cancers gives significantly worse performance compared to pre-training on a smaller dataset of 7 cancers (including the target cancer). We posit that pre-training on a smaller, more focused dataset allows the feature extractor to make better use of the limited feature space to better discriminate between subtle differences in the input patch.

In this paper, we first establish the local well-posednesss for the Cauchy problem of a modified Camassa-Holm (MOCH) equation in critical Besov spaces $B^{\frac 1 p}_{p,1}$ with $1\leq p<+\infty.$ The obtained results improve considerably the recent result in \cite{Luo1}. Then we show the persiscence property of MOCH.

In this paper, we concentrate on the Novikov equation. We provide a description of the solution in a neighborhood of each singular point.

Gravity is the driving force of star formation. Although gravity is caused by the presence of matter, its role in complex regions is still unsettled. One effective way to study the pattern of gravity is to compute the accretion it exerts on the gas by providing gravitational acceleration maps. A practical way to study acceleration is by computing it using 2D surface density maps, yet whether these maps are accurate remains uncertain. Using numerical simulations, we confirm that the accuracy of the acceleration maps $\mathbf a_{\rm 2D}(x,y)$ computed from 2D surface density are good representations for the mean acceleration weighted by mass. Due to the under-estimations of the distances from projected maps, the magnitudes of accelerations will be over-estimated $|\mathbf a_{\rm 2D}(x,y)| \approx 2.3 \pm 1.8 \; |\mathbf a_{\rm 3D}^{\rm proj}(x,y)|$, where $\mathbf a_{\rm 3D}^{\rm proj}(x,y)$ is mass-weighted projected gravitational acceleration, yet $\mathbf a_{\rm 2D}(x,y)$ and $ \mathbf a_{\rm 3D}^{\rm proj}(x,y)$ stay aligned within 20$^{\circ}$. Significant deviations only occur in regions where multiple structures are present along the line of sight. The acceleration maps estimated from surface density provide good descriptions of the projection of 3D acceleration fields. We expect this technique useful in establishing the link between cloud morphology and star formation, and in understanding the link between gravity and other processes such as the magnetic field. A version of the code for calculating surface density gravitational potential is available at \url{https://github.com/zhenzhen-research/phi_2d}., Comment: Accepted by MNRAS

Optically addressable spin defects in silicon carbide (SiC) have emerged as attractable platforms for various quantum technologies. However, the low photon count rate significantly limits their applications. We strongly enhanced the brightness by 7 times and spin-control strength by 14 times of single divacancy defects in 4H-SiC membranes using surface plasmon generated by gold film coplanar waveguides. The mechanism of the plasmonic-enhanced effect is further studied by tuning the distance between single defects and the surface of the gold film. A three-energy-level model is used to determine the corresponding transition rates consistent with the enhanced brightness of single defects. Lifetime measurements also verified the coupling between defects and surface plasmons. Our scheme is low-cost, without complicated microfabrication and delicate structures, which is applicable for other spin defects in different materials. This work would promote developing spin defect-based quantum applications in mature SiC materials.

This brief aims at the issue of globally composite-learning-based neural fast finite-time (F-FnT) tracking control for a class of uncertain systems in strict-feedback form subject to nonlinearly periodic disturbances. First, uncertain dynamics with periodic parameters are identified by incorporating Fourier series expansion (FSE) into an intelligent estimator, which leverages the feedback of newly designed prediction errors in updating weights to boost learning performance. Then, a novel switching mechanism is constructed to fulfill smooth switching from the composite FSE-based neural controller to robust control law when the inputs of the intelligent estimator transcend the valid approximation domain. By fusing the switching mechanism with an improved F-FnT backstepping algorithm, the globally F-FnT boundedness of all variables in the closed-loop system is guaranteed. Finally, a simulation study is conducted to evince the availability of the theoretical result., Comment: 5 pages, 3 figures

Predicting panoramic indoor lighting from a single perspective image is a fundamental but highly ill-posed problem in computer vision and graphics. To achieve locale-aware and robust prediction, this problem can be decomposed into three sub-tasks: depth-based image warping, panorama inpainting and high-dynamic-range (HDR) reconstruction, among which the success of panorama inpainting plays a key role. Recent methods mostly rely on convolutional neural networks (CNNs) to fill the missing contents in the warped panorama. However, they usually achieve suboptimal performance since the missing contents occupy a very large portion in the panoramic space while CNNs are plagued by limited receptive fields. The spatially-varying distortion in the spherical signals further increases the difficulty for conventional CNNs. To address these issues, we propose a local-to-global strategy for large-scale panorama inpainting. In our method, a depth-guided local inpainting is first applied on the warped panorama to fill small but dense holes. Then, a transformer-based network, dubbed PanoTransformer, is designed to hallucinate reasonable global structures in the large holes. To avoid distortion, we further employ cubemap projection in our design of PanoTransformer. The high-quality panorama recovered at any locale helps us to capture spatially-varying indoor illumination with physically-plausible global structures and fine details., Comment: 10 pages, 11 figures

In this work, we give the sharp upper bound for the number of cliques in graphs with bounded odd circumferences. This generalized Tur\'an-type result is an extension of the celebrated Erd\H{o}s and Gallai theorem and a strengthening of Luo's recent result. The same bound for graphs with bounded even circumferences is a trivial application of the theorem of Li and Ning.

Let $H$ be a graph and $p$ be an integer. The edge blow-up $H^p$ of $H$ is the graph obtained from replacing each edge in $H$ by a copy of $K_p$ where the new vertices of the cliques are all distinct. Let $C_k$ and $P_k$ denote the cycle and path of length $k$, respectively. In this paper, we find sharp upper bounds for $ex(n,K_3,C_3^3)$ and the exact value for $ ex(n,K_3,P_3^3)$ and determine the graphs attaining these bounds.

Let $f_k(n,H)$ denote the maximum number of edges not contained in any monochromatic copy of~$H$ in a $k$-coloring of the edges of $K_n$, and let $ex(n,H)$ denote the Tur\'an number of $H$. In place of $f_2(n,H)$ we simply write $f(n,H)$. Keevash and Sudakov proved that $f(n,H)=ex(n,H)$ if $H$ is an edge-critical graph or $C_4$ and asked if this equality holds for any graph $H$. All known exact values of this question require $H$ to contain at least one cycle. In this paper we focus on acyclic graphs and have the following results: (1) We prove $f(n,H)=ex(n,H)$ when $H$ is a spider or a double broom. (2) A \emph{tail} in $H$ is a path $P_3=v_0v_1v_2$ such that $v_2$ is only adjacent to $v_1$ and $v_1$ is only adjacent to $v_0,v_2$ in $H$. We obtain a tight upper bound for $f(n,H)$ when $H$ is a bipartite graph with a tail. This result provides the first bipartite graphs which answer the question of Keevash and Sudakov in the negative. (3) Liu, Pikhurko and Sharifzadeh asked if $f_k(n,T)=(k-1)ex(n,T)$ when $T$ is a tree. We provide an upper bound for $f_{2k}(n,P_{2k})$ and show it is tight when $2k-1$ is prime. This provides a negative answer to their question.