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We classify superpotentials for the Hamiltonian system corresponding to the cohomogeneity one gradient Ricci soliton equations. Aside from recovering known examples of superpotentials for steady solitons, we find a new superpotential on a specific case of the B\'erard-Bergery-Calabi ansatz. The latter is used to obtain an explicit formula for a steady complete soliton with an equidistant family of hypersurfaces given by circle bundles over $S^2\times S^2$. There are no superpotentials in the non-steady case in dimensions greater than 2, even if polynomial coefficients are allowed. We also briefly discuss generalised first integrals and the limitations of some known methods of finding them., Comment: 28 pages

Deciphering oracle bone scripts plays an important role in Chinese archaeology and philology. However, a significant challenge remains due to the scarcity of oracle character images. To overcome this issue, we propose Diff-Oracle, a novel approach based on diffusion models to generate a diverse range of controllable oracle characters. Unlike traditional diffusion models that operate primarily on text prompts, Diff-Oracle incorporates a style encoder that utilizes style reference images to control the generation style. This encoder extracts style prompts from existing oracle character images, where style details are converted into a text embedding format via a pretrained language-vision model. On the other hand, a content encoder is integrated within Diff-Oracle to capture specific content details from content reference images, ensuring that the generated characters accurately represent the intended glyphs. To effectively train Diff-Oracle, we pre-generate pixel-level paired oracle character images (i.e., style and content images) by an image-to-image translation model. Extensive qualitative and quantitative experiments are conducted on datasets Oracle-241 and OBC306. While significantly surpassing present generative methods in terms of image generation, Diff-Oracle substantially benefits downstream oracle character recognition, outperforming all existing SOTAs by a large margin. In particular, on the challenging OBC306 dataset, Diff-Oracle leads to an accuracy gain of 7.70% in the zero-shot setting and is able to recognize unseen oracle character images with the accuracy of 84.62%, achieving a new benchmark for deciphering oracle bone scripts.

Phantom scalar theories are widely considered in cosmology, but rarely at the quantum level, where they give rise to negative-energy ghost particles. These cause decay of the vacuum into gravitons and photons, violating observational gamma-ray limits unless the ghosts are effective degrees of freedom with a cutoff $\Lambda$ at the few-MeV scale. We update the constraints on this scale, finding that $\Lambda \lesssim 19$ MeV. We further explore the possible coupling of ghosts to a light, possibly massless, hidden sector particle, such as a sterile neutrino. Vacuum decays can then cause the dark matter density of the universe to grow at late times. The combined phantom plus dark matter fluid has an effective equation of state $w < -1$, and functions as a new source of dark energy. We derive constraints from cosmological observables on the rate of vacuum decay into such a phantom fluid. We find a mild preference for the ghost model over the standard cosmological one, and a modest amelioration of the Hubble and $S_8$ tensions., Comment: v1: 36 pages, 20 figures; v2: 37 pages, modified and expanded comment on big rip singularity in the conclusions, fixed typos, added extra references; this version matches the published one

AI has made significant progress in solving math problems, but geometry problems remain challenging due to their reliance on both text and diagrams. In the text description, symbolic characters such as "$\triangle$ABC" often serve as a bridge to connect the corresponding diagram. However, by simply tokenizing symbolic characters into individual letters (e.g., 'A', 'B' and 'C'), existing works fail to study them explicitly and thus lose the semantic relationship with the diagram. In this paper, we develop a symbolic character-aware model to fully explore the role of these characters in both text and diagram understanding and optimize the model under a multi-modal reasoning framework. In the text encoder, we propose merging individual symbolic characters to form one semantic unit along with geometric information from the corresponding diagram. For the diagram encoder, we pre-train it under a multi-label classification framework with the symbolic characters as labels. In addition, we enhance the geometry diagram understanding ability via a self-supervised learning method under the masked image modeling auxiliary task. By integrating the proposed model into a general encoder-decoder pipeline for solving geometry problems, we demonstrate its superiority on two benchmark datasets, including GeoQA and Geometry3K, with extensive experiments. Specifically, on GeoQA, the question-solving accuracy is increased from 60.0\% to 64.1\%, achieving a new state-of-the-art accuracy; on Geometry3K, we reduce the question average solving steps from 6.9 down to 6.0 with marginally higher solving accuracy., Comment: ACM MM 2023 Accepted

Anomaly detection under open-set scenario is a challenging task that requires learning discriminative fine-grained features to detect anomalies that were even unseen during training. As a cheap yet effective approach, data augmentation has been widely used to create pseudo anomalies for better training of such models. Recent wisdom of augmentation methods focuses on generating random pseudo instances that may lead to a mixture of augmented instances with seen anomalies, or out of the typical range of anomalies. To address this issue, we propose a novel saliency-guided data augmentation method, SaliencyCut, to produce pseudo but more common anomalies which tend to stay in the plausible range of anomalies. Furthermore, we deploy a two-head learning strategy consisting of normal and anomaly learning heads, to learn the anomaly score of each sample. Theoretical analyses show that this mechanism offers a more tractable and tighter lower bound of the data log-likelihood. We then design a novel patch-wise residual module in the anomaly learning head to extract and assess the fine-grained anomaly features from each sample, facilitating the learning of discriminative representations of anomaly instances. Extensive experiments conducted on six real-world anomaly detection datasets demonstrate the superiority of our method to competing methods under various settings.

We analytically extend the 5D Myers-Perry metric through the event and Cauchy horizons by defining Eddington-Finkelstein-type coordinates. Then, we use the orthonormal frame formalism to formulate and perform separation of variables on the massive Dirac equation, and analyse the asymptotic behaviour at the horizons and at infinity of the solutions to the radial ordinary differential equation (ODE) thus obtained. Using the essential self-adjointness result of Finster and R\"oken and Stone's formula, we obtain an integral spectral representation of the Dirac propagator for spinors with low masses and suitably bounded frequency spectra in terms of resolvents of the Dirac Hamiltonian, which can in turn be expressed in terms of Green's functions of the radial ODE., Comment: Minor corrections and improvements, published version, 23 pages

We consider the massive Dirac equation in the exterior region of the 5-dimensional Myers-Perry black hole. Using the resulting ODEs obtained from the separation of variables of the Dirac equation, we construct an integral spectral representation for the solution of the Cauchy problem with compactly supported smooth initial data. We then prove that the probability of presence of a Dirac particle to be in any compact region of space decays to zero as $t\to\infty$, in analogy with the case of the Dirac operator in the Kerr-Newman geometry., Comment: Minor corrections and improvements, published version, 15 pages

In the last a few decades, deep neural networks have achieved remarkable success in machine learning, computer vision, and pattern recognition. Recent studies however show that neural networks (both shallow and deep) may be easily fooled by certain imperceptibly perturbed input samples called adversarial examples. Such security vulnerability has resulted in a large body of research in recent years because real-world threats could be introduced due to vast applications of neural networks. To address the robustness issue to adversarial examples particularly in pattern recognition, robust adversarial training has become one mainstream. Various ideas, methods, and applications have boomed in the field. Yet, a deep understanding of adversarial training including characteristics, interpretations, theories, and connections among different models has still remained elusive. In this paper, we present a comprehensive survey trying to offer a systematic and structured investigation on robust adversarial training in pattern recognition. We start with fundamentals including definition, notations, and properties of adversarial examples. We then introduce a unified theoretical framework for defending against adversarial samples - robust adversarial training with visualizations and interpretations on why adversarial training can lead to model robustness. Connections will be also established between adversarial training and other traditional learning theories. After that, we summarize, review, and discuss various methodologies with adversarial attack and defense/training algorithms in a structured way. Finally, we present analysis, outlook, and remarks of adversarial training.