arXiv subjects
Publications and source records attributed to Xingjian Zhou.
In this paper, we establish general monotone quantities and sharp mass-capacity inequalities related to $p$-capacitary functions in $3$-dimensional asymptotically flat half-spaces of simple topology with nonnegative scalar curvature and nonnegative boundary mean curvature. These inequalities attain equality on a Schwarzschild half-space outside a rotationally symmetric half sphere.
Neural network (NN) verification aims to formally verify properties of NNs, which is crucial for ensuring the behavior of NN-based models in safety-critical applications. In recent years, the community has developed many NN verifiers and benchmarks to evaluate them. However, existing benchmarks typically lack ground-truth for hard instances where no current verifier can verify the property and no counterexample can be found. This makes it difficult to validate the soundness of a verifier, when it claims verification on such challenging instances that no other verifier can handle. In this work, we develop a new benchmark for NN verification, named SoundnessBench, specifically for testing the soundness of NN verifiers. SoundnessBench consists of instances with deliberately inserted counterexamples that are hidden from adversarial attacks commonly used to find counterexamples. Thereby, it can identify false verification claims when hidden counterexamples are known to exist. We design a training method to produce NNs with hidden counterexamples and systematically construct our SoundnessBench with instances across various model architectures, activation functions, and input data. We demonstrate that our training effectively produces hidden counterexamples and our SoundnessBench successfully identifies bugs in state-of-the-art NN verifiers. Our code is available at https://github.com/mvp-harry/SoundnessBench and our dataset is available at https://huggingface.co/datasets/SoundnessBench/SoundnessBench.
In this paper, we deal with an overdetermined problem for the $k$-Hessian equation ($1\leq k<\frac n2$) in the exterior domain and prove the corresponding ball characterizations. Since that Weinberger type approach seems to fail to solve the problem, we give a new perspective to solve exterior overdetermined problem by combining two integral identities and geometric inequalities inspired by Brandolini-Nitsch-Salani's results \cite{BNS}. Meanwhile, we establish general monotone formulas to derive geometric inequalities related to $k$-admissible solution $u$ in $\mathbb R^n\setminusΩ$, where $Ω$ is smooth, $k$-convex and star-shaped domain, which constructed by Ma-Zhang\cite{MZ} and Xiao\cite{xiao}.
In this paper, we derive general monotone quantities and geometric inequalities associated with $p$-capacitary functions in asymptotically flat $3$-manifolds with simple topology and nonnegative scalar curvature. The inequalities become equalities on the spatial Schwarzschild manifolds outside rotationally symmetric spheres. This generalizes Miao's result \cite{M} from $p=2$ to $p\in (1, 3)$. As applications, we recover mass-to-$p$-capacity and $p$-capacity-to-area inequalities due to Bray-Miao \cite{BM} and Xiao \cite{Xiao}.
Certain solutions of autonomous PDEs without any boundary conditions describing the spatiotemporal evolution of a dependent variable in an unbounded spatial domain can be characterised as a travelling wave moving with constant speed. In the simplest case, such PDEs can be reduced to a single autonomous second order ODE with one dependent variable. For certain parameter values it has been shown using perturbations methods in combination with ansätze that numerous such second order ODEs have analytical travelling wave solutions described by a simple sigmoid function. However, this methodology provides no leverage on the problem of finding a generalised class of models possessing such analytical travelling wave solutions. The most efficient methods for both finding analytical solutions and constructing classes of ODEs are based on Lie symmetries which are transformations known as one parameter $\mathcal{C}^{\infty}$ diffeomorphisms mapping solutions to other solutions. Recently, analytical solutions of a second order ODE encapsulating numerous oscillatory models as well as some of the previously mentioned travelling wave models with simple analytical solutions have been found by means of a two dimensional Lie algebra. Based on this Lie algebra, we construct the most general class of integrable autonomous second order ODEs for which these symmetries are manifest. Moreover, we show that a sub-class of second order ODEs has simple analytical travelling wave solutions described by a sigmoid function. Lastly, we characterise the action of the two symmetries in this Lie algebra on these simple analytical travelling wave solutions and we relate our sub-class of ODEs to previously known integrable travelling wave models.