arXiv subjects

Jingshi Cui

Publications and source records attributed to Jingshi Cui.

10 recordsLinked to original sources

arXiv2026

Horizontal inverse mean curvature flow in the Heisenberg group

Huisken and Ilmanen [J. Differential Geom., 2001] created the theory of weak solutions for inverse mean curvature flows (IMCF) of hypersurfaces on Riemannian manifolds, and proved successfully a Riemannian version of the Penrose inequality. The present paper investigates and constructs a sub-Riemannian version of the theory of weak solutions for inverse mean curvature flows of surfaces in the first Heisenberg group $\mathbb{H}^1$. The level set formulation of the IMCF in $\mathbb{H}^1$ is given by (0.1), where $Ω\subset \mathbb{H}^1$ is an open set with smooth boundary, and $Ω^{c} = \mathbb{H}^1\setminus Ω= \{ u \leq 0\}$ is bounded. Let $w_p = \exp \left( \frac{u_p}{1-p}\right)$ and $w_p$ satisfies (0.2). Following the argument by Moser, the key ingredient in proving the existence of weak solutions to (0.1) is to establish a uniform interior estimate for $|\nabla_{0} u_p|$. However, due to the lack of boundary continuity of $|\nabla_{0} u_p| = (p-1)\frac{|\nabla_{0} w_p|}{w_p} \in C^{0,β}(Ω)$ ($0< β<1, p>1$) by Zhong and Mukherjee [Anal. PDE, 2021], the standard method in [R. Moser, J. Eur. Math. Soc., 2007] cannot be applied to obtain a uniform interior estimate for $|\nabla_{0} u_p|$. Fortunately, the present paper discovers two refined inequalities: Harnack inequality and Lipschitz estimate for $w_p$, which allow one to obtain interior estimates for $|\nabla_{0} u_p|$ independent of $p$. By further combining them with Arzel$\grave{\rm a} $-Ascoli theorem, the weak solution of (0.1) can then be generated as the limit of $u_p$ as $p \to 1$, where $w_p = \exp \left( \frac{u_p}{1-p}\right)$ and $w_p$ is of solutions to (0.2). As an important application of the IMCF in $\mathbb{H}^1$, a positive answer to an open problem posed in [F. Montefalcon, Ann. Mat. Pura Appl. (4), 2014]:Heintze-Karcher inequality in $\mathbb{H}^1$ is provided.

math.DG↗

arXiv2025

Legendrian curve flow in Sasakian sub-Riemannian 3-manifolds

In this paper, we introduce a kind of inverse mean curvature flow (1.2) in a Sasakian sub-Riemannian 3-manifold $M$ for Legendrian curves, which slightly differs from the classical one, and confirm that this flow preserves the Legendrian condition and increases the length of curves. We establish the long-time existence of the flow (1.2) when the Webster scalar curvature $W$ of $M$ satisfies $ W \in (-\infty, \bar{W}_{0} )\cup \{ 0\} \cup (W_{0}, +\infty)$, where $\bar{W}_{0} <0$ and $W_{0} >0$ are constants. Moreover, we derive that the local limit curve (the asymptotic behavior) along the flow (1.2) is a geodesic of vanishing curvature when $W \geq 0$, wherea it is a geodesic of nonvanishing curvature when $W$ is a negative constant. Specially, in the first Heisenberg group $\mathbb{M}(0)$, we further construct a length-preserving flow (1.3) via a dilation of the flow (1.2) and show that closed Legendrian curves converge to Euclidean helices with vertical axis. By exploiting the properties of the flow (1.3), we establish a Minkowski-type formula for Legendrian curves in $\mathbb{M}(0)$ and provide a new proof of the fact that the total curvature of $γ\subset \mathbb{M}(0)$ with strictly positive curvature equals $2π$.

math.DG↗

arXiv2025

ScenGAN: Attention-Intensive Generative Model for Uncertainty-Aware Renewable Scenario Forecasting

To address the intermittency of renewable energy source (RES) generation, scenario forecasting offers a series of stochastic realizations for predictive objects with superior flexibility and direct views. Based on a long time-series perspective, this paper explores uncertainties in the realms of renewable power and deep learning. Then, an uncertainty-aware model is meticulously designed for renewable scenario forecasting, which leverages an attention mechanism and generative adversarial networks (GANs) to precisely capture complex spatial-temporal dynamics. To improve the interpretability of uncertain behavior in RES generation, Bayesian deep learning and adaptive instance normalization (AdaIN) are incorporated to simulate typical patterns and variations. Additionally, the integration of meteorological information, forecasts, and historical trajectories in the processing layer improves the synergistic forecasting capability for multiscale periodic regularities. Numerical experiments and case analyses demonstrate that the proposed approach provides an appropriate interpretation for renewable uncertainty representation, including both aleatoric and epistemic uncertainties, and shows superior performance over state-of-the-art methods.

cs.LG↗

arXiv2025

Joint Planning and Operations of Wind Power under Decision-dependent Uncertainty

We study a joint wind farm planning and operational scheduling problem under decision-dependent uncertainty. The objective is to determine the optimal number of wind turbines at each location to minimize total cost, including both investment and operational expenses. Due to the stochastic nature and geographical heterogeneity of wind power, fluctuations across dispersed wind farms can partially offset one another, thereby influencing the distribution of aggregated wind power generation-a phenomenon known as the smoothing effect. Effectively harnessing this effect requires strategic capacity allocation, which introduces decision-dependent uncertainty into the planning process. To address this challenge, we propose a two-stage distributionally robust optimization model with a decision-dependent Wasserstein ambiguity set, in which both the distribution and the radius are modeled as functions of the planning decisions, reflecting the statistical characteristics of wind power resources. Then, we reformulate the model as a mixed-integer second-order cone program, and the optimal objective value provides a probabilistic guarantee on the out-of-sample performance. To improve computational efficiency, we develop a constraint generation based solution framework that accelerates the solution procedure by hundreds of times. Numerical experiments using different datasets validate the effectiveness of the solution framework and demonstrate the superior performance of the proposed model.

math.OC↗

arXiv2025

Spatiotemporal Prediction of Electric Vehicle Charging Load Based on Large Language Models

The rapid growth of EVs and the subsequent increase in charging demand pose significant challenges for load grid scheduling and the operation of EV charging stations. Effectively harnessing the spatiotemporal correlations among EV charging stations to improve forecasting accuracy is complex. To tackle these challenges, we propose EV-LLM for EV charging loads based on LLMs in this paper. EV-LLM integrates the strengths of Graph Convolutional Networks (GCNs) in spatiotemporal feature extraction with the generalization capabilities of fine-tuned generative LLMs. Also, EV-LLM enables effective data mining and feature extraction across multimodal and multidimensional datasets, incorporating historical charging data, weather information, and relevant textual descriptions to enhance forecasting accuracy for multiple charging stations. We validate the effectiveness of EV-LLM by using charging data from 10 stations in California, demonstrating its superiority over the other traditional deep learning methods and potential to optimize load grid scheduling and support vehicle-to-grid interactions.

eess.SP↗

arXiv2025

Locally constrained flows and sharp Michael-Simon inequalities in hyperbolic space

Brendle [6] successfully establishes the sharp Michael-Simon inequality for mean curvature on Riemannian manifolds with nonnegative sectional curvature ($\mathcal{K} \geq 0$), and the proof relies on the Alexandrov-Bakelman-Pucci method. Nevertheless, this result cannot be extended to hyperbolic space $\mathbb{H}^{n+1}$ ($\mathcal{K} = -1$), as demonstrated by Counterexample 1.7. In the present paper, we propose Conjectures 1.8 and 1.9 concerning the hyperbolic version of the sharp Michael-Simon type inequality for $k$-th mean curvatures. However, the proof method in \cite{B21} failed to verify the validity of these conjectures. Recently, the authors [12] proved Conjectures 1.8 and 1.9 only for $h$-convex hypersurfaces by means of the Brendle-Guan-Li's flow. This paper aims to utilize other types of curvature flows to prove Conjectures 1.8 and 1.9 for hypersurfaces with weaker convexity conditions. For $k = 1$, we first investigate a new locally constrained mean curvature flow (1.9) in $\mathbb{H}^{n+1}$ and prove its longtime existence and exponential convergence. Then, the sharp Michael-Simon type inequality for mean curvature of starshaped hypersurfaces in $\mathbb{H}^{n+1}$ is confirmed through the flow (1.9). For $k \geq 2$, the sharp Michael-Simon inequality for $k$-th mean curvatures of starshaped, strictly $k$-convex hypersurfaces in $\mathbb{H}^{n+1}$ is proven using the locally constrained inverse curvature flow (1.11) introduced by Scheuer and Xia [31].

math.DG↗

arXiv2024

An expanding curvature flow and the (p,q)-Christoffel-Minkowski problems

The present paper introduces a new class of geometric measures, the k-th (p,q)-mixed curvature measures, and a natural correspondence-(p,q)-Christoffel-Minkowski problem is proposed. The (p,q)-Christoffel-Minkowski problem posed here can be regarded as a natural generalization of the L_p Christoffel-Minkowski problem and Lp dual Minkowski problem. We investigate and arrive at the existence of smooth solution to the (p,q)-Christoffel-Minkowski problem by a type of expanding curvature flow. Furthermore, the uniqueness result of solutions to the (p,q)-Christoffel-Minkowski problem shall be discussed.

math.DG↗

arXiv2024

Michael-Simon type inequalities in hyperbolic space $\mathbb{H}^{n+1}$ via Brendle-Guan-Li's flows

In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space $\mathbb{H}^{n+1}$ based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li, provided that $M$ is $h$-convex and $f$ is a positive smooth function, where $λ^{'}(r)=\rm{cosh}$$r$. In particular, when $f$ is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang. Further, we also establish and confirm a new sharp Michael-Simon inequality for the $k$-th mean curvatures in $\mathbb{H}^{n+1}$ by virtue of the Brendle-Guan-Li's flow, provided that $M$ is $h$-convex and $Ω$ is the domain enclosed by $M$. In particular, when $f$ is of constant and $k$ is odd, (0.2) is exactly the weighted Alexandrov-Fenchel inequalities proven by Hu, Li, and Wei.

math.DG↗

arXiv2019

Robust Data-driven Profile-based Pricing Schemes

To enable an efficient electricity market, a good pricing scheme is of vital importance. Among many practical schemes, customized pricing is commonly believed to be able to best exploit the flexibility in the demand side. However, due to the large volume of consumers in the electricity sector, such task is simply too overwhelming. In this paper, we first compare two data driven schemes: one based on load profile and the other based on user's marginal system cost. Vulnerability analysis shows that the former approach may lead to loopholes in the electricity market while the latter one is able to guarantee the robustness, which yields our robust data-driven pricing scheme. Although k-means clustering is in general NP-hard, surprisingly, by exploiting the structure of our problem, we design an efficient yet optimal k-means clustering algorithm to implement our proposed scheme.

cs.CE↗

arXiv2019

Vulnerability Analysis for Data Driven Pricing Schemes

Data analytics and machine learning techniques are being rapidly adopted into the power system, including power system control as well as electricity market design. In this paper, from an adversarial machine learning point of view, we examine the vulnerability of data-driven electricity market design. More precisely, we follow the idea that consumer's load profile should uniquely determine its electricity rate, which yields a clustering oriented pricing scheme. We first identify the strategic behaviors of malicious users by defining a notion of disguising. Based on this notion, we characterize the sensitivity zones to evaluate the percentage of malicious users in each cluster. Based on a thorough cost benefit analysis, we conclude with the vulnerability analysis.

cs.LG↗