Danping Yang

Danping Yang, Hong Wang

We study the Dirichlet boundary-value problem of steady-state two-sided variable-coefficient conservative space-fractional diffusion equations. We show that the Galerkin weak formulation, which was proved to be coercive and continuous for a constant-coefficient analogue of the problem, loses its coercivity. We characterize the solution to the variable-coefficient problem in terms of the solutions of second-order diffusion equations along with a two-sided fractional integral equation. We then derive a Petrov-Galerkin formulation for this problem and prove that the weak formulation is weakly coercive and so the problem is well posed. We then prove high-order regularity estimates of the true solution in a properly chosen norm of Riemann-Liouville derivatives.

Youquan Xian, Xueying Zeng, Duancheng Xuan et al.

Blockchain smart contracts have catalyzed the development of decentralized applications across various domains, including decentralized finance. However, due to constraints in computational resources and the prevalence of data silos, current smart contracts face significant challenges in fully leveraging the powerful capabilities of Large Language Models (LLMs) for tasks such as intelligent analysis and reasoning. To address this gap, this paper proposes and implements a universal framework for integrating LLMs with blockchain data, {\sysname}, effectively overcoming the interoperability barriers between blockchain and LLMs. By combining semantic relatedness with truth discovery methods, we introduce an innovative data aggregation approach, {\funcname}, which significantly enhances the accuracy and trustworthiness of data generated by LLMs. To validate the framework's effectiveness, we construct a dataset consisting of three types of questions, capturing Q\&A interactions between 10 oracle nodes and 5 LLM models. Experimental results demonstrate that, even with 40\% malicious nodes, the proposed solution improves data accuracy by an average of 17.74\% compared to the optimal baseline. This research not only provides an innovative solution for the intelligent enhancement of smart contracts but also highlights the potential for deep integration between LLMs and blockchain technology, paving the way for more intelligent and complex applications of smart contracts in the future.