Multi-horseshoe dense property and intermediate entropy property of ergodic measures with same level

Katok conjectured that for every $C^{2}$ diffeomorphism $f$ on a Riemannian manifold $X$, the set $\{h_{\mu}(f):\mu \text{ is an ergodic measure for } (X,f)\}$ includes $[0, h_{top}(f))$. In this paper we obtained a refined Katok's conjecture on intermediate metric entropies of ergodic measures with same level that for a transitive locally maximal hyperbolic set or a transitive two-side subshit of finite type, one has $$\mathrm{Int}(\{h_{\mu}(f):\mu\in M_{erg}(f,X) \text{ and }\int\varphi d\mu=a\})=\mathrm{Int}(\{h_{\mu}(f):\mu\in M(f,X) \text{ and }\int\varphi d\mu=a\})$$ for any $a\in \left(\inf_{\mu\in M(f,X)}\int\varphi d\mu, \, \sup_{\mu\in M(f,X)}\int\varphi d\mu\right)$ and any continuous function $\varphi$. In this process, we establish 'multi-horseshoe' entropy-dense property and use it to get the goal combined with conditional variational principles

Semi-analytical stiffness model of bolted joints in machine tools considering the coupling effect

This study proposes an improved semi-analytical approach for contact stiffness modeling of bolted joints in a machine tool system. First, nonlinear contact stress distribution within a single-bolted joint is obtained from the simulation results of finite element analysis software. Second, employing the Hertz contact theory and fractal theory, the contact stiffness model of a single asperity is formulated, affording analytical expressions for normal and tangential contact stiffnesses of a single-bolted joint by integrating multi-asperities in the contact area. Subsequently, considering two test specimens as illustrations, the mode shapes and natural frequencies of the proposed model and modal analysis tests are compared, and the influence of coupling effects between two adjacent bolts is illustrated. The maximum error in the natural frequencies of the proposed approach is < 2.73% relative to the experimental results. Finally, the measurements of frequency response functions on a box-in-box precision horizontal machine tool are conducted to demonstrate the accuracy and efficiency of the proposed model. The proposed model is highly efficient in revealing the influence of microcontact factors on the contact stiffness of bolted joints and in guiding the optimal functional design of bolt arrangements under the framework of virtual machine tools

Assessing Large Language Models in Mechanical Engineering Education: A Study on Mechanics-Focused Conceptual Understanding

This study is a pioneering endeavor to investigate the capabilities of Large Language Models (LLMs) in addressing conceptual questions within the domain of mechanical engineering with a focus on mechanics. Our examination involves a manually crafted exam encompassing 126 multiple-choice questions, spanning various aspects of mechanics courses, including Fluid Mechanics, Mechanical Vibration, Engineering Statics and Dynamics, Mechanics of Materials, Theory of Elasticity, and Continuum Mechanics. Three LLMs, including ChatGPT (GPT-3.5), ChatGPT (GPT-4), and Claude (Claude-2.1), were subjected to evaluation against engineering faculties and students with or without mechanical engineering background. The findings reveal GPT-4's superior performance over the other two LLMs and human cohorts in answering questions across various mechanics topics, except for Continuum Mechanics. This signals the potential future improvements for GPT models in handling symbolic calculations and tensor analyses. The performances of LLMs were all significantly improved with explanations prompted prior to direct responses, underscoring the crucial role of prompt engineering. Interestingly, GPT-3.5 demonstrates improved performance with prompts covering a broader domain, while GPT-4 excels with prompts focusing on specific subjects. Finally, GPT-4 exhibits notable advancements in mitigating input bias, as evidenced by guessing preferences for humans. This study unveils the substantial potential of LLMs as highly knowledgeable assistants in both mechanical pedagogy and scientific research.Comment: 30 pages, 7 figures, and 1 tabl

Evaluating the Potential of Leading Large Language Models in Reasoning Biology Questions

Recent advances in Large Language Models (LLMs) have presented new opportunities for integrating Artificial General Intelligence (AGI) into biological research and education. This study evaluated the capabilities of leading LLMs, including GPT-4, GPT-3.5, PaLM2, Claude2, and SenseNova, in answering conceptual biology questions. The models were tested on a 108-question multiple-choice exam covering biology topics in molecular biology, biological techniques, metabolic engineering, and synthetic biology. Among the models, GPT-4 achieved the highest average score of 90 and demonstrated the greatest consistency across trials with different prompts. The results indicated GPT-4's proficiency in logical reasoning and its potential to aid biology research through capabilities like data analysis, hypothesis generation, and knowledge integration. However, further development and validation are still required before the promise of LLMs in accelerating biological discovery can be realized