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Background In order to assess skeletal expansion, alveolar bone bending, and dental tipping after maxillary expansion, linear and angular measurements have been performed utilizing different craniofacial references. Since the expansion with midfacial skeletal expander (MSE) is archial in nature, the aim of this paper is to quantify the differential components of MSE expansion by calculating the fulcrum locations and applying a novel angular measurement system. Methods Thirty-nine subjects with a mean age of 18.2 ± 4.2 years were treated with MSE. Pre- and post-expansion CBCT records were superimposed and compared. The rotational fulcrum of the zygomaticomaxillary complex was identified by localizing the interfrontal distance and modified interfrontal distance. Based on the fulcrum, a novel angular measurement method is presented and compared with a conventional linear method to assess changes of the zygomaticomaxillary complex, dentoalveolar bone, and maxillary first molars. Results From 39 patients, 20 subjects have the rotational fulcrum of the zygomaticomaxillary complex at the most distant points of the interfrontal distance (101.6 ± 4.7 mm) and 19 subjects at the most distant points of the modified interfrontal distance (98.9 ± 5.7 mm). Linear measurements accounted for 60.16% and 56.83% of skeletal expansion, 16.15% and 16.55% of alveolar bone bending, and 23.69% and 26.62% of dental tipping for right and left side. Angular measurements showed 96.58% and 95.44% of skeletal expansion, 0.34% and 0.33% alveolar bone bending, and 3.08% and 4.23% of dental tipping for the right and left sides. The frontozygomatic, frontoalveolar, and frontodental angles were not significant different (P > 0.05). Conclusions In the coronal plane, the center of rotation for the zygomaticomaxillary complex was located at the most external and inferior point of the zygomatic process of the frontal bone or slightly above and parallel to the interfrontal distance. Due to the rotational displacement of the zygomaticomaxillary complex, angular measurements should be a preferred method for assessing the expansion effects, instead of the traditional linear measurement method.

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In the field of vehicle system dynamics, it is significant to propose an appropriate quantitative indicator for system stability or dynamics characteristics. Lyapunov exponents method is an excellent quantitative indicator for analysing nonlinear system characteristics. It was used to studied the stable region of nonlinear vehicle plane motion system. However, the effect of Lyapunov exponents method in revealing the global dynamics characteristics has not been fully studied. Aiming at this key problem, this paper analyses the global characteristics of Lyapunov exponents with different degrees of freedom nonlinear models. The results show that Lyapunov exponents can well reflect the global layer characteristics for vehicle system. However, the value characteristic under different DOF models is not unified. The research and analysis in this paper supplement the quantitative analysis theory for vehicle system dynamics.