On the definition of Alexandrov space

This paper shows that, in the definition of Alexandrov space with lower ([BGP]) or upper ([AKP]) curvature bound, the original conditions can be replaced with much weaker ones, which can be viewed as comparison versions of the second variation formula in Riemannian geometry (and thus if we define Alexandrov spaces using these weakened conditions, then the original definition will become a local version of Toponogov's Comparison Theorem on such spaces). As an application, we give a new proof for the Doubling Theorem by Perel'man.Comment: 12 page

EFFECTS OF SHOD AND BAREFOOT CONDITIONS ON MEDIAL LONGITUDINAL ARCH ANGLE DURING RUNNING

The structure of the medial longitudinal arch (MLA) affects the spring-like function of the foot and is crucial to running performance. The purpose of this study was to investigate the differences in the MLA angle between barefoot and shod conditions by using a high-speed dual fluoroscopic imaging system (DFIS). Computed tomography was taken of each participant’s right foot for the construction of 3D models and local coordinate systems. Fifteen participants ran with or without running shoes at 3 m/s±5% speed. We recorded foot kinematics using DFIS. After the process of 3D-2D registration, MLA angles were calculated. Compared to barefoot, wearing shoes 1) decreased the initial landing MLA angle, maximum MLA angle and range of motion of the MLA angle (p \u3c 0.05); 2) decreased the MLA angles during 0%-70% of the stance phase (p \u3c 0.05). It suggests that shoes limit the MLA compression and recoil and its spring-like function

EFFECTS OF SHOD AND BAREFOOT RUNNING ON THE IN VIVO KINEMATICS OF THE FIRST METATARSOPHALANGEAL JOINT

The purpose of this study is to investigate the differences of the first metatarsophalangeal joint’s 6 degree-of-freedom (6DOF) kinematics during shod and barefoot conditions by using a high-speed dual fluoroscopic imaging system (DFIS). Fifteen healthy male runners were recruited. Computed tomography (CT) scans were taken of each participant’s right foot for the construction of 3D models and local coordinate system. The fluoroscopic images of the right foot during the stance period were acquired under shod and barefoot condition with rearfoot strike pattern Radiographic images were acquired at 100 Hz while the participants ran at a speed of 3±5% m/s in a track and 6DOF kinematics were calculated by 2D-3D registration. Paired sample t-test was used to compare the kinematic characteristics of the first MTPJ 6DOF kinematics between shod and barefoot. Compared with barefoot, wearing shoes 1) decreased the peak medial, posterior, and superior translation of the first MTPJ during stance (P ＜ 0.05); 2) decreased maximum extension angle, minimum extension angle, and flexion/extension range of motion of the first MTPJ during stance (P ＜ 0.05); 3) increased minimum adduction angle of the first MTPJ during stance (P ＜ 0.05). It suggests that shoes may affect the function of the first MTPJ and increase the risk of hallux valgus. Our study makes up for the deficiency of traditional motion measurement methods that only focus on the sagittal flexion and extension movement of the first MTPJ and provides a more comprehensive understanding of the potential relationship between joint motion and injurie

Impact of biogenic SOA loading on the molecular composition of wintertime PM2.5 in urban Tianjin: an insight from Fourier transform ion cyclotron resonance mass spectrometry

Biomass burning is one of the key sources of urban aerosols in the North China Plain, especially in winter when the impact of secondary organic aerosols (SOA) formed from biogenic volatile organic compounds (BVOCs) is generally considered to be minor. However, little is known about the influence of biogenic SOA loading on the molecular composition of wintertime organic aerosols. Here, we investigated the water-soluble organic compounds in fine particles (PM2.5) from urban Tianjin by ultrahigh-resolution Fourier transform ion cyclotron resonance mass spectrometry (FT-ICR MS). Our results show that most of the CHO and CHON compounds were derived from biomass burning; they contain O-poor and highly unsaturated compounds with aromatic rings, which are sensitive to photochemical reactions, and some of which probably contribute to light-absorbing chromophores. Under moderate to high SOA loading conditions, the nocturnal chemistry is more efficient than photooxidation to generate secondary CHO and CHON compounds with high oxygen content. Under low SOA-loading, secondary CHO and CHON compounds with low oxygen content are mainly formed by photochemistry. Secondary CHO compounds are mainly derived from oxidation of monoterpenes. But nocturnal chemistry may be more productive to sesquiterpene-derived CHON compounds. In contrast, the number- and intensity-weight of S-containing groups (CHOS and CHONS) increased significantly with the increase of biogenic SOA-loading, which agrees with the fact that a majority of the S-containing groups are identified as organosulfates and nitrooxy-organosulfates that are derived from the oxidation of BVOCs. Terpenes may be potential major contributors to the chemical diversity of organosulfates and nitrooxy-organosulfates under photo-oxidation. While the nocturnal chemistry is more beneficial to the formation of organosulfates and nitrooxy-organosulfates under low SOA-loading. The SOA-loading is an important factor associating with the oxidation degree, nitrate group content and chemodiversity of nitrooxy-organosulfates. Furthermore, our study suggests that the hydrolysis of nitrooxy-organosulfates is a possible pathway for the formation of organosulfates.</p