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Let φ : N → (0 , 1 ] be a positive function. We consider the size of the set E f (φ) : = { β > 1 : | T β n (x) − f (β) | < φ (n) i. o. n } , where " i. o. n " stands for "infinitely often", and f : (1 , ∞) → [ 0 , 1 ] is a Lipschitz function. For any x ∈ (0 , 1 ] , it is proved that the Hausdorff measure of E f (φ) fulfill a dichotomy law according to lim sup n → ∞ log φ (n) n = − ∞ or not, where T β is the β -transformation. In ergodic theory, the phenomenon of shrinking targets is crucial for understanding the long-term behavior of systems. By studying the shrinking target problem of the β dynamical system, we can reveal the relationship between randomness and determinism, which is significant for constructing more complex mathematical models. Moreover, there is a close connection between the β transformation and number theory. Investigating the contraction target problem helps uncover new properties and patterns in number theory, advancing the development of this field. In this work, we establish a significant relationship between the decay rate of the positive function φ (n) and the structural properties of the set E f (φ) . Specifically, we show that: The Hausdorff dimension of E f (φ) either vanishes or is positive based on the behavior of φ (n) as n approaches infinity. The establishment of this dichotomy can help us more effectively understand the geometric characteristics and dynamical behavior of the system, thereby aiding our acceptance and comprehension of complex theories. Researching this shrinking target problem can help us uncover new properties in number theory, leading to a better understanding of the structure of numbers and promoting the development of related fields in number theory. [ABSTRACT FROM AUTHOR]

It can effectively improve gas drainage efficiency by arranging the surface drilling in the “high-level annular body” above the working surface. Based on the working condition of 17171(1) of Pansan Coal Mine, COSFLOW numerical simulation software was used to study and determine the range and location of the “high-level annular body” for gas drainage. The results show that: the height of the “high-level annular body” is within 108 m and 20 m above the mining coal seam. The front area adjacent to the working face is 140 m in advance direction of the working face. The outer boundary inclination angle extending to the roof is 70°, and the inner boundary inclination angle is 80°. During the recovery of 17171(1) working face, the average gas drainage flow volume of the surface drilling was 18.27 m3/min, which was 14.2% higher than the traditional method; the maximum value of the gas concentration for surface drilling drainage is 98% and the average is 50%, which meets the concentration requirements well for gas utilization.

This paper investigates the averaging principle for a class of McKean–Vlasov Stochastic Differential Equations (MVSDEs) with Lévy noise and Hölder continuous coefficients. By employing the Yamada–Watanabe approximation technique, we establish the convergence of the original equation's solution to the solution of the averaging equation in the L p (Ω ; L ∞ ([ 0 , T ] , ℝ)) sense with p > 0. [ABSTRACT FROM AUTHOR]

The inter-satellite laser ranging interferometer is one of the core components of future gravity missions to achieve high ranging precision. This work builds a preliminary breadboard of the off-axis optical bench design, which integrates the merits of the off-axis optical bench design of GRACE Follow-On mission and other on-axis designs. The study finds that the displacement noise between two optical benches has been reduced to 20 n m / Hz at a frequency of 10 mHz, and the differential wavefront sensing noise has been suppressed to 1 0 − 5 rad / Hz at 1 kHz as well. In addition, the tilt-to-length coupling noise related to the piston effect is restricted within 30 μm/rad, and the relative parallelism error of the transmitting beam and receiving beam is less than 4.5%. The results show that this off-axis optical bench design is an important candidate for China's future gravity missions. [ABSTRACT FROM AUTHOR]

The primary objective of this work was to delve into the potential therapeutic advantages and dissect the molecular mechanisms of salidroside in enhancing erectile function in rats afflicted with diabetic microvascular erectile dysfunction (DMED), addressing both the whole-animal and cellular dimensions.We established a DMED model in Sprague‒Dawley (SD) rats and conducted in vivo experiments. The DMED rats were administered varying doses of salidroside, the effects of which on DMED were compared. Erectile function was evaluated by applying electrical stimulation to the cavernous nerves and measuring intracavernous pressure in real time. The penile tissue underwent histological examination and Western blotting. Hydrogen peroxide (H2O2) was employed in the in vitro trial to induce an oxidative stress for the purpose of identifying alterations in cell viability. The CCK-8 assay was used to measure the viability of corpus cavernous smooth muscle cells (CCSMCs) treated with vs. without salidroside. Flow cytometry was utilized to detect alterations in intracellular reactive oxygen species (ROS). Apoptosis was assessed through Western blotting and TdT-mediated dUTP nick-end labelling (TUNEL). Animal and cellular experiments indicate that the Nrf2/HO-1 signalling pathway may be upregulated by salidroside, leading to the improvement of erectile function in diabetic male rats by alleviating oxidative stress and reducing apoptosis in corpus cavernosum tissue. [ABSTRACT FROM AUTHOR]