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In this work, a system of three masses on the vertices of equilateral triangle is investigated. This system is known in the literature as a planar system. We first give a description to the system by constructing its classical Lagrangian. Secondly, the classical Euler-Lagrange equations (i.e., the classical equations of motion) are derived. Thirdly, we fractionalize the classical Lagrangian of the system, and as a result, we obtain the fractional Euler-Lagrange equations. As the final step, we give the numerical simulations of the fractional model, a new model which is based on Caputo fractional derivative.

The possible positions of an equilateral triangle whose vertices are located on the support sides of a generic triangle are studied. Using complex coordinates, we show that there are infinitely many such configurations, then we prove that the centroids of these equilateral triangles are collinear, defining two lines perpendicular to the Euler’s line of the original triangle. Finally, we obtain the complex coordinates of the intersection points and study some particular cases.

The issue of pedestrian-level wind environments around high-rise buildings is closely related to the comfort and safety of human settlements. In this paper, we study the effects of different wind direction angles and spacing ratios on the wind environment at pedestrian heights around buildings arranged in an equilateral triangle configuration. Three-dimensional steady-state numerical simulation was employed, with the standard k-ε model selected as the turbulence model. Wind speed ratios and different area ratio parameters are used to quantitatively express the degree and range of influence of wind speed by buildings. The results show that the maximum wind speed ratio at the corner of a building is greatly affected by the wind direction angle, with 45°, 135°, and 157.5° being the unfavorable wind direction angles. Conversely, the area ratio of different areas is greatly affected by the spacing ratio. As the spacing ratio increases, the mutual interference effect between buildings weakens, resulting in a better pedestrian wind environment. Owing to the unique layout of the building group, different degrees of ventilation corridors are formed among the three buildings. The wind speed amplification effect in the corridors is more significant, and the areas with poor wind environments are concentrated in these corridors.

Novel method for semi-analytical solving of equations of a trapped dynamics for a planetoid m 4 close to the plane of mutual motion of main bodies around each other (in case of a special type of Bi-Elliptic Restricted 4-Bodies Problem) is presented. We consider here three primaries m 1 , m 2 , m 3 orbiting around their center of mass on elliptic orbits which are permanently forming Lagrangian configuration of an equilateral triangle. Our aim is to obtain approximate coordinates of quasi-planar trajectory of the infinitesimal planetoid m 4 , when the primaries have masses equal to 1/3 (not stable configuration of the Lagrange solution). Results are as follows: (1) equations for coordinates { x ¯ , y ¯ } are described by system of coupled second-order ODEs with respect to true anomaly f and (2) expression for z ¯ stems from solving second-order Riccati ordinary differential equation that determines the quasi-periodical oscillations of planetoid m 4 not far from invariant plane { x ¯ , y ¯ , 0 } .

To observe lower-frequency gravitational waves (GWs), it is effective to utilize a large spacecraft formation baseline, spanning hundreds of thousands to millions of kilometers. To overcome the limitations of a gravitational-wave observatory (GWO) on specific orbits, a scientific observation mode and a non-scientific observation mode for GWOs are proposed. For the non-scientific observation mode, this paper designs equilateral triangle and equilateral tetrahedral array formations for a space-based GWO near a collinear libration point. A stable configuration is the prerequisite for a GWO; however, the motion near the collinear libration points is highly unstable. Therefore, the output regulation theory is applied. By leveraging the tracking aspect of the theory, the equilateral triangle and equilateral tetrahedral array formations are achieved. For an equilateral triangle array formation, two geometric configuration design methods are proposed, addressing the fuel consumption required for initialization and maintenance. To observe GWs in different directions and avoid configuration/reconfiguration, the multi-layer equilateral tetrahedral array formation is given. Additionally, the control errors are calculated. Finally, the effectiveness of the control method is demonstrated using the Sun–Earth circular-restricted three-body problem (CRTBP) and the ephemeris model located at Lagrange point L1.

In the information age, the new wave of the information technology revolution has profoundly changed our mode of production and way of life. Pseudo human settlements (PHS), consisting of digits and information, have become increasingly important in human settlements (HS) systems, and become a strong support for the high-quality development of global HS. Against this background, clarifying the spatiotemporal heterogeneity and driving mechanisms of the coupling and coordination between the PHS and real human settlements (RHS) is of great significance to the high-quality development of HS and providing a reasonable explanation of today’s man–land relationship. Therefore, we developed a theoretical framework system for describing PHS–RHS coupling and coordination based on multi-source data such as internet socialization, public utility, and remote sensing images, etc. Taking the urban agglomeration in the middle reaches of the Yangtze River (UAMRYR), which is the key region consolidating China’s “two horizontal and three vertical” urbanization strategy, as a case study area, we have comprehensively analyzed the spatiotemporal heterogeneity of the coupling and coordination of PHS and RHS and its driving mechanism in UAMRYR during the period of 2011–2021, by comprehensively applying the modified coupling coordination degree (CCD) and other models. The results show are as follows: (1) Temporal process—The CCD exhibited a reverse L-shaped increasing trend. The CCD class varied significantly, with the extremely uncoordinated and severely uncoordinated classes present at the beginning of the study period and disappearing toward the end of the study period, while the well coordinated and highly coordinated classes were absent at the beginning of the study period and appeared toward the end of the study period. (2) Spatial pattern—The CCD exhibited an equilateral triangle-shaped, core–margin spatial pattern and a characteristic of core polarization. Overall, the spatial distribution of the CCD exhibited a characteristic of “high in the central region, low in the eastern and western regions, and balanced in the south–north direction”. (3) Dynamic evolution—The CCD increased more rapidly in the north-eastern direction than in the south-western direction; the CCD exhibited north-eastward migration and dispersion, and the spatial variability decreased. (4) Driving mechanisms—The primary factors affecting the CCD varied significantly over time. The living system was dominant in the PHS, whereas the human system was dominant in the RHS. The PHS had a greater effect than the RHS on the CCD. The study broadens the research scope of human settlements geography, establishes a scientific foundation for advancing urban HS construction in the UAMRYR, and offers theoretical support for the high-quality development of cities in the UAMRYR.

Antennas are vital in the internet of things (IoT) for enabling telemedicine and healthcare communication between devices and networks. They receive and transmit signals, extending range, improving efficiency, and reducing power consumption. Antennas are versatile and can be integrated into devices or added as external modules. Their flexibility and adaptability are important in applications involving humans, as they can bend and conform to the shape of the body. Overall, antennas are a crucial and adaptable component of IoT technology. The first thing that needs to be done is to determine the frequency at which the antenna should operate for the problem at hand and design an antenna that can work at those resonant frequencies. In this study, equilateral triangular-shaped compact microstrip antennas (ETMAs) were chosen, and their resonance frequencies were calculated using the Gaussian process regression method (GPR). For this purpose, 630 ETMA were simulated, and a dataset was created utilizing the antenna characteristics and resonant frequencies. Support vector machines (SVM), artificial neural networks (ANN), and GPR models were trained on the obtained data set. To validate the performance of the trained models, two ETMAs with an outer length of 50 mm and an inner slot length of 5 mm were fabricated utilizing polylactic acid (PLA) and felt-based substrates with copper tape as the conducting material. The accuracy of the resonant frequency estimation using the GPR approach for the fabricated antennas is 2.833% and 1.706% for the PLA- and felt-based antennas, respectively, when compared to the measurement results. The GPR model trained in this study has an accuracy of 0.470% and 0.662% when compared to simulations in the literature and measurement results, respectively. In addition, one of the designed antennas is in wearable form, and the other is PLA, produced with a low-cost 3D printer, allowing continuous monitoring of patients with high cancer risk. In this article, an easier and cheaper microstrip patch antenna that can be used for imaging and telemedicine applications is designed with a copper band on one flexible and one rigid substrate, and its performance is analyzed experimentally.

The flow induced vibration and energy extraction of an equilateral triangle prism elastically mounted in a water channel are investigated experimentally at different system damping ratios ζtotal with the constant oscillating mass Mosc and system stiffness K. A power take-off system with a variable damping function is developed. The translation-rotation equation of the vibration system deduced in the study indicates that the total oscillating mass includes the material mass, and the equivalent mass due to the rotation of the gears and rotor. Besides, increasing load resistance can result in a decrease in ζtotal when K and Mosc remain unchanged. The prism experiences, in turn, soft galloping, hard galloping 1 and hard galloping 2 with increasing ζtotal. As ζtotal increases up to 0.335, only the vortex-induced vibration is observed because the extremely high ζtotal prevents the prism from galloping. The response amplitude decreases with the increasing ζtotal. In addition, higher ζtotal promotes the galloping to start at a higher reduced velocity. The galloping characteristics of the prism, including large amplitude responses in an extremely large range of flow velocities, excellent vibration stationarity, and steady vibration frequencies, are beneficial for improving energy conversion. The prism can extract hydraulic energy for the flow velocity U > 0.610 m/s. The harnessed power Pout and the energy conversion efficiency ηout increase with increasing ζtotal in the galloping zone. The maximum Pout and ηout reach 53.56 W and 40.44%, respectively. The optimal system damping ratio for extracting energy is the maximum system damping ratio that the prism can overcome to experience stable galloping.

Let n be a positive and squarefree integer. We show that the equilateral triangle can be dissected into n·k2 congruent triangles for some k if and only if n≤3, or at least one of the curves Cn:y2=x(x-n)(x+3n) and C-n:y2=x(x+n)(x-3n) has a rational point with y≠0. We prove that if p is a positive prime such that p≡7 (mod 24), then Cp and C-p do not have such points. Consequently, for these primes the equilateral triangle cannot be dissected into p·k2 congruent triangles for any k.

We show that every tiling of a convex set in the Euclidean plane \\[\\mathbb {R}^2\\] by equilateral triangles of mutually different sizes contains arbitrarily small tiles. The proof is purely elementary up to the discussion of one family of tilings of the full plane \\[\\mathbb {R}^2\\], which is based on a surprising connection to a random walk on a directed graph.