Considering the basic role of numbers in Mathematics, Science, and Technology the author developed a new structure of numbers named as ‘Theory of Dynamics of Numbers.’ According to the Theory of ...Dynamics of Numbers, the author defined 0 (zero) is the starting point of any number and also defined 0 (zero) as a neutral number. The numbers can move in infinite directions from the starting point 0 (zero) and back to 0 (zero). The author has defined the three types of numbers: 1) Neutral Numbers, 2) Count Up Numbers, and 3) Count Down Numbers. These three types of numbers cover the entire numbers in the number system where there is no necessity for the concept of imaginary numbers. Introducing this new concept the author solved the quadratic equation in one unknown (say x) in the form ax2 + bx + c = 0, even if the numerical value of the discriminant b2 – 4ac < 0 in real numbers without using the concept of imaginary numbers. Already the author solved the quadratic equation x2 + 1 = 0 and proved that √ -1 = -1 by using the Theory of Dynamics of Numbers. The Theory of Dynamics of Numbers is a more powerful tool than that of the real and imaginary number system to explain the truth of nature.
Staircase approximation in the simulation of the arbitrarily irregular surface will cause artificial interference of wavefield. The body-fit coordinate transformation can accurately simulate the ...arbitrarily irregular surface and avoid the staircase approximation. The finite difference method (FDM) based on body-fit coordinate system (BFCS) can simulate the propagation process of wavefield in the media with an undulating surface. Currently, the collocated grid is usually used in BFCS. However, adopting this method to simulate waves based on the first-order velocity-stress wave equation in the BFCS will cause oscillation due to the odd-even decoupling; therefore, additional filtering operations are required. To solve the above problem, we propose collocate-staggered grid (CSG) to simulate the first-order velocity-stress wave equation in the 2-D BFCS. CSG uses staggered grid scheme to calculate the equation parameters and adopts collocated grid scheme to store these values. We use second-order accuracy spatial and temporal difference for numerical tests. The results show that the elastic wavefield modeling method in the BFCS using CSG difference scheme has the characteristics of high accuracy and stable. This method is suitable for the wavefield simulation in a medium with a large Poisson's ratio.
As one of the crucial carriers for large-scale deep underground energy storage, salt caverns have great prospects for development. Pillar in salt cavern energy storage (SCES) refers to the preserved ...rock mass between adjacent salt caverns, which plays a crucial role in maintaining the stability of the SCES. Given the limited research on the analytical solutions for assessing the pillar stability of SCES, this study proposes a stability analysis method that includes horizontal and vertical profiles. In the analysis of the horizontal profile, elasticity theory and the Mohr-Coulomb strength criterion are comprehensively utilized to determine the critical operating pressure of the cavern, the range of plastic zones, and the stress distribution within the pillar. In the analysis of the vertical profile, the shear stresses and additional axial stress caused by uncoordinated deformation at the interfaces are analyzed. Through case analyses involving different operating modes and pillar widths and by comparing the results with the results of numerical simulations, the following conclusions are drawn: 1) Damage is most likely to occur on one side of the pillar adjacent to the cavern, which is crucial for cavern stability; 2) Maintaining an alternating pattern of high and low pressure (asynchronous injection-production) in two adjacent caverns rather than maintaining equal pressure in two caverns (synchronous injection-production) can reduce the critical operating pressure of the SCES to some extent, thereby improving energy storage efficiency; and 3) When the width of the pillar is low, the radial stress may decrease at the center of the pillar, potentially leading to plastic failure. 4) Uncoordinated deformation causes an increase in the range of the plastic zone around the cavern and increases the likelihood of damage within the interlayer. These findings provide important theoretical methods and guidance for the stability assessment of pillars in SCES.
In 3D-analysis of the calcaneus, a consistent coordinate system aligned with the original anatomical directions is crucial for pre- and postoperative analysis. This importance stems from the ...calcaneus's key role in weight-bearing and biomechanical alignment. However, defining a reliable coordinate system based solely on fractured or surgically reconstructed calcanei presents significant challenges. Given its anatomical prominence and consistent orientation, the talus offers a potential solution to this challenge. Our work explores the feasibility of talus-derived coordinate systems for 3D-modeling of the calcaneus across its various conditions. Four methods were tested on nonfractured, fractured and surgically reconstructed calcanei, utilizing Principal Component Analysis, anatomical landmarks, bounding box, and an atlas-based approach. The methods were compared with a self-defined calcaneus reference coordinate system. Additionally, the impact of deviation of the coordinate system on morphological measurements was investigated. Among methods for constructing nonfractured calcanei coordinate systems, the atlas-based method displayed the lowest Root Mean Square value in comparison with the reference coordinate system. For morphological measures like Böhler's Angle and the Critical angle of Gissane, the atlas talus-based system closely aligned with ground truth, yielding differences of 0.6° and 1.2°, respectively, compared to larger deviations seen in other talus-based coordinate systems. In conclusion, all tested methods were feasible for creating a talus derived coordinate system. A talus derived coordinate system showed potential, offering benefits for morphological measurements and clinical scenarios involving fractured and surgically reconstructed calcanei. Further research is recommended to assess the impact of these coordinate systems on surgical planning and outcomes.
The accuracy assessment of terrestrial reference frames (TRFs) at coordinate system level is a key task to ensure their successful use in Earth studies, satellite navigation and other geodetic ...positioning applications. Currently, the TRF quality specifications for the most demanding users dictate that the origin, orientation and scale should be determined at an accuracy level of 1 mm, and they should remain stable over time at a rate of 0.1 mm/yr. To evaluate the conformity of the internal accuracy of modern TRFs to such requirements, an appropriate mapping is needed to convert frame coordinate errors (and their CV matrix) in a terrestrial network to matching errors (and their CV matrix) in the realized coordinate system. Several projection schemes may be considered for this mapping problem, all of which aim at extracting the correlated part of the estimation error in TRF coordinates that is describable by small random perturbations in their coordinate system. The goal of the present paper is to investigate the inference problem of frame accuracy at coordinate system level, and to discuss not only the theoretical aspects of the required covariance projectors, but also the practical impact on the results obtained by their implementation in space geodetic solutions. For this purpose, a relevant case study is performed to evaluate the accuracy of the realized origin, orientation and scale in the ITRF frame series based on the formal CV matrices for their estimated positions and velocities in the four technique subnetworks (DORIS, SLR, VLBI, GNSS).
Most rigid formation controllers reported in the literature aim to only stabilize a rigid formation shape, while the formation orientation is not controlled. This paper studies the problem of ...controlling rigid formations with prescribed orientations in both 2-D and 3-D spaces. The proposed controllers involve the commonly-used gradient descent control for shape stabilization, and an additional term to control the directions of certain relative position vectors associated with certain chosen agents. In this control framework, we show the minimal number of agents which should have knowledge of a global coordinate system (2 agents for a 2-D rigid formation and 3 agents for a 3-D rigid formation), while all other agents do not require any global coordinate knowledge or any coordinate frame alignment to implement the proposed control. The exponential convergence to the desired rigid shape and formation orientation is also proved.
The curved trajectory and long synthetic aperture time of medium-Earth-orbit (MEO) synthetic aperture radar (SAR) lead to a 2-D spatial variation in the signals. Traditional methods treat the range ...and azimuth variations separately and usually suffer from high computational complexities. In this article, we investigate the Doppler rate distribution across a large scene and exploit an optimal imaging coordinate system, in which the MEO SAR signals satisfy the azimuth-shift-invariant property. Thus, the additional processing of the azimuth spatial variation in MEO SAR imaging algorithms can be avoided, and the efficiency of the image formation processor can be obviously improved. The Doppler linearization is used to address the higher-order Doppler parameters to achieve more precise focusing, and at the same time, addresses the azimuth time shift caused by the changes of signal distribution. Finally, processing results of simulated stripmap-mode data with the 2-m resolution are presented to validate the proposed algorithm.
In the working of the existing reference coordinate system transformed into the coordinate system of the geocentric coordinate system, many experts and scholars have in-depth research on the ...transformation methods such as geodetic control point results (including GNSS base station coordinates, GNSS control point coordinates, triangle point coordinates), basic geographic information data results (including DLG, DOM, DEM, DRG, DSM, etc.) and other spatial information data. On the basis of these studies, many provinces and cities have completed the coordinate system transformation of surveying and mapping results in China. This paper expounds the method of transforming the existing reference coordinate system into the geocentric coordinate system, summarizes and sorts out the common quality problems in the process of different data and different methods, and analyses the causes of the problems and the impact on the transforming results. Based on the above work, The paper provides reference and suggestions for the transforming work, which is aimed to improve the transformation quality in the future.
Managing convergence and diversity is essential in the design of multiobjective particle swarm optimization (MOPSO) in search of an accurate and well distributed approximation of the true ...Pareto-optimal front. Largely due to its fast convergence, particle swarm optimization incurs a rapid loss of diversity during the evolutionary process. Many mechanisms have been proposed in existing MOPSOs in terms of leader selection, archive maintenance, and perturbation to tackle this deficiency. However, few MOPSOs are designed to dynamically adjust the balance in exploration and exploitation according to the feedback information detected from the evolutionary environment. In this paper, a novel method, named parallel cell coordinate system (PCCS), is proposed to assess the evolutionary environment including density, rank, and diversity indicators based on the measurements of parallel cell distance, potential, and distribution entropy, respectively. Based on PCCS, strategies proposed for selecting global best and personal best, maintaining archive, adjusting flight parameters, and perturbing stagnation are integrated into a self-adaptive MOPSO (pccsAMOPSO). The comparative experimental results show that the proposed pccsAMOPSO outperforms the other eight state-of-the-art competitors on ZDT and DTLZ test suites in terms of the chosen performance metrics. An additional experiment for density estimation in MOPSO illustrates that the performance of PCCS is superior to that of adaptive grid and crowding distance in terms of convergence and diversity.
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