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Summarizing some old research on the dynamics of a pointall body along its own trajectory, this paper established the differential relationships between the principal curvatures of a 3D curve, that is the normal curvature and the torsional curvature, and its Cartesian coordinates. The differential system thus derived is actually a dynamical system of a representative point of the curve moving along it. This dynamic system is analyzed to see the possibilities of finding analytical solutions in finite terms, using Frobenius' integrability theorem for the general case and usual integration methods for the particular case consisting of the constant ratio between the two curvatures.

The method of non-standard algebraic extensions based on the use of additional formal solutions of the reduced equations is extended to the case corresponding to three-dimensional space. This method differs from the classical one in that it leads to the formation of algebraic rings rather than fields. The proposed approach allows one to construct a discrete coordinate system in which the role of three basis vectors is played by idempotent elements of the ring obtained by a non-standard algebraic extension. This approach allows, among other things, the identification of the symmetry properties of objects defined through discrete Cartesian coordinates, which is important, for example, when using advanced methods of digital image processing. An explicit form of solutions of the equations is established that allow one to construct idempotent elements for Galois fields GFp such that p−1 is divisible by three. The possibilities of practical use of the proposed approach are considered; in particular, it is shown that the use of discrete Cartesian coordinates mapped onto algebraic rings is of interest from the point of view of improving UAV swarm control algorithms.

For the graph’s vertex coloring, it is required that for every vertex in a graph, the colors used in its open neighborhood or closed neighborhood must be able to form a continuous integer interval. A coloring is called an open neighborhood interval vertex coloring or a closed neighborhood interval vertex coloring of a graph if the neighborhood satisfying the condition is open or closed. In this paper, the interval vertex coloring of cartesian products and strong products of two paths is studied, and the low bound of the interval chromatic number is given.

The 17th and 18th century opposition between Cartesian and Newtonian science is often depicted as a contest between a priorism and speculation on one hand, and observation and mathematical proof on the other, one of which won out. This is a simplification. In 17th century England Cartesian natural philosophy, including the vortex theory of the planetary orbits, was (intentionally) easy to understand. It was seen however as poorly disguised atheism and widely disparaged on that account by influential theologians. Amongst the 18th century French philosophes, this aspect of Cartesianism was hardly a problem. Newtonianism now denoted an appetite for exciting experimental demonstrations as against the dead letter of scientific books, including not only René Descartes’s Principles but also Isaac Newton’s (for most readers) largely impenetrable Principia .

The authors present the results of the research combining the methods of approximation theory and optimal decision theory. Namely, a solution to the optimization problem for the biharmonic Poisson integral in the upper half-plane is considered one of the most optimal solutions to the biharmonic equation in Cartesian coordinates. The approximate properties of the biharmonic Poisson operator in the upper half-plane on the classes of quasi-smooth functions are obtained in the form of an exact equality for the deviation of quasi-smooth functions from the positive operator under consideration.

The kinetic origin of the cubic terms found in electromagnetic reduced fluid models of magnetized plasmas is identified and analyzed. The transition from cartesian to guiding center variables in kinetic systems is responsible for the introduction of such cubic terms, and the derivation of a set of such terms is explicitly shown. The role of these terms in fluid simulations, and their importance for energy conservation of cubic terms in both frameworks is discussed

Terahertz video synthetic aperture radar (THz-ViSAR) has tremendous research and application value due to its high resolution and high frame rate imaging benefits. However, it requires more efficient imaging algorithms. Thus, a novel multistage back projection fast imaging algorithm for the THz-ViSAR system is proposed in this paper to enable continuous playback of images like video. The radar echo data of the entire aperture is first divided into multiple sub-apertures, as with the fast-factorized back projection algorithm (FFBP). However, there are two improvements in sub-aperture imaging. On the one hand, the back projection algorithm (BPA) is replaced by the polar format algorithm (PFA) to improve the sub-aperture imaging efficiency. The imaging process, on the other hand, uses the global Cartesian coordinate system rather than the local polar coordinate system, and the wavenumber domain data of the full aperture are obtained step by step through simple splicing and fusion, avoiding the amount of two-dimensional (2D) interpolation operations required for local polar coordinate system transformation in FFBP. Finally, 2D interpolation for full-resolution images is carried out to image the ground object targets in the same coordinate system due to the geometric distortion caused by linear phase error (LPE) and the mismatch of coordinate systems in different imaging frames. The simulation experiments of point targets and surface targets both verify the effectiveness and superiority of the proposed algorithm. Under the same conditions, the running time of the proposed algorithm is only about 6% of FFBP, while the imaging quality is guaranteed.

The high-speed operation of unbalanced machines may cause vibrations that lead to noise, wear, and fatigue that will eventually limit their efficiency and operating life. To restrain such vibrations, a complete balancing must be performed. This paper presents the complete balancing optimization of a six-bar mechanism with the use of counterweights. A novel method based on fully Cartesian coordinates (FCC) is proposed to represent such a balanced mechanism. A multiobjective optimization problem was solved using the Differential Evolution (DE) algorithm to minimize the shaking force (ShF) and the shaking moment (ShM) and thus balance the system. The Pareto front is used to determine the best solutions according to three optimization criteria: only the ShF, only the ShM, and both the ShF and ShM. The dimensions of the counterweights are further fine-tuned with an analysis of their partial derivatives, volumes, and area–thickness relations. Numerical results show that the ShF and ShM can be reduced by 76.82% and 77.21%, respectively, when importance is given to either of them and by 45.69% and 46.81%, respectively, when equal importance is given to both. A comparison of these results with others previously reported in the literature shows that the use of FCC in conjunction with DE is a suitable methodology for the complete balancing of mechanisms.

To derive a closed-form analytical solution to the swing equation describing the power system dynamics, which is a nonlinear second order differential equation. No analytical solution to the swing equation has been identified, due to the complex nature of power systems. Two major approaches are pursued for stability assessments on systems: (1) computationally simple models based on physically unacceptable assumptions, and (2) digital simulations with high computational costs. The motion of the rotor angle that the swing equation describes is a vector function. Often, a simple form of the physical laws is revealed by coordinate transformation. The study included the formulation of the swing equation in the Cartesian coordinate system, which is different from conventional approaches that describe the equation in the polar coordinate system. Based on the properties and operational conditions of electric power grids referred to in the literature, we identified the swing equation in the Cartesian coordinate system and derived an analytical solution within a validity region. The estimated results from the analytical solution derived in this study agree with the results using conventional methods, which indicates the derived analytical solution is correct. An analytical solution to the swing equation is derived without unphysical assumptions, and the closed-form solution correctly estimates the dynamics after a fault occurs.

A simple and efficient finite-difference scheme is developed to calculate seismic wave propagation in a partial spherical shell model of a three-dimensionally (3-D) heterogeneous global Earth structure for modeling on regional or sub-global scales where the effects of the Earth’s spherical geometry cannot be ignored. This scheme solves the elastodynamic equation in the quasi - Cartesian coordinate form similar to the local Cartesian one, instead of the spherical polar coordinate form, with a staggered-grid finite-difference method in time domain (FDTD) that is one of the most popular numerical methods in seismic-motion simulations for local-scale models. The proposed scheme may be a local-friendly approach for modeling on a sub-global scale to link regional-scale and local-scale simulations. It can be easily implemented using an available 3-D Cartesian FDTD local-scale modeling code by changing a very small part of the code. We implement the scheme in an existing Cartesian FDTD code and demonstrate the accuracy and validity of the present scheme and the feasibility to apply it to real large simulations through numerical examples. Graphical abstract .