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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.

In a scientific enquiry it is common to study the behaviour of combined systems after studying the systems individually. The graphical structures are combined in many ways. The Cartesian product is one of the way of combining graphs to get more general structures from the simple structures. In this paper, we study a-domination number of the Cartesian product of some simple path semigraphs.

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.

Light effects have been frequently used in volume rendering because they can depict the shapes of objects more realistically. Global illumination reflects light intensity values at relevant pixel positions of reconstructed images based on the considerations of scattering and extinction phenomena. However, in the cases of ultrasound volumes that do not use Cartesian coordinates, internal lighting operations generate errors owing to the distorted direction of light propagation, and thus increase the amount of light and its effects according to the position of the volume inside. In this study, we present a novel global illumination method with calibrated light along the progression direction in accordance with volume ray casting in non-Cartesian coordinates. In addition, we reduce the consumption of lighting operation in these lighting processes using a light-distribution template. Experimental results show the volume rendering outcomes in non-Cartesian coordinates that realistically visualize the global illumination effect. The light scattering effect is expressed uniformly in the top and bottom areas where many distortions are generated in the ultrasound coordinates by using the light template kernels adaptively. Our method can effectively identify dark areas that are invisible owing to differences in brightness at the upper and lower regions of the ultrasound coordinates. Our method can be used to realistically show the shapes of the fetus during relevant examinations with ultrasonography. •Global illumination reflects light result by considering the scattering effect.•Internal lighting distortion generate error in the ultrasound coordinates.•Provide novel scattering method with calibrated direction of light progression.•Reduce the operation amount for light consumption using light-distribution template.