Search Results (13)

In this paper, we study the convergence and complex dynamics of a novel higher-order multipoint iteration scheme to solve nonlinear equations. The approach is based upon utilizing cubic interpolation in the second step of the King–Werner method to improve its convergence order from $2.414$ to 3 and the efficiency index from $1.554$ to $1.732$, which is higher than the efficiency of optimal fourth- and eighth-order iterative schemes. The proposed method is validated through numerical and dynamic experiments concerning the absolute error, approximated computational order, regions of convergence, and CPU time (sec) on the real-world problems, including Kepler’s equation, isentropic supersonic flow, and law of population growth, demonstrating superior performance compared to some existing well-known methods. Commonly, regions of convergence of iterative methods are investigated and compared by plotting attractor basins of iteration schemes in the complex plane on polynomial functions of the type $z^n-1$. However, in this paper, the attractor basins of the proposed method are investigated on diverse nonlinear functions. The proposed scheme creates portraits of basins of attraction faster with wider convergence areas outperforming existing well-known iteration schemes. Full article

The Kepler optimization algorithm (KOA) is a metaheuristic algorithm based on Kepler’s laws of planetary motion and has demonstrated outstanding performance in multiple test sets and for various optimization issues. However, the KOA is hampered by the limitations of insufficient convergence accuracy, weak global search ability, and slow convergence speed. To address these deficiencies, this paper presents a multi-strategy fusion Kepler optimization algorithm (MKOA). Firstly, the algorithm initializes the population using Good Point Set, enhancing population diversity. Secondly, Dynamic Opposition-Based Learning is applied for population individuals to further improve its global exploration effectiveness. Furthermore, we introduce the Normal Cloud Model to perturb the best solution, improving its convergence rate and accuracy. Finally, a new position-update strategy is introduced to balance local and global search, helping KOA escape local optima. To test the performance of the MKOA, we uses the CEC2017 and CEC2019 test suites for testing. The data indicate that the MKOA has more advantages than other algorithms in terms of practicality and effectiveness. Aiming at the engineering issue, this study selected three classic engineering cases. The results reveal that the MKOA demonstrates strong applicability in engineering practice. Full article

Although Kepler’s laws can be empirically proven by applying Newton’s laws to the dynamics of two particles attracted by gravitational interaction, an explicit formula for the motion as a function of time remains undefined. This paper proposes a quasi-analytical solution to address this challenge. It approximates the real dynamics of celestial bodies with a satisfactory degree of accuracy and minimal computational cost. This problem is closely related to Kepler’s equation, as solving the equations of motion as a function of time also provides a solution to Kepler’s equation. The results are presented for each planet of the solar system, including Pluto, and the solution is compared against real orbits. Full article

In recent years, the rapid development of exoplanet research has provided us with an opportunity to better understand planetary systems in the universe and to search for signs of life. In order to further investigate the prevalence of habitable exoplanets and to validate planetary formation theories, as well as to comprehend planetary evolution, we have utilized confirmed exoplanet data obtained from the NASA Exoplanet Archive database, including data released by telescopes such as Kepler and TESS. By analyzing these data, we have selected a sample of planets around F, G, K, and M-type stars within a radius range of 1 to 20 $R_{\oplus }$ and with orbital periods ranging from 0.4 days to 400 days. Using the IDEM method based on these data, we calculated the overall formation rate, which is estimated to be 2.02%. Then, we use these data to analyze the relationship among planet formation rates, stellar metallicity, and stellar gravitational acceleration ($logg$). We firstly find that the formation rate of giant planets is higher around metal-rich stellars, but it inhibits the formation of gas giants when $logg$ > 4.5, yet the stellar metallicity seems to have no effect on the formation rate of smaller planets. Secondly, the host stellar gravitational acceleration affects the relationship between planet formation rate and orbital period. Thirdly, there is a robust power-law relationship between the orbital period of smaller planets and their formation rate. Finally, we find that, for a given orbital period, there is a positive correlation between the planet formation rate and the $logg$. Full article

On the basis of a general action principle, we revisit the scale invariant field equation using the cotensor relations by Dirac (1973). This action principle also leads to an expression for the scale factor $\lambda$, which corresponds to the one derived from the gauging condition, which assumes that a macroscopic empty space is scale-invariant, homogeneous, and isotropic. These results strengthen the basis of the scale-invariant vacuum (SIV) paradigm. From the field and geodesic equations, we derive, in current time units (years, seconds), the Newton-like equation, the equations of the two-body problem, and its secular variations. In a two-body system, orbits very slightly expand, while the orbital velocity keeps constant during expansion. Interestingly enough, Kepler’s third law is a remarkable scale-invariant property. Full article

In the study of systems’ dynamics the presence of symmetry dramatically reduces the complexity, while in chemistry, symmetry plays a central role in the analysis of the structure, bonding, and spectroscopy of molecules. In a more general context, the principle of equivalence, a principle of local symmetry, dictated the dynamics of gravity, of space-time itself. In certain instances, especially in the presence of symmetry, we end up having to deal with an equation with multiple roots. A variety of optimal methods have been proposed in the literature for multiple roots with known multiplicity, all of which need derivative evaluations in the formulations. However, in the literature, optimal methods without derivatives are few. Motivated by this feature, here we present a novel optimal family of fourth-order methods for multiple roots with known multiplicity, which do not use any derivative. The scheme of the new iterative family consists of two steps, namely Traub-Steffensen and Traub-Steffensen-like iterations with weight factor. According to the Kung-Traub hypothesis, the new algorithms satisfy the optimality criterion. Taylor’s series expansion is used to examine order of convergence. We also demonstrate the application of new algorithms to real-life problems, i.e., Van der Waals problem, Manning problem, Planck law radiation problem, and Kepler’s problem. Furthermore, the performance comparisons have shown that the given derivative-free algorithms are competitive with existing optimal fourth-order algorithms that require derivative information. Full article

The flattening of spiral-galaxy rotation curves is unnatural in view of the expectations from Kepler’s third law and a central mass. It is interesting, however, that the radius-independence velocity is what one expects in one less dimension. In our three-dimensional space, the rotation curve is natural if, outside the galaxy’s center, the gravitational potential corresponds to that of a very prolate ellipsoid, filament, string, or otherwise cylindrical structure perpendicular to the galactic plane. While there is observational evidence (and numerical simulations) for filamentary structure at large scales, this has not been discussed at scales commensurable with galactic sizes. If, nevertheless, the hypothesis is tentatively adopted, the scaling exponent of the baryonic Tully–Fisher relation due to accretion of visible matter by the halo comes out to reasonably be 4. At a minimum, this analytical limit would suggest that simulations yielding prolate haloes would provide a better overall fit to small-scale galaxy data. Full article

A recent study of a sample of wide binary star systems from the Hipparcos and Gaia catalogues has found clear evidence of a gravitational anomaly of the same kind as that appearing in galaxies and galactic clusters. Instead of a relative orbital velocity decaying as the square root of the separation, $\Delta V\propto r^{-1/2}$, it was shown that an asymptotic constant velocity is reached for distances of order $0.1$ pc. If confirmed, it would be difficult to accommodate this breakdown of Kepler’s laws within the current dark matter (DM) paradigm because DM does not aggregate in small scales, so there would be very little DM in a $0.1$ pc sphere. In this paper, we propose a simple non-Newtonian model of gravity that could explain both the wide binaries anomaly and the anomalous rotation curves of galaxies as codified by the Tully-Fisher relation. The required extra potential can be understood as a Klein-Gordon field with a position-dependent mass parameter. The extra forces behave as $1/r$ on parsec scales and r on Solar system scales. We show that retrograde anomalous perihelion precessions are predicted for the planets. This could be tested by precision ephemerides in the near future. Full article

The Voronoi entropy is a mathematical tool for quantitative characterization of the orderliness of points distributed on a surface. The tool is useful to study various surface self-assembly processes. We provide the historical background, from Kepler and Descartes to our days, and discuss topological properties of the Voronoi tessellation, upon which the entropy concept is based, and its scaling properties, known as the Lewis and Aboav–Weaire laws. The Voronoi entropy has been successfully applied to recently discovered self-assembled structures, such as patterned microporous polymer surfaces obtained by the breath figure method and levitating ordered water microdroplet clusters. Full article

The epistemological rupture of Copernicus, the laws of planetary motions of Kepler, the comprehensive physical observations of Galileo and Huygens, the conception of relativity, and the physical theory of Newton were components of an extremely fertile and influential cognitive environment that prompted the restless Leibniz to shape an innovative theory of space and time. This theory expressed some of the concerns and intuitions of the scientific community of the seventeenth century, in particular the scientific group of the Academy of Sciences of Paris, but remained relatively unknown until the twentieth century. After Einstein, however, the relational theory of Leibniz gained wider respect and fame. The aim of this article is to explain how Leibniz foresaw relativity, through his critique of contemporary mechanistic philosophy. Full article

Multiwavelength blazar variability is produced by noise-like processes with the power-law form of power spectral density (PSD). We present the results of our detailed investigation of multiwavelength ( $\gamma$ -ray and optical) light curves covering decades to minutes timescales, of two BL Lac objects namely, PKS 0735+178 and OJ 287. The PSDs are derived using discrete Fourier transform (DFT) method. Our systematic approach reveals that OJ 287 is, on average, more variable than PKS 0735+178 at both optical and $\gamma$ -ray energies on the corresponding time scales. On timescales shorter than ∼10 days, due to continuous and dense monitoring by the Kepler satellite, a steepening of power spectrum is observed for OJ 287. This indicates the necessity of an intermittent process generating variability on intra-night timescales for OJ 287. Full article

Many writings on information mix information on a given system (I_S), measurable information content of a given system (I_M), and the (also measurable) information content that we communicate among us on a given system (I_C). They belong to different levels and different aspects of information. The first (I_S) involves everything that one possibly can, at least potentially, know about a system, but will never learn completely. The second (I_M) contains quantitative data that one really learns about a system. The third (I_C) relates rather to the language (including mathematical) by which we transmit information on the system to one another, rather than to the system itself. The information content of a system (I_M —this is what we generally mean by information) may include all (relevant) data on each element of the system. However, we can reduce the quantity of information we need to mediate to each other (I_C), if we refer to certain symmetry principles or natural laws which the elements of the given system correspond to. Instead of listing the data for all elements separately, even in a not very extreme case, we can give a short mathematical formula that informs about the data of the individual elements of the system. This abbreviated form of information delivery includes several conventions. These conventions are protocols that we have learnt before, and do not need to be repeated each time in the given community. These conventions include the knowledge that the scientific community accumulated earlier when discovered and formulated the symmetry principle or the law of nature, the language in which those regularities were discovered and formulated, for example, the symmetry principle or the law of nature, the language in which those regularities were formulated and then accepted by the community, and the mathematical marks and abbreviations that are known only for the members of the given scientific community. We do not need to repeat the rules of the convention each time, because the conveyed information includes them, and it is there in our minds behind our communicated data on the information content. I demonstrate this by using two examples, Kepler’s laws, and the law of correspondence between the DNA codons’ triplet structure and the individual amino acids which they encode. The information content of the language by which we communicate the obtained information cannot be identified with the information content of the system that we want to characterize, and moreover, it does not include all the possible information that we could potentially learn about the system. Symmetry principles and natural laws may reduce the information we need to communicate about a system, but we must keep in mind the conventions that we have learnt about the abbreviating mechanism of those principles, laws, and mathematical descriptions. Full article

We analyse in a common framework the properties of the Voronoi tessellations resulting from regular 2D and 3D crystals and those of tessellations generated by Poisson distributions of points, thus joining on symmetry breaking processes and the approach to uniform random distributions of seeds. We perturb crystalline structures in 2D and 3D with a spatial Gaussian noise whose adimensional strength is α and analyse the statistical properties of the cells of the resulting Voronoi tessellations using an ensemble approach. In 2D we consider triangular, square and hexagonal regular lattices, resulting into hexagonal, square and triangular tessellations, respectively. In 3D we consider the simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC) crystals, whose corresponding Voronoi cells are the cube, the truncated octahedron, and the rhombic dodecahedron, respectively. In 2D, for all values α>0, hexagons constitute the most common class of cells. Noise destroys the triangular and square tessellations, which are structurally unstable, as their topological properties are discontinuous in α=0. On the contrary, the honeycomb hexagonal tessellation is topologically stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise with α<0.12. Basically, the same happens in the 3D case, where only the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. In both 2D and 3D cases, already for a moderate amount of Gaussian noise (α>0.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations are indistinguishable. When α>2, results converge to those of Poisson-Voronoi tessellations. In 2D, while the isoperimetric ratio increases with noise for the perturbed hexagonal tessellation, for the perturbed triangular and square tessellations it is optimised for specific value of noise intensity. The same applies in 3D, where noise degrades the isoperimetric ratio for perturbed FCC and BCC lattices, whereas the opposite holds for perturbed SCC lattices. This allows for formulating a weaker form of the Kelvin conjecture. By analysing jointly the statistical properties of the area and of the volume of the cells, we discover that also the cells shape heavily fluctuates when noise is introduced in the system. In 2D, the geometrical properties of n-sided cells change with α until the Poisson-Voronoi limit is reached for α>2; in this limit the Desch law for perimeters is shown to be not valid and a square root dependence on n is established, which agrees with exact asymptotic results. Anomalous scaling relations are observed between the perimeter and the area in the 2D and between the areas and the volumes of the cells in 3D: except for the hexagonal (2D) and FCC structure (3D), this applies also for infinitesimal noise. In the Poisson-Voronoi limit, the anomalous exponent is about 0.17 in both the 2D and 3D case. A positive anomaly in the scaling indicates that large cells preferentially feature large isoperimetric quotients. As the number of faces is strongly correlated with the sphericity (cells with more faces are bulkier), in 3D it is shown that the anomalous scaling is heavily reduced when we perform power law fits separately on cells with a specific number of faces. Full article