Eighth Order Research Articles

General Three-Body Problem in Conformal-Euclidean Space: New Properties of a Low-Dimensional Dynamical System

Despite the huge number of studies of the three-body problem in physics and mathematics, the study of this problem remains relevant due to both its wide practical application and taking into account its fundamental importance for the theory of dynamical systems. In addition, one often has to answer the cognitive question: is irreversibility fundamental for the description of the classical world? To answer this question, we considered a reference classical dynamical system, the general three-body problem, formulating it in conformal Euclidean space and rigorously proving its equivalence to the Newtonian three-body problem. It has been proven that a curved configuration space with a local coordinate system reveals new hidden symmetries of the internal motion of a dynamical system, which makes it possible to reduce the problem to a sixth-order system instead of the eighth order. An important consequence of the developed representation is that the chronologizing parameter of the motion of a system of bodies, which we call internal time, differs significantly from ordinary time in its properties. In particular, it more accurately describes the irreversible nature of multichannel scattering in a three-body system and other chaotic properties of a dynamical system. The paper derives an equation describing the evolution of the flow of geodesic trajectories, with the help of which the entropy of the system is constructed. New criteria for assessing the complexity of a low-dimensional dynamical system and the dimension of stochastic fractal structures arising in three-dimensional space are obtained. An effective mathematical algorithm is developed for the numerical simulation of the general three-body problem, which is traditionally a difficult-to-solve system of stiff ordinary differential equations.

Devising an analytical method for solving the eighth-order Kolmogorov equations for an asymmetric Markov chain

The object of research is a complex system of three subsystems, which function independently of each other and are in a working or failed state. There is a need to analytically model and manage the Markov random process in the system, varying the intensity of their development-restoration and degradation-destruction flows. In the study, an analytical method for solving Kolmogorov equations of the eighth order for an asymmetric Markov chain was devised. The corresponding Kolmogorov equations of the eighth order have an ordered transition probability matrix. The distribution of the eight roots of this equation in the complex plane has central symmetry. The results are analytical solutions for the probabilities of the eight states of the Markov chain in time in the form of ordered determinants with respect to the indices of the eight roots and the indices of the eight states, including the column vector of the initial conditions. Symmetry has been established in the distribution on the complex plane of eight real, negative roots of the characteristic Kolmogorov equation centered at the point defined as Re ϑ = –a7/8, where a7 is the coefficient of the characteristic equation of the eighth degree at the seventh power. Formulas expressing eight roots of the characteristic Kolmogorov equation have been heuristically derived, one of which is zero, due to the intensities of failures and recovery of three subsystems, the eight states of which in general make up an asymmetric Markov chain. For structures consisting of three independently functioning processes, the random process of the transition of the structure through eight possible states with a known initial state is determined in time. An analytical solution to Kolmogorov differential equations of the eighth order for an asymmetric state graph is proposed in harmonic form for the purpose of analysis and synthesis of a random Markov process in a triple system.

Continuum-extrapolated high-order baryon fluctuations

Fluctuations play a key role in the study of QCD phases. Lattice QCD is a valuable tool to calculate them, but going to high orders is challenging. Up to the fourth order, continuum results have been available since 2015. We present the first continuum results for sixth-order baryon fluctuations for temperatures between T=130 and 200 MeV and eighth order at T=145 MeV in a fixed volume. Comparison with earlier studies with imaginary chemical potential suggests that the volume effect is under control for T<145 MeV. Our results are in sharp contrast with well-known results in the literature obtained at finite lattice spacing. Published by the American Physical Society 2024

QCD critical point, Lee-Yang edge singularities, and Padé resummations

I analyze the trajectory of the Lee-Yang edge singularities of the QCD equation of state in the complex baryon chemical potential (μB) plane for different values of the temperature by using the recent lattice results for the Taylor expansion coefficients up to eighth order in μB and various resummation techniques that blend in Padé expansions and conformal maps. By extrapolating from this information, I estimate for the location of the QCD critical point: Tc≈100 MeV, μc≈580 MeV. I also estimate the crossover slope at the critical point to be α1≈9∘ and further constrain the nonuniversal mapping parameters between the three-dimensional Ising model and QCD equations of state. Published by the American Physical Society 2024

Anharmonic incommensurate structure modulation in Ni-Mn-Ga martensite exhibiting highly mobile twin boundaries

We perform neutron diffraction on a bulk Ni50.0Mn27.5Ga22.5 single crystal to investigate the evolution of its five-layered modulated (10M) martensite structure, from 300 K down to 10 K. Close to martensite transformation, the modulation is nearly commensurate, with q = 0.402 in a modulation vector (q, q, 0). Upon cooling, the shift in diffraction satellites indicates a transition to an incommensurate modulation with increasing q. However, the observed fifth-order diffraction satellites below 260 K and even more complex satellite landscape at 10 K cannot be explained by incommensurability alone. Our analysis reveals that the modulation function is highly anharmonic, encompassing Fourier components up to the eighth order. We have developed a single-parameter description of the evolution of modulation across the entire temperature interval of the 10M phase existence. Surprisingly, the structure evolution from commensurate to incommensurate modulation has no significant effect on twin boundary mobility.

A design method for filter to achieve arbitrary distribution of multiple transmission zeros without cross-coupling

This paper proposes a method for introducing transmission zeros (TZs) in filters using dual-mode resonators. By loading a dual-mode resonator as a second-order extracted-pole unit (EPU) at the input or output of the filter, a pair of finite TZs can be introduced without cross-coupling. This approach reduces the size of the filter while generating two finite transmission zeros compared to conventional methods. Since the dual-mode resonator is characterized by its separable odd and even modes, and by adjusting the resonant frequency of the EPU, it can achieve an arbitrary distribution of TZs. It is worth noting that the EPU does not affect the insertion loss of the filter because it operates mainly outside the passband. Furthermore, this work delves into the EPU’s mechanism for generating TZs and the method of adjusting TZ position by designing filters of second, fourth, sixth, and eighth orders, providing a solid foundation for the creation of higher-order filter circuits.

Phi4tools: compilation of Feynman diagrams for Landau-Ginzburg-Wilson theories

Scalar field theories with quartic interactions are of central interest in the study of second-order phase transitions. For three-dimensional theories, numerous studies make use of the fixed-dimensional perturbative computation of [B. Nickel, D. Meiron, and G. Baker Jr, Compilation of 2-pt and 4-pt graphs for continuous spin model, University of Guelph report (1977)], unfortunately left unpublished. We independently verify the results of Nickel et al., and we extend the computation to the eighth order in the coupling constant. The results of our calculations, together with the tools developed, are made available in Phi4tools, a user-friendly package that allows displaying the information about the individual Feynman diagrams, including the numerical values for the diagrams for zero, two, and four-point functions. We also provide the perturbative series up to order eight for the renormalization-group functions for the O(N) and cubic anisotropic models.

2-битные метаповерхности Патчаратнама-Берри с генерацией, интерференцией и гашением орбитального углового момента для широкополосного снижения ЭПР

The radar cross-section (RCS) reduction using non-absorbing digital Patcharatnam-Berry (PB) metasurfaces (MSs), consisting of 2-bit modules with generation, interference and cancellation of vortex electromagnetic waves with orbital angular momentum (OAM) is studied. Each module has a helical PB-phase profile and possesses vortex scattering. Module unit cells contain a shielded dielectric substrate and meta-particles in the form of triple-coupled split-ring resonators. The modules are distinguished by 2-bit coding of the initial tilt angle of the meta-particles 0º, 45º, 90º and 135º. The purpose of the work is to numerically analyze the generation of antiphase vortex beams (VBs) by OAM-modules and to assess the influence of interference and cancellation of antiphase vortices on the structure and order of OAM in the resulting field and the RCS reduction of three digital MSs in the form of 2´2 lattice of 2-bit OAM-modules of different sizes with different topological charges. The scattering characteristics are studied using HFSS for normally incident circularly polarized waves using of the finite element method. The simulation showed that the interference of phase vortices of four OAM-modules forms the main axial vortex and N additional interference side near and far vortices (N depends on the size of the MS). Interference vortex beams create phase singularities and zero the resulting far field in N side directions over the entire forward hemisphere. The intense OAM modes of the main vortex beams are of the fourth or eighth order. The OAM modes of adjacent interference VBs, as a rule, differ in sign. The proposed 2-bit MSs provide wider-angle phase cancellation of vortex fields (compared to 1-bit MSs) and, as a consequence, wider-angle diffuse scattering of the resulting field of the MS, which is important for RCS reduction.

Investigation of W and M shaped solitons in an optical fiber for eighth order nonlinear Schrödinger (NLS) equation

Investigation of W and M shaped solitons in an optical fiber for eighth order nonlinear Schrödinger (NLS) equation

Numerical methods of fourth, sixth and eighth orders convergence for solving third order nonlinear ODEs

In this paper, we design numerical methods of fourth, sixth and eighth orders convergence for solving BVPs of fully third order nonlinear differential equations. The methods are based on the use of high order quadrature formulas for computing integrals containing Green function and its derivatives at each iteration of the iterative method on continuous level for finding the solutions of the BVPs. We prove that the order of the methods is equal to the order of quadrature methods used. Many examples confirm the theoretical conclusion.

Compact freeform-surface-based Offner imaging spectrometer with both a long-slit and broadband.

Current imaging spectrometers with conventional optical elements face major challenges in achieving a large field of view (FOV), broadband and compact structure simultaneously. In this paper, a compact freeform-surface-based Offner imaging spectrometer with both a long-slit and a broadband (CISLS) is proposed. To keep a long slit and an anastigmatic imaging, the slit off-axis amount of the initial system is within a specific range theoretically. While to achieve a compact structure, the slit off-axis amount should be away from the specific range and as small as possible. Based on the vector aberration theory and the analytical study, Zernike polynomial terms Z5 and Z6 introduce the astigmatism independent of FOV. They are utilized to well balance the astigmatism when the slit off-axis amount is away from the specific range, helping a miniaturization of the system. Other Zernike polynomial terms below the eighth order introduce the astigmatism related to FOV. They contribute to balancing the astigmatism that produced with the increasing of the FOV, thus achieving a wide FOV. The design results show that the proposed CISLS with a high spectral resolution of 2.7 nm achieves a long slit of 30 mm in length but a small size of only 60 mm × 64 mm × 90 mm in volume under a broadband from 400 nm to 1000 nm.

Numerical Solution of Eighth Order Two Point Boundary Value Problems by Taylor Series Method

A Taylor series method which is taught at undergraduate levels and which has been hitherto used for the solution of initial value problem is successfully used in this work for the solutions of eighth-order two point boundary-value problems. The method is based on successive differentiation of the governing equation to obtain high order derivatives and then evaluated at the boundary point x=a. The solution is expressed inform a Taylor series with the unknown coefficients at a point x=a. By applying boundary conditions at x=b in the Taylor series solution, the system of unknown coefficient is obtained. After solving the system, then unknown coefficient are determined. The procedure is applied on both linear and nonlinear boundary-value problems. A comparison of the results obtained by the present method with results obtained by other methods reveals that the present method is simple, effective and also is in good agreement with the previous result and exact solution as showing in the tables and figures.

Nine-Stage Runge–Kutta–Nyström Pairs Sharing Orders Eight and Six

We explore second-order systems of non-stiff initial-value problems (IVPs), particularly those cases where the first derivatives are absent. These types of problems are of significant interest and have applications in various domains, such as astronomy and physics. Runge–Kutta–Nyström (RKN) pairs stand out as highly effective methods of addressing these IVPs. In order to create a pair with eighth and sixth orders, we need to address a certain known set of equations concerning the coefficients. When constructing such pairs for use in double-precision arithmetic, we often need to meet various conditions. Primarily, we aim to maintain small coefficient magnitudes to prevent a loss of accuracy. Nevertheless, in the context of quadruple precision, we can tolerate larger coefficients. This flexibility enables us to establish pairs with eighth and sixth orders that exhibit significantly reduced truncation errors. Traditionally, these pairs are constructed to go through eight stages per step. Here, we propose using nine stages per step. Then we have available more coefficients in order to further reduce truncation errors. As a result, we construct a novel pair that, as anticipated, achieves superior performance compared to equivalent-order pairs in various significant problem scenarios.

Extended Higher Order Iterative Method for Nonlinear Equations and its Convergence Analysis in Banach Spaces

In this article, a novel higher order iterative method for solving nonlinear equations is developed. The new iterative method obtained from fifth order Newton-Özban method attains eighth order of convergence by adding a single step with only one additional function evaluation. The method is extended to Banach spaces and its local as well as semi-local convergence analysis is done under generalized continuity conditions. The existence and uniqueness results of solution are also provided along with radii of convergence balls. From the numerical experiments, it can be inferred that the proposed method is more accurate and effective in high precision computations than existing eighth order methods. The computation of error analysis and norm of functions demonstrate that proposed method takes a lead over the considered methods.

Constructing optimal fourth and eighth order iterative methods by using variant of Newton’s method

In this paper, we have presented an optimal fourth order iterative method and an optimal eighth order iterative method without memory using weight functions. In terms of computational point of view, our first method require three evaluations (two function and one first derivatives) per iteration to get fourth order and the second method require four evaluations (three functions and one derivatives) per iteration to get eighth order. Hence, these methods have high efficiency indices 1.587 and 1.682 respectively. Some numerical examples are tested to know the performance of the new methods which verifies the theoretical results.

Eighth order, Numerov-like schemes with coefficients tailored for superior performance on ODE systems with oscillatory solutions

<p>Second order Ordinary Differential Equations (ODE) were considered. Numerov-like techniques employing effectively seven stages per step and sharing eighth algebraic order were under examination for numerically solving them. The coefficients of these methods were contingent on four independent parameters. To tackle issues with oscillatory solutions, we typically aimed to fulfill specific criteria such as minimizing phase-lag, expanding the periodicity interval, or even neutralizing amplification errors. These latter attributes stemmed from a test problem mimicking an ideal trigonometric trajectory. Here, we suggested training the coefficients of the chosen method family across a broad spectrum of pertinent problems. Following this training using the differential evolution method, we identified a particular method that surpassed others in this category across an even broader array of oscillatory problems.</p>

Algorithms of predictor-corrector type with convergence and stability analysis for solving nonlinear systems

<p>Many researchers have proposed iterative algorithms for nonlinear equations and systems of nonlinear equations; similarly, in this paper, we developed two two-step algorithms of the predictor-corrector type. A combination of Taylor's series and the composition approach was used. One of the algorithms had an eighth order of convergence and a high-efficiency index of approximately 1.5157, which was higher than that of some existing algorithms, while the other possessed fourth-order convergence. The convergence analysis was carried out in both senses, that is, local and semi-local convergence. Various complex polynomials of different degrees were considered for visual analysis via the basins of attraction. We analyzed and compared the proposed algorithms with other existing algorithms having the same features. The visual results showed that the modified algorithms had a higher convergence rate compared to existing algorithms. Real-life systems related to chemistry, astronomy, and neurology were used in the numerical simulations. The numerical simulations of the test problems revealed that the proposed algorithms surpassed similar existing algorithms established in the literature.</p>

An Eighth Order A-Stable Rational Integrator

An Eighth Order A-Stable Rational Integrator

Iterative method improving Newton's method with higher order

In the present article, we have constructed and looked at improving two- and three-step iterative methods to locate simple roots of non-linear equations. It appears that the three-step iterative method converges to the eighth order. The whole aim of this research is to derive and present a new modified iterative method of higher order and to obtain less iteration than the classical Newton method for solving nonlinear equations of simple roots. The present technique has to evaluate three functions and two first derivatives in each iteration. The advantage of the proposed method has been observed to have at least better performance effective and more stability by comparing with the other methods for the same or less than order. Also, noted that our method gives better results in terms of the number of iterations. To demonstrate the efficacy and popularity, different numerical illustrations are provided for the recommended techniques.

Virial coefficients of hard, two-dimensional, convex particles up to the eighth order

We calculated virial coefficients for hard ellipses, stadia, rectangles, planar lenses and rhombi up to the eighth order employing Mayer-sampling Monte Carlo simulations. We improved and extended available data for ellipses, stadia, and rectangles and provide virial coefficients for planar lenses and rhombi. The influence of the aspect ratio ν on the virial coefficients is discussed, including hard discs as the limit of ellipses, stadia, and planar lenses and hard squares as the limit of rectangles and rhombi with aspect ratio ν → 1 . In two dimensions, both in the limit ν → 0 and ν → ∞ , anisotropic figures approach hard needles. The dependence of reduced virial coefficients B ~ i = B i / B 2 i − 1 on the shape parameter α = B 2 ∗ − 1 , with B 2 ∗ = B 2 / V P and V P the particle area is analysed. The reduced virial coefficients B ~ i ( α ) are approximated as functions of the shape parameter employing third-order polynomials, enabling an interpolation for all aspect ratios ν.