Traditional Approximation Research Articles

Detecting deep axisymmetric toroidal magnetic fields in stars

Context. Asteroseismology has revealed small core-to-surface rotation contrasts in stars in the whole Hertzsprung–Russell diagram. This is the signature of strong transport of angular momentum (AM) in stellar interiors. One of the plausible candidates to efficiently carry AM is magnetic fields with various topologies that could be present in stellar radiative zones. Among them, strong axisymmetric azimuthal (toroidal) magnetic fields have received a lot of interest. Indeed, if they are subject to the so-called Tayler instability, the accompanying triggered Maxwell stresses can transport AM efficiently. In addition, the electromotive force induced by the fluctuations of magnetic and velocity fields could potentially sustain a dynamo action that leads to the regeneration of the initial strong axisymmetric azimuthal magnetic field. Aims. The key question we aim to answer is whether we can detect signatures of these deep strong azimuthal magnetic fields. The only way to answer this question is asteroseismology, and the best laboratories of study are intermediate-mass and massive stars with external radiative envelopes. Most of these are rapid rotators during their main sequence. Therefore, we have to study stellar pulsations propagating in stably stratified, rotating, and potentially strongly magnetised radiative zones, namely magneto-gravito-inertial (MGI) waves. Methods. We generalise the traditional approximation of rotation (TAR) by simultaneously taking general axisymmetric differential rotation and azimuthal magnetic fields into account. Both the Coriolis acceleration and the Lorentz force are therefore treated in a non-perturbative way. Using this new formalism, we derive the asymptotic properties of MGI waves and their period spacings. Results. We find that toroidal magnetic fields induce a shift in the period spacings of gravity (g) and Rossby (r) modes. An equatorial azimuthal magnetic field with an amplitude of the order of 105 G leads to signatures that are detectable in period spacings for high-radial-order g and r modes in γ Doradus (γ Dor) and slowly pulsating B (SPB) stars. More complex hemispheric configurations are more difficult to observe, particularly when they are localised out of the propagation region of MGI modes, which can be localised in an equatorial belt. Conclusions. The magnetic TAR, which takes into account toroidal magnetic fields in a non-perturbative way, is derived. This new formalism allows us to assess the effects of the magnetic field in γ Dor and SPB stars on g and r modes. We find that these effects should be detectable for equatorial fields thanks to modern space photometry using observations from Kepler, TESS CVZ, and PLATO.

Many-body approach to superfluid nuclei in axial geometry

Starting from a general many-body fermionic Hamiltonian, we derive the equations of motion (EOM) for nucleonic propagators in a superfluid system. The resulting EOM is of the Dyson type formulated in the basis of Bogoliubov's quasiparticles. As the leading contributions to the dynamical kernel of this EOM in strongly-coupled regimes contain phonon degrees of freedom in various channels, an efficient method of calculating phonon's characteristics is required to successfully model these kernels. The traditional quasiparticle random phase approximation (QRPA) solvers are typically used for this purpose in nuclear structure calculations, however, they become very prohibitive in nonspherical geometries. In this paper, by linking the notion of the quasiparticle-phonon vertex to the variation of the Bogoliubov's Hamiltonian, we show that the recently developed finite-amplitude method (FAM) can be efficiently employed to compute the vertices within the FAM-QRPA. To illustrate the validity of the method, calculations based on the relativistic density-dependent point-coupling Lagrangian are performed for the single-nucleon states in heavy and medium-mass nuclei with axial deformations. The cases of $^{38}\mathrm{Si}$ and $^{250}\mathrm{Cf}$ are presented and discussed.

Density functional theory of waterwith the machine-learned DM21 functional.

The delicate interplay between functional-driven and density-driven errors in density functional theory (DFT) has hindered traditional density functional approximations (DFAs) from providing an accurate description of water for over 30 years. Recently, the deep-learned DeepMind 21 (DM21) functional has been shown to overcome the limitations of traditional DFAs as it is free of delocalization error. To determine if DM21 can enable a molecular-level description of the physical properties of aqueous systems within Kohn-Sham DFT, we assess the accuracy of the DM21 functional for neutral, protonated, and deprotonated water clusters. We find that the ability of DM21 to accurately predict the energetics of aqueous clusters varies significantly with cluster size. Additionally, we introduce the many-body MB-DM21 potential derived from DM21 data within the many-body expansion of the energy and use it in simulations of liquid water as a function of temperature at ambient pressure. We find that size-dependent functional-driven errors identified in the analysis of the energetics of small clusters calculated with the DM21 functional result in the MB-DM21 potential systematically overestimating the hydrogen-bond strength and, consequently, predicting a more ice-like local structure of water at room temperature.

An Enriched Finite Element Method with Appropriate Interpolation Cover Functions for Transient Wave Propagation Dynamic Problems

A novel enriched finite element method (EFEM) was employed to analyze the transient wave propagation problems. In the present method, the traditional finite element approximation was enriched by employing the appropriate interpolation covers. We mathematically and numerically showed that the present EFEM possessed the important monotonic convergence property with the decrease of the used time steps for transient wave propagation problems when the unconditional stable Newmark time integration scheme was used for time integration. This attractive property markedly distinguishes the present EFEM from the traditional FEM for transient wave propagation problems. Two typical numerical examples were given to demonstrate the capabilities of the present method.

A Robust Pseudo-Three-Dimensional Computational Fluid Dynamic Approach for Industrial Applications

Abstract Two-dimensional instead of three-dimensional computational fluid dynamic solutions of flow problems are quite often used in industry to facilitate short design turn-around times with varying degrees of success. A simple and robust approach for improving the accuracy of two-dimensional computational fluid dynamics solutions for problems involving internal flow passages in industrial applications is presented. The technique utilizes an approximation to the shearing stresses that act in the fully three-dimensional case but are ignored in the traditional two-dimensional approximation. Although the technique does not fully account for all the three-dimensional effects in such flows, it gives a reasonable estimate of the operation of devices with internal flows, even those involving transients. The usefulness and accuracy of the method are demonstrated through the application of the method to predict the performance of a supersonic fluidic oscillator for industrial design purposes. This brief provides industrial designers with a simple and robust tool for improving the accuracy of their computational fluid dynamic simulations.

Nonparaxial pulse propagation in a planar waveguide with Kerr–like and quintic nonlinearities; computational simulations

This article investigates some novel solitary wave solutions of the cubic-quintic nonlinear Helmholtz equation by implementing two recent analytical (Khater II and generalized exponential methods) techniques. As a result of seeing the nonparaxial effect, this model was developed to show how a pulse propagates in a planar waveguide with Kerr-like and quintic nonlinearities as well as spatial dispersion. When the traditional approximation of a slowly changing envelope fails, this effect takes over. Some novel soliton wave solutions are obtained and formulated in distinct forms such as rational, hyperbolic, and trigonometric forms. The obtained solutions are represented through various figures’ types to show the physical and dynamical behavior of the nonparaxial pulse propagation. The model’s features are demonstrated, such as its intriguing bright, anti-bright, periodic, singular, gray, chirped anti-dark, and dark solitary waves depending upon the nature of nonlinearities. The solutions’ originality is discussed by comparing it with those constructed in recently published articles. All built results are checked their accuracy by putting them back into the original model by employing Mathematica 12.

Oscillating dynamical friction on galactic bars by trapped dark matter

ABSTRACT The dynamic evolution of galactic bars in standard ΛCDM models is dominated by angular momentum loss to the dark matter haloes via dynamical friction. Traditional approximations to dynamical friction are formulated using the linearized collisionless Boltzmann equation and have been shown to be valid in the fast limit, i.e. for rapidly slowing bars. However, the linear assumption breaks down within a few dynamical periods for typical slowly evolving bars, which trap a significant amount of disc stars and dark matter in resonances. Recent observations of the Galactic bar imply this slow regime at the main bar resonances. We formulate the time-dependent dynamical friction in the slow limit and explore its mechanism in the general slow regime with test-particle simulations. Here, angular momentum exchange is dominated by resonantly trapped orbits which slowly librate around the resonances. In typical equilibrium haloes, the initial phase-space density within the trapped zone is higher at lower angular momentum. Since the libration frequency falls towards the separatrix, this density contrast winds up into a phase-space spiral, resulting in a dynamical friction that oscillates with ∼Gyr periods and damps over secular time-scales. We quantify the long-term behaviour of this torque with secular perturbation theory, and predict two observable consequences: (i) The phase-space spirals may be detectable in the stellar disc where the number of windings encodes the age of the bar. (ii) The torque causes weak oscillations in the bar’s pattern speed, overlaying the overall slowdown – while not discussed, this feature is visible in previous simulations.

Enhanced Feature Selection Based on Integration Containment Neighborhoods Rough Set Approximations and Binary Honey Badger Optimization.

This article appoints a novel model of rough set approximations (RSA), namely, rough set approximation models build on containment neighborhoods RSA (CRSA), that generalize the traditional notions of RSA and obtain valuable consequences by minifying the boundary areas. To justify this extension, it is integrated with the binary version of the honey badger optimization (HBO) algorithm as a feature selection (FS) approach. The main target of using this extension is to assess the quality of selected features. To evaluate the performance of BHBO based on CRSA, a set of ten datasets is used. In addition, the results of BHOB are compared with other well-known FS approaches. The results show the superiority of CRSA over the traditional RS approximations. In addition, they illustrate the high ability of BHBO to improve the classification accuracy overall the compared methods in terms of performance metrics.

The multiple kink solutions and interaction mechanism with help of the coupled Burgers' equation

This paper presents a study of the traditional approximation potential vorticity equation. A nonlinear forced Zakharov-Kuznetsov equation by using the perturbation expansion method and multiple temporal and spatial scales method is obtained. And the multiple kink solutions of the forced Zakharov-Kuznetsov equation are obtained and the one- and two-soliton solutions are given by making use of the simplest equation method, namely the coupled Burgers’ equation, which is of a low-order than obtained equation and the multiple kink solutions of the equation are already given by utilizing the simplified Hirota's method. The results show that the proposed method is a powerful mathematical method to solve the nonlinear evolution equation arising in various areas of nonlinear science. Based on the one- and two-soliton solutions, graphical figures are portrayed by taking definite values to parameters and the features of the variable parameters are discussed in detail. Some results are given about interaction of two-soliton solutions and we found the wave amplitude of the two-soliton is influenced with the Coriolis parameters. Meanwhile, the soliton of the model with the external source is also obtained.

Off-resonant dynamics of the Rabi model under a deep-strong slow driving

We investigate the dynamics of a two-level system driven by a deep-strong slow sinusoidal field using a novel counterrotating-hybridized rotating-wave (CHRW) method. By two successive unitary transformations, we obtain a transformed Hamiltonian which possesses a simple rotating-wave interaction with a renormalized driving strength and a modified tunneling strength. The modified CHRW method allows us to directly calculate the analytical results of the dynamics and Bloch-Siegert shift. We show that the results given by the modified CHRW method are in good agreement with the numerical solutions of the Rabi model with a moderate strong driving strength in the off-resonance case. It turns out that the modified CHRW method is beyond the traditional rotating-wave approximation in some interesting parameter regions. As the modified CHRW method is mathematically simple and physically clear, it may be applied to driven dissipative systems which are difficult to solve numerically.

Self-interacting dark matter in cosmology: accurate numerical implementation and observational constraints

This paper presents a systematic and accurate treatment of the evolution of cosmological perturbations in self-interacting dark matter models, for particles which decoupled from the primordial plasma while relativistic. We provide a numerical implementation of the Boltzmann hierarchies developed in a previous paper [JCAP, 09 (2020) 041] in a publicly available Boltzmann code and show how it can be applied to realistic DM candidates such as sterile neutrinos either under resonant or non-resonant production mechanisms, and for different field mediators. At difference with traditional fluid approximations — also known as a c eff-c vis parametrizations — our approach follows the evolution of phase-space perturbations under elastic DM interactions for a wide range of interaction models, including the effects of late kinetic decoupling. Finally, we analyze the imprints left by different self interacting models on linear structure formation, which can be constrained using Lyman-α forest and satellite counts. We find new lower bounds on the particle mass that are less restrictive than previous constraints.

A Pseudo-Spectral Fourier Collocation Method for Inhomogeneous Elliptical Inclusions with Partial Differential Equations

Inhomogeneous elliptical inclusions with partial differential equations have aroused appreciable concern in many disciplines. In this paper, a pseudo-spectral collocation method, based on Fourier basis functions, is proposed for the numerical solutions of two- (2D) and three-dimensional (3D) inhomogeneous elliptic boundary value problems. We describe how one can improve the numerical accuracy by making some extra “reconstruction techniques” before applying the traditional Fourier series approximation. After the particular solutions have been obtained, the resulting homogeneous equation can then be calculated using various boundary-type methods, such as the method of fundamental solutions (MFS). Using Fourier basis functions, one does not need to use large matrices, making accrual computations relatively fast. Three benchmark numerical examples involving Poisson, Helmholtz, and modified-Helmholtz equations are presented to illustrate the applicability and accuracy of the proposed method.

Improved Quasi-Steady-State Approximation Analysis of Stefan Problems Under 2nd-Kind Boundary Conditions

Improved quasi-steady state approximation solutions were obtained for Stefan problem under the second kind boundary condition, both in Cartesian and cylindrical coordinate, based on traditional quasi-steady state approximation and the first law of thermodynamics. For Cartesian coordinate condition, the solution has high accuracy, and it is convenient for practical use for its explicit form. For cylindrical coordinate solution, the provided approximation solution is the only solution reported in the literature. The proposed improved solutions take sensible heat into consideration and greatly improve the accuracy of traditional method, it enriched the analysis method for Stefan problem, which has definite physical meaning and can be served as reference for quick preliminary calculation of practical problems.

MuGRA: A Scalable Multi-Grained Reconfigurable Accelerator Powered by Elastic Neural Network

A massive core computing architecture is developed for accelerating arbitrary calculations in fully parallel with high speed and low cost. The proposed architecture is reconfigurable in fine-grained (arbitrary functions), mid-grained (flexible function feature, accuracy, and number of operands), and coarse-grained (organization of cores). By implementing a large scale of novel bisection neural network (BNN) on hardware, the re-configuration is conducted by partitioning entire BNN into any specific pieces without redundancy. Each piece of BNN retrieves the arbitrary function approximately. By reconfiguring the BNN topology in software, we can easily adjust dimensions of the computing kernel without rewiring, and achieve a wide range of trade-offs between accuracy and efficiency in hardware. In this manner, the multi-grained reconfigurable accelerator (MuGRA) is achieved. Since MuGRA is flexible in all grained levels, various configurations for each validation are demonstrated with rich options of performance-cost matrix. From the FPGA implementation results, compared with other traditional function approximation methods, our method provides fewer parameter storage requirements. The comparison against related works proves that our accelerator effectively reduces the calculation latency with slight accuracy loss.

Template banks based on Zn and An* lattices

Matched filtering is a traditional method used to search a data stream for signals. If the source (and hence its $n$ parameters) are unknown, many filters must be employed. These form a grid in the $n$-dimensional parameter space, known as a template bank. It is often convenient to construct these grids as a lattice. Here, we examine some of the properties of these template banks for $\mathbb{Z}^n$ and $A_n^*$ lattices. In particular, we focus on the distribution of the mismatch function, both in the traditional quadratic approximation and in the recently-proposed spherical approximation. The fraction of signals which are lost is determined by the even moments of this distribution, which we calculate. Many of these quantities we examine have a simple and well-defined $n\to\infty$ limit, which often gives an accurate estimate even for small $n$. Our main conclusions are the following: (i) a fairly effective template-based search can be constructed at mismatch values that are shockingly high in the quadratic approximation; (ii) the minor advantage offered by an $A_n^*$ template bank (compared to $\mathbb{Z}^n$) at small template separation becomes even less significant at large mismatch. So there is little motivation for using template banks based on the $A_n^*$ lattice.

Power law bond price and yield approximation

PurposeThis paper determines a simple transformation that nearly linearizes the bond price formula. The transformed price can be used to derive a highly accurate approximation of the change in a bond price resulting from a change in interest rates.Design/methodology/approachA logarithmic transformation exactly linearizes the price function for a zero coupon bond and a reciprocal transformation exactly linearizes the price function for a perpetuity. A power law transformation combines aspects of both types of transformations and provides a superior approximation of the bond price sensitivity for both short-term and long-term bonds.FindingsIt is demonstrated that the new formula, based on power-law transformation, is a much better approximation than either the traditional duration-convexity approximation and the more recently developed approximations based on logarithmic transformation of the price function.Originality/valueThe new formula will be used by risk managers to perform stress-testing on bond portfolios. The new formula can easily be inverted, making it possible to relate the distribution of prices (which are observable in the market) to the distribution of yields (which are numerical solutions that are not directly observable).

A new limited memory method for unconstrained nonlinear least squares

This paper suggests a new limited memory trust region algorithm for large unconstrained black box least squares problems, called LMLS. Main features of LMLS are a new non-monotone technique, a new adaptive radius strategy, a new Broyden-like algorithm based on the previous good points, and a heuristic estimation for the Jacobian matrix in a subspace with random basis indices. Our numerical results show that LMLS is robust and efficient, especially in comparison with solvers using traditional limited memory and standard quasi-Newton approximations.

Relativistic motion on Gaussian quantum steering for two-mode localized Gaussian states

Realistic quantum systems always exhibit gravitational and relativistic features. In this paper, we investigate the properties of Gaussian steering and its asymmetry by the localized two-mode Gaussian quantum states, instead of the traditional single-mode approximation method in the relativistic setting. We find that the one-side Gaussian quantum steering will monotonically decrease with increasing observers of acceleration. Meanwhile, our results also reveal the interesting behavior of the Gaussian steering asymmetry, which increases for a specific range of accelerated parameter and then gradually approaches to a finite value. Such finding is well consistent and explained by the well-known Unruh effect, which could significantly destroy the one-side Gaussian quantum steering. Finally, our results could also be applied to the dynamical studies of Gaussian steering between the Earth and satellites, since the effects of acceleration are equal to the effects of gravity according to the equivalence principle.

The traditional approximation of rotation for rapidly rotating stars and planets

Context. We examine the dynamics of low-frequency gravito-inertial waves (GIWs) in differentially rotating deformed radiation zones in stars and planets by generalising the traditional approximation of rotation (TAR). The TAR treatment was built on the assumptions that the star is spherical (i.e. its centrifugal deformation is neglected) and uniformly rotating. However, it has been generalised in our previous work by including the effects of the centrifugal deformation using a non-perturbative approach. In the meantime, TAR has been generalised in spherical geometry to take the differential rotation into account. Aims. We aim to carry out a new generalisation of the TAR treatment to account for the differential rotation and the strong centrifugal deformation simultaneously. Methods. We generalise our previous work by taking into account the differential rotation in the derivation of our complete analytical formalism that allows the study of the dynamics of GIWs in differentially and rapidly rotating stars. Results. We derived the complete set of equations that generalises the TAR, simultaneously taking the full centrifugal acceleration and the differential rotation into account. Within the validity domain of the TAR, we derived a generalised Laplace tidal equation for the horizontal eigenfunctions and asymptotic wave periods of the GIWs, which can be used to probe the structure and dynamics of differentially rotating deformed stars with asteroseismology. Conclusions. A new generalisation of the TAR, which simultaneously takes into account the differential rotation and the centrifugal acceleration in a non-perturbative way, was derived. This generalisation allowed us to study the detectability and the signature of the differential rotation on GIWs in rapidly rotating deformed stars and planets. We found that the effects of the differential rotation in early-type deformed stars on GIWs is theoretically largely detectable in modern space photometry using observations from Kepler and TESS.

R-L ALGORITHM: AN APPROXIMATION ALGORITHM FOR FRACTAL SIGNALS BASED ON FRACTIONAL CALCULUS

An important characteristic of a fractal signal is that its graph is not smooth in any small interval. This indicates the difficulty of the approximation of fractal signals, because traditional approximation methods normally require some certain smoothness of the approximated function. However, recent studies have shown that fractal functions that satisfy the Hölder condition can be linearly changed in the fractal dimension of their graphs by the fractional calculus, which implies that we can use the fractional calculus to make graphs of fractal signals smoother, and then approximate these fractal signals. This paper first gives our research background and related theories of fractals and the fractional calculus, and then introduces the main research content, including the following aspects: (1) Proposing a fractal signal approximation algorithm, the R-L algorithm, explaining and deriving how to implement this algorithm. (2) Aiming at a specific fractal signal, carrying out an approximation experiment, and confirming the R-L algorithm is better than a direct approximation. (3) According to the experimental process and conclusions, introducing our future work, such as further optimization of the R-L algorithm, and showing the R-L algorithm may have the predictive capability outside the sampling interval. The R-L algorithm can effectively modify the smoothness of graphs of the fractal signal, so that more types of approximation algorithms can be selected with a better approximation effect.