Analytical Formulas Research Articles

Dipole–dipole-interaction-induced entanglement between two-dimensional ferromagnets

We investigate the viability of dipole–dipole interaction as a means of entangling two distant ferromagnets. To this end, we make use of the Bogoliubov transformation as a symplectic transformation. We show that the coupling of the uniform magnon modes can be expressed using four squeezing parameters, which we interpret in terms of hybridization, one-mode, and two-mode squeezing. We utilize the expansion in terms of the squeezing parameters to obtain an analytic formula for the entanglement in the magnon ground state using the logarithmic negativity as entanglement measure. Our investigation predicts that for infinitely large two-dimensional ferromagnets, the dipole–dipole interaction does not lead to significant long-range entanglement. However, in the case of finite ferromagnets, finite entanglement can be expected.

Analysis of two-dimensional planar gratings and metasurfaces in the quasi-static approximation

Currently, composite materials based on metal gratings with a period D much smaller than the wavelength l are widely used. With the use of such materials, it is possible to control the amplitude, phase and polarization of an electromagnetic wave, which opens wide possibilities for the creation of new types of antennas, radio-absorbing coatings, etc. This paper considers gratings of metallic strip or patches on the surface of a thin metal-backed dielectric slab to control the phase of the reflected wave. In the quasi-static approximation, the input impedance of such materials can be calculated using simple analytical formulas. This paper investigates the accuracy of the quasi-static approximation at different angles of incidence of the electromagnetic wave on two-dimensional gratings. Using the equivalent circuits method, analytical expressions for the input impedance of metasurfaces are obtained. The dispersion properties of composite materials “inductive or capacitive grating on the surface of a thin metal-backed dielectric slab” are investigated, and a technique for determining the resonant frequency, which corresponds to the maximum of the input impedance of the metasurface, is presented. The obtained results show that at any angle of incidence, it is possible to evaluate the condition of the quasi-static approximation applicability for two-dimensional gratings and for metasurfaces based on such gratings. If this condition is satisfied, that simple analytical expressions can be used to calculate the input impedance. The use of these expressions allows us to determine the frequency at which the input impedance reaches its maximum. It is established that for inductive and capacitive gratings the impedance boundary condition of M.A. Leontovich is equivalent to the averaged boundary condition of M.I. Kontorovich. It is obtained that for grating in free space, quasi-static approximation corresponds to the M.A. Leontovich impedance condition. The dispersion properties of composite materials “inductive or capacitive grating on a thin metal-backed dielectric slab” are investigated. It is shown that the phase transition of the reflection coefficient through zero occurs at maximum values of the input impedance. Using the considered method of equivalent circuits, the presented results can be easily generalized to any type of multilayer metasurfaces.

Asymptotical convergence of solutions of boundary value problems for singularly perturbed higher-order integro-differential equations

The article considers a two-point boundary value problem for a linear integro-differential equation of the n + m order with small parameters for m higher derivatives, provided that the roots of the additional characteristic equation are negative. The aim of the study is to obtain asymptotic estimates of the solution, to find out the asymptotic behavior of solutions in the vicinity of points where additional conditions are set, as well as to construct a degenerate problem, the solution of which tend to the solution of the initial perturbed boundary value problem. The Cauchy function and boundary functions of a boundary value problem for a singularly perturbed homogeneous differential equation are constructed, and their asymptotic estimates are obtained. Using the Cauchy functions and boundary functions, an analytical formula for solutions to the boundary value problem is obtained. A theorem on an asymptotic estimate for the solution of the considered boundary value problem is proved. The asymptotic behavior of the solution with respect to a small parameter and the order of growth of its derivatives are established. It is shown that the solution of the boundary value problem under consideration at the left end of this segment has the phenomenon of an initial jump and the order of this jump is determined. A modified degenerate boundary value problem containing initial jumps of the solution and the integral term is constructed.

Lagrangian descriptors with uncertainty

Lagrangian descriptors provide a global dynamical picture of the geometric structures for arbitrarily time-dependent flows with broad applications. This paper develops a mathematical framework for computing Lagrangian descriptors when uncertainty appears. The uncertainty originates from estimating the underlying flow field as a natural consequence of data assimilation or statistical forecast. It also appears in the resulting Lagrangian trajectories. The uncertainty in the flow field directly affects the path integration of the crucial nonlinear positive scalar function in computing the Lagrangian descriptor, making it fundamentally different from many other diagnostic methods. Despite being highly nonlinear and non-Gaussian, closed analytic formulae are developed to efficiently compute the expectation of such a scalar function due to the uncertain velocity field by exploiting suitable approximations. A rapid and accurate sampling algorithm is then built to assist the forecast of the probability density function (PDF) of the Lagrangian trajectories. Such a PDF provides the weight to combine the Lagrangian descriptors along different paths. Simple but illustrative examples are designed to show the distinguished behavior of using Lagrangian descriptors in revealing the flow field when uncertainty appears. Uncertainty can either completely erode the flow3 structure or barely affect the underlying geometry of the flow field. The method is also applied for eddy identification, indicating that uncertainty has distinct impacts on detecting eddies at different time scales. Finally, when uncertainty is incorporated into the Lagrangian descriptor for inferring the source target, the likelihood criterion provides a very different conclusion from the deterministic methods.

Easy bootstrap for the 3D Ising model: a hybrid approach of the lightcone bootstrap and error minimization methods

As a simple lattice model that exhibits a phase transition, the Ising model plays a fundamental role in statistical and condensed matter physics. The Ising transition is realized by physical systems, such as the liquid-vapor transition. Its continuum limit also furnishes a basic example of interacting quantum field theories and universality classes. Motivated by a recent hybrid bootstrap study of the quantum quartic oscillator, we revisit the conformal bootstrap approach to the 3D Ising model at criticality, without resorting to positivity constraints. We use at most 10 nonperturbative crossing constraints at low derivatives from the Taylor expansion around a crossing symmetric point. The high-lying contributions are approximated by simple analytic formulae deduced from the lightcone singularity structure. Surprisingly, the low-lying properties are determined to good accuracy by this computationally very cheap approach. For instance, the results for the two relevant scaling dimensions (∆σ, ∆ϵ) ≈ (0.518153, 1.41278) are close to the most precise rigorous bounds obtained at a much higher computational cost.

CGNSDE: Conditional Gaussian neural stochastic differential equation for modeling complex systems and data assimilation

A new knowledge-based and machine learning hybrid modeling approach, called conditional Gaussian neural stochastic differential equation (CGNSDE), is developed to facilitate modeling complex dynamical systems and implementing analytic formulae of the associated data assimilation (DA). In contrast to the standard neural network predictive models, the CGNSDE is designed to effectively tackle both forward prediction tasks and inverse state estimation problems. The CGNSDE starts by exploiting a systematic causal inference via information theory to build a simple knowledge-based nonlinear model that nevertheless captures as much explainable physics as possible. Then, neural networks are supplemented to the knowledge-based model in a specific way, which not only characterizes the remaining features that are challenging to model with simple forms but also advances the use of analytic formulae to efficiently compute the nonlinear DA solution. These analytic formulae are used as an additional computationally affordable loss to train the neural networks that directly improve the DA accuracy. This DA loss function promotes the CGNSDE to capture the interactions between state variables and thus advances its modeling skills. With the DA loss, the CGNSDE is more capable of estimating extreme events and quantifying the associated uncertainty. Furthermore, crucial physical properties in many complex systems, such as the translate-invariant local dependence of state variables, can significantly simplify the neural network structures and facilitate the CGNSDE to be applied to high-dimensional systems. Numerical experiments based on chaotic systems with intermittency and strong non-Gaussian features indicate that the CGNSDE outperforms knowledge-based regression models, and the DA loss further enhances the modeling skills of the CGNSDE.

Pricing Asian options under the mixed fractional Brownian motion with jumps

The mixed fractional Brownian motion (mfBm) has gained popularity in finance because it can effectively model long-range dependence, self-similarity, and is arbitrage-free. This paper focuses on mfBm with jumps modeled by the Poisson process and derives an analytical formula for valuing geometric Asian options. Additionally, approximate closed-form solutions for pricing arithmetic Asian options and arithmetic Asian power options are obtained. Numerical examples are provided to demonstrate the accuracy of these formulas, which rely on a convenient approximation of the option strike price. The proposed approximation demonstrates significantly higher computational efficiency compared to Monte Carlo simulation.

Dynamic response of girder bridges with corrugated steel webs subjected to moving loads

Girders with corrugated steel webs (CSWs), replacing traditional concrete webs in prestressed concrete box girders, exhibit a unique dynamic behavior due to the accordion effect and the thinner dimension of CSWs. The study aims to investigate the dynamic response of girder bridges with CSWs under moving loads. Utilizing the Timoshenko beam theory and modal analysis method, the analytical formula for the dynamic response, considering shear deformation and rotational inertia, was developed. The proposed analytical formula was validated through finite element simulation. Employing dimensionless parameters, a parametric analysis was conducted to examine the effects of multi-modes, loading position and shear deformation on the dynamic response of girders with CSWs. The results highlight the significant influence of the shear parameter αS and the speed parameter SE1 on the magnitude of the impact factor. Precise calculations for deflection and bending moment only require consideration of low-order modes, whereas accurate estimation of shear force requires attention to higher-order modes. In particular, a high-frequency resonance phenomenon is identified when the speed parameter SE1 and the shear parameter αS conform to the formula “πSE1αS = 1”. The suggested impact factor formulas for midpoint deflection, midpoint bending moment and end shear force have been developed and categorized into three groups based on the magnitude of the shear parameter αS.

Optimization on frequency constraints with FFT using automatic differentiation on hybrid ODE applications

PurposeThis study aims to optimize electrical systems represented by ordinary differential equations and events, using their frequency spectrum is an important purpose for designers, especially to calculate harmonics.Design/methodology/approachThis paper presents a methodology to achieve this, by using a gradient-based optimization algorithm. The paper proposes to use a time simulation of the electrical system, and then to compute its frequency spectrum in the optimization loop.FindingsThe paper shows how to proceed efficiently to compute the frequency spectrum of an electrical system to include it in an optimization loop. Derivatives of the frequency spectrum such as the optimization inputs can also be calculated. This is possible even if the sized system behavior cannot be defined a priori, e.g. when there are static converters or electrical devices with natural switching.Originality/valueUsing an efficient sequential quadratic programming optimizer, automatic differentiation is used to compute the model gradients. Frequency spectrum derivatives with respect to the optimization inputs are calculated by an analytical formula. The methodology uses a “white-box” approach so that automatic differentiation and the differential equations simulator can be used, unlike most state-of-the-art simulators.

Numerical Calculation and Analytic Formulas of Yarkovsky Effect Parameter of Circumsolar Asteroids

Numerical Calculation and Analytic Formulas of Yarkovsky Effect Parameter of Circumsolar Asteroids

Features of determining the estimated length of positive arches made of laminated wood

The issue of loss of stability of arched structures is very important in engineering practice and design. This issue is especially key from the point of view of reliability and safety and economic losses, since structures of this type are usually used in buildings with a consequence class of at least CC2. [1].Laminated timber arches are long-span com-pression-bent elements in which the loss of stability plays a major role in determining the cross-sectional dimensions. In contrast to beams, an arch has a much larger ratio between the length of the element and its cross section, which in turn increases the risk of loss of stability. The loss of stability of an arch and its cor-rectly determined design length are directly related. The design length is defined as the value at which an arch can lose its stability under the influence of external loads. The main aspects of this relationship are: the value of the design length of the arch; mechanisms of loss of stabil-ity; dependence on the material and shape; ac-curacy of engineering calculations.It should be noted that recommendations for determining the design length of an arch are contained exclusively in the methodological lit-erature and outdated regulatory documents, such as [7]. In the latest current regulatory doc-ument [8], this issue is resolved by checking the strength and stability of the element, without limiting the flexibility. Since the flexibility of the element depends on its design length, the is-sue of determining this parameter remains open and relies directly on the design engineer, or rather, on his experience and competence. This issue is very relevant, given that a minor error in determining the design length significantly affects the results of testing compressed-bent elements.As a result of the work carried out, we analyzed the determination of the design lengths of hollow arches with different shapes and types of loading using two calculation methods: analytical formulas and the finite element method using the LIRA-SAPR-2019 software. Based on the calculations, recommendations for determining the design lengths of arches are given.

Mixed noise and posterior estimation with conditional deepGEM

We develop an algorithm for jointly estimating the posterior and the noise parameters in Bayesian inverse problems, which is motivated by indirect measurements and applications from nanometrology with a mixed noise model. We propose to solve the problem by an expectation maximization (EM) algorithm. Based on the current noise parameters, we learn in the E-step a conditional normalizing flow that approximates the posterior. In the M-step, we propose to find the noise parameter updates again by an EM algorithm, which has analytical formulas. We compare the training of the conditional normalizing flow with the forward and reverse Kullback–Leibler divergence, and show that our model is able to incorporate information from many measurements, unlike previous approaches.

Unsteady solute dispersion of electro-osmotic flow of micropolar fluid in a rectangular microchannel

This study scrutinizes the two-dimensional concentration distribution for a solute cloud containing a micropolar fluid in a rectangular microchannel under the influence of an applied electric field. The concentration distribution is obtained up to second order approximation using Mei's homogenization method. Analytical formulas are derived for dispersion coefficient, mean and two-dimensional concentration distributions. This study also includes the analytical expressions for electric potential, velocity, and microrotation profiles. This study discusses the impact of coupling number, couple stress parameter, electric double layer thickness, and Péclet number on solute concentration distribution. The results of fluid velocity and dispersion coefficient are validated with available works in the literature. The non-Newtonian parameter and electric double layer thickness are shown to have a significant impact on dispersion. Our study reveals that concentration distribution rises but spreading of solute reduces when the coupling number increases. This is also true when the Debye length decreases. It is also obtained that the solute spreads more in the Newtonian fluid case compared to the micropolar fluid case. Finally, coupling number and electric double layer thickness show a symmetric pattern to the indicator function for the transverse concentration variation rate. The findings of this work have broad implications in deoxyribonucleic acid analysis, chemical mixing, and separation.

Coherence of the frequency-difference autoproduct deduced from high-frequency acoustic fields scattered from a rough sea surfacea).

The prevalence of random scattering from a rough ocean surface increases with increasing χ=kh cos θ, where k is the acoustic wavenumber, h is the root-mean-square surface height, and θ is the incidence angle. Generally, when χ≫1, coherence between incident and surface-scattered fields is lost. However, such coherence may be recovered when χ≫1 by considering the frequency-difference autoproduct of the surface-scattered field, a quadratic product of complex fields at nearby frequencies. Herein, the autoproduct's coherent reflection coefficient for χ> 20 is determined from surface-scattered sound fields obtained from 50 independent realizations of the rough ocean surface measured in pelagic waters off the coast of California in January 1992. The recordings were made with a source at a depth of 147 m that broadcasted 30 and 40 kHz signals to a single receiver 576 m away at depth of 66 m. An analytic formula for the coherent reflection coefficient of the frequency-difference autoproduct, based on the Kirchhoff approximation and a Gaussian surface autocorrelation function, compares favorably with measurements. Improved agreement with the single-receiver measurements is possible via a minor adjustment to the surface autocorrelation length. The adjustment identified here matches that determined previously from horizontal spatial coherence estimates utilizing the experiment's eight-element receiving array.

Inverse Kinematics Analysis of a 2-DOF Stabilized Platform using Simulink and Simscape

Abstract This paper presents a parallel manipulator with two universal-prismatic-universal (UPU)-type active limbs and one revolute-spherical (RS)-type passive limb which is used as a stabilizing platform based on the shipboard helicopter’s landing process and its kinematics are studied systematically. First, a 3D model of this stabilized platform is constructed, and its degrees of freedom are analyzed. Second, analytic formulas for solving the inverse kinematics and calculating the required lengths of each limb are derived and modeled using Simulink®. Third, a Simscape™ model is constructed for the proposed design, and finally, the comparison of the results between the two models is done.

Analytical perturbations of relativistic images in Kerr space-time

Light rays passing very close to black holes may wind several times before escaping. For any given electromagnetic source around the black hole, a distant observer would thus observe two infinite sequences of images on either side of the black hole. These images are generated by light rays performing an increasing numbers of loops. The strong deflection limit provides a simple analytic formalism to describe such higher order images for spherically symmetric metrics, while for axially symmetric black holes one typically resorts to numerical approaches. Here we present the leading order perturbation to higher order images when the black hole spin is turned on. We show that the images slide around the black hole shadow as an effect of space-time dragging. We derive analytical formulae for their shifts and the perturbation of their time delays. We also discuss how such simple analytical formulae for images by Kerr black holes can be of great help in many applications.

An analytical method to estimate the accuracy representation index of circular and elliptical disc models for rectangular fractures with application to seepage analysis

Constructing a suitable discrete fracture network (DFN) to represent rock fractures proves vital for tackling engineering projects involving rock masses. Among the commonly used models for constructing DFN, the circular disc model (CDM) and elliptical disc model (EDM) stand out. The selection between these models has been facilitated by the introduction of the Accuracy Representation Index (ARI), which quantitatively assesses their suitability. The existing methods for calculating ARI based on Monte Carlo simulations are time-consuming. In order to quickly estimate the ARI for CDM and EDM concerning any length-width ratio (kr) of rectangular fracture, the paper derives analytical formulas based on the geometric relationships between circles and ellipses with rectangles. These formulas show that: (a) for the CDM, ARI is only related to kr, and its value decreases as kr increases; (b) for the EDM, ARI is dependent on the relationship between kr and the long-short axis length ratio (ke) of the ellipse, and it has been theoretically proven that ARI reaches its maximum value of 0.91 when kr is equal to ke. Moreover, we apply ARI to the study of rock mass seepage characteristics, and the results show that ARI could reflect the error rate of permeability of the CDM and EDM for rock masses with rectangular fractures.

Bootstrapping cascaded random matrix models: Correlations in permutations of matrix products.

Random matrix theory is a useful tool in the study of the physics of multiple scattering systems, often striking a balance between computation speed and physical rigour. Propagation of waves through thick disordered media, as arises, for example, in optical scattering or electron transport, typically necessitates cascading of multiple random matrices drawn from an underlying ensemble for thin media, greatly increasing the computational burden. Here we propose a dual pool based bootstrapping approach to speed up statistical studies of scattering in thick random media. We examine how potential matrix reuse in a pool based approach can impact statistical estimates of population averages. Specifically, we discuss how both bias and additional variance in the sample mean estimator are introduced through bootstrapping. In the diffusive scattering regime, the extra estimator variance is shown to originate from samples in which cascaded transfer matrices are permuted matrix products. Through analysis of the combinatorics and cycle structure of permutations we quantify the resulting correlations. Proofs of several analytic formulas enumerating the frequency with which correlations of different strengths occur are derived. Extension to the ballistic regime is briefly considered.

On the sound field of an oscillating elliptical disk in free space with an open and closed back.

An oscillating elliptical disk in free space has a uniform surface velocity distribution and is therefore a rigid sound radiator. The radiated sound also represents that scattered from a stationary disk in the presence of a normal plane wave. Using the Fourier (Hankel) Green's function in cylindrical coordinates, together with the monopole Kirchhoff-Helmholtz boundary integral, rigorous analytical formulas are derived for calculating the surface pressure distribution, radiation impedance, on-axis pressure, and, directivity pattern of an elliptical open-back disk in free space, and these are plotted over a range of normalized frequencies. The directivity is compared with that of a rigid circular disk in free space, and asymptotic low-frequency approximations are derived for the radiation impedance and on-axis pressure. Using the Gutin concept, the radiated sound is combined with that from a rigid elliptical disk in an infinite baffle to obtain the radiation impedance, on-axis pressure, and, directivity pattern of a closed-back elliptical disk in free space. The latter has a uniform velocity distribution on the front surface and zero velocity over the entire back surface.

Modeling the longitudinal wave in a nanorod based on a novel theory of elastic waves with surface effects

This paper investigates the impact of surface effects on the propagation behavior of longitudinal waves in a nanorod. A theoretical model has been established on the basis of a newly proposed theory of elastic waves with surface effects. The surface effects comprise two components: the effect of surface energy and the effect of surface inertia. An analytical formula for the longitudinal wave velocity of a nanorod has been derived. Two inherent lengths at nanoscale have been deduced to characterize these two types of surface effects. The results indicate that the longitudinal wave in a nanorod is still nondispersive. However, an attractive phenomenon uncovered is that when the size of a rod reduces to the inherent lengths at nanoscale, the longitudinal wave velocity becomes size-dependent due to the effects of surface energy and surface inertia. The former increases the longitudinal wave velocity, whereas the latter decreases it. This can be understood as the former equivalently increasing the stiffness of the nanorod, whereas the latter enhancing its effective density. On the other hand, when the rod is at the macroscale, the longitudinal wave velocity degenerates to the classical velocity for a macroscopic rod without any surface effects. The current findings not only enhance our understanding of the size-dependent wave velocity of longitudinal waves in nanorods but also facilitate precisely designing the elastic wave nanodevices.