Approximate Analytical Expressions Research Articles

Forced Response of a High-Static-Low-Dynamic (HSLD) Stiffness Isolator with Active Stiffness Control

The High-Static-Low-Dynamic (HSLD) stiffness vibration isolators have been exploited in many engineering applications due to its capability in having a wider isolation bandwidth, while maintaining the high static load capacities. However, it will lead to a large payload oscillation at the static equilibrium position, if the source of vibration is an oscillating force originating within the payload. In this case, the considerably large resultant motion of the payload will change the system nonlinearity. An active stiffness control for reducing the displacement amplitude of the payload oscillation subjected to a harmonic force excitation is proposed in this paper. The dynamic model of an actively stiffened HSLD stiffness isolator is introduced, and the approximate analytical expression for forced response is obtained using the Harmonic Balance Method (HBM). The obtained forced response curve has demonstrated that the active stiffness control is able to reduce the system’s force response, particularly at low frequencies with an approximation of 50%. The nonlinearity of the system becomes smaller as the active stiffness control is applied.

Note on warped compactification. Finite brane potentials and non-Hermiticity

We study radius stabilization in the Randall-Sundrum model without assuming any unnaturally large stabilizing scalar potential parameter at the boundary branes (γ) by the frequently used superpotential method. Employing a perturbative expansion in 1/γ2 and the backreaction parameter, we obtain approximate analytical expressions for the radion mass and wavefunction. We validate them through a dedicated numerical analysis, which solves the linearized coupled scalar and metric field equations exactly. It is observed that the radion mass decreases with decreasing γ. Below a critical value of γ, the radion becomes tachyonic, suggesting destabilization of the extra dimension. We also address the issue of non-Hermiticity of the differential operator that determines the radion and Kaluza-Klein (KK) mode wavefunctions in the finite γ limit. It is accomplished by finding an explicit form of the general scalar product that re-establishes the orthogonality in the KK decomposition.

Remaining useful life prediction based on multi-stage Wiener process and Bayesian information criterion

Equipment will go through multiple degradation stages under complex operating conditions, and the single-stage degradation model cannot accurately describe the degradation process of the equipment at different stages, resulting in inaccurate remaining service life prediction results and reliability analysis. Therefore, this paper establishes a multi-stage Wiener degradation process model that considers measurement errors and includes three different forms of drift functions. First, by calculating the Bayesian information criterion (BIC) values of these three degradation models separately and analysing the variation trends of the BIC values, a method for detecting change-points is proposed to achieve stage division. Next, by comparing the BIC values of the three models, a method for adaptively selecting the optimal model for each stage is proposed. Then, based on the results of stage division and optimal model selection, approximate analytical expressions for the RUL of each stage are derived, and parameter estimation is performed using maximum likelihood estimation (MLE). Finally, the RUL prediction study using the proposed method is carried out through simulation cases and practical cases. The results show that the accuracy of the proposed method is higher than the existing research methods, verifying the effectiveness of the proposed method.

A density of states-based approach to determine temperature-dependent aggregation rates.

Here, we establish an approach to determine temperature-dependent aggregation rates in terms of thermostatistical quantities, which can be obtained directly from flat-histogram and statistical temperature algorithms considering the density of states of the system. Our approach is validated through simulations of an Ising-like model with anisotropically interacting particles at temperatures close to its first-order phase transition. Quantitative comparisons between the numerically obtained forward and reverse rates to approximate analytical expressions corroborate its use as a model-independent approach.

Open-loop quantum control of small-size networks for high-order cumulants and cross-correlations sensing

Quantum control techniques are one of the most efficient tools for attaining high-fidelity quantum operations and a convenient approach for quantum sensing and quantum noise spectroscopy. In this work, we investigate dynamical decoupling while processing an entangling two-qubit gate based on an Ising-xx interaction, each qubit affected by pure dephasing classical correlated 1/f-noises. To evaluate the gate error, we used the Magnus expansion introducing generalized filter functions that describe decoupling while processing and allow us to derive an approximate analytic expression as a hierarchy of nested integrals of noise cumulants. The error is separated in contributions of Gaussian and non-Gaussian noise, with the corresponding generalized filter functions calculated up to the fourth order. By exploiting the properties of selected pulse sequences, we show that it is possible to extract the second-order statistics (spectrum and cross-spectrum) and to highlight non-Gaussian features contained in the fourth-order cumulant. We discuss the applicability of these results to state-of-the-art small networks based on solid-state platforms.

Collective photon emission in solid state environments: Concatenating non-Markovian and Markovian dynamics

Collective light emission and multiqubit dynamics of solid-state quantum emitters are affected both by their coupling to the light field and to lattice vibrations. The effect of phonons on quantum emitters is twofold: polaron formation is described by ultrafast non-Markovian dynamics, while slower dephasing is well described by exponential decay. Of the two temperature-dependent processes, the effect of the former on the collective emission and the entanglement decay of emitters is usually not modeled, and also the latter is sometimes neglected. Here we propose and compare two methods that are efficient also for several emitters: the first method concatenates the fast and slow phonon dynamics, and the second is the polaron method. For a single quantum emitter, we show that the dynamical equations are identical in both methods, while predictions for two or more emitters also agree very well. Both of our methods incorporate non-Markovian dynamics due to phonons demonstrating the temperature sensitivity of the collective photon emission. Utilizing a simplified Markovian model instead may not be accurate enough especially for quantum information applications: for example, we show how the Markovian model may considerably overestimate the two-emitter concurrence, except at very low temperatures. Our concatenation and polaron methods can be applied to an arbitrary number and type of quantum emitters, and beyond the bulk GaAs environment that we consider here. Especially the concatenation method can take phonon effects into account at the same computational cost as modeling the emitter-photon interaction alone. Finally, we present approximate analytical expressions for the collective emission spectrum for N emitters on a one-dimensional chain. Published by the American Physical Society 2024

Mathematical analysis of batch reactor performance for the enzymatic synthesis of cephalexin: Laplace Homotopy perturbation method and Adomian decomposition method

In this article, a mathematical model of diffusion reaction in kinetically controlled cephalexin synthesis in the batch reactor with penicillin acylase immobilized in glyoxyl-agarose is analyzed. The kinetic model is a non-linear non-stead-state reaction–diffusion equation with non-linear terms related to the Fick’s law. We have presented the approximate analytical expression for the non-steady-state non-linear reaction–diffusion equation by utilizing the Laplace homotopy perturbation method (LHPM) and for steady-state by using LHPM and Adomian decomposition method (ADM). The approximate analytical solution obtained from these methods proved that they are fit for every values of the reaction–diffusion and kinetic parameters. We also present the numerical solution of the considered reaction–diffusion equation by using pdepe tool in MATLAB software. When comparing the semi-analytical solution with the numerical solution, a satisfactory result is noted for all the possible values of the parameters. The closed-form analytical expressions for the effectiveness factor in non-steady and steady-state conditions are obtained and analyzed using LHPM and ADM.

Mathematical analysis of non-linear boundary-value problems in reaction-diffusion model of chitosan-alginate microsphere using homotopy perturbation and Akbari-Ganji methods

This article describes the non-linear reaction-diffusion model, which depicts the behaviour of hydrogen peroxide production and glucose oxidation in the chitosan-alginate microsphere. Analytical solutions of glucose, oxygen, gluconic acid and hydrogen peroxide concentrations in planar coordinates under steady-state circumstances are obtained for all reaction-diffusion parameters. The non-linear boundary value problem's approximate analytical expressions are obtained using asymptotic techniques based on the homotopy perturbation and Akbari-Ganji method. The exact and commonly used MATLAB program provided a numerical simulation. It is demonstrated that the resulting analytical expressions strongly agree with the numerical outcomes reported in the literature. Furthermore, determining hydrogen peroxide flux and pH profiles within the microspheres are also discussed. The theoretical findings make it possible to forecast and enhance the efficiency of enzyme kinetics.

A Theoretical Approach to Understand the Nonlinear Processes in Enzymatic Electrochemical Biosensors.

The enzymatic biosensors' response can be monitored based on the results of nonlinear differential equations. The nonlinear reaction-diffusion equations proposed for this enzyme-based electrochemical biosensor include a nonlinear term associated with Michaelis-Menten kinetics. Herein, the system of nonlinear reaction-diffusion equations is solved using a modified homotopy perturbation method. For all values of the rate constants, the approximate analytical expressions for the concentration profiles, current, sensitivity, and gradient of biosensor have been determined. Performance factors of an enzymatic electrochemical biosensor, such as response time, sensitivity, accuracy, and resistance, are discussed. The analytical results and numerically simulated outcomes using Matlab software have been compared.

Efficient path integral approach via analytical asymptotic expansion for nonlinear systems under Gaussian white noise

In this paper an efficient formulation of the Path integral (PI) approach is developed for determining the response probability density functions (PDFs) and first-passage statistics of nonlinear oscillators subject to stationary and time-modulated external Gaussian white noise excitations. Specifically, the evolution of the response PDF is obtained in short time steps, by using a discrete version of the Chapman-Kolmogorov equation and assuming a Gaussian form for the conditional response PDF. Next, the technique involves proceeding to treating the problem via an analytical asymptotic expansion procedure, namely the Laplace’s method of integration. In this manner, the repetitive double integrals involved in the standard implementation of the PI approach are evaluated in a closed form, while the response and first-passage PDFs are obtained by mundane step-by-step application of the derived approximate analytical expression. It is shown that the herein proposed formulation can drastically decrease the associated computational cost by several orders of magnitude, as compared to both the standard PI technique and Monte Carlo solution (MCS) approach. A number of nonlinear oscillators are considered in the numerical examples. Notably, for these systems both response PDFs and first-passage probabilities are presented, whereas comparisons with pertinent MCS data demonstrate the efficiency and accuracy of the technique.

Mathematical Modelling and Application of Analytical Methods for A Non-Linear EC2E Mechanism in Rotating Disk Electrode

Rotating disk electrode is the hydrodynamic technique used in the process of analyzing electroanalytic works. This paper deals with the mathematical model describing a non-linear EC2E mechanism that arises in a rotating disk electrode. This model is based on the system of non-linear reaction-convection-diffusion equations. EC2E mechanism has an application in finding the shape of current curves in the system of chronoamperometry in addition to steady-state voltammetry. The approximate analytical expression for the concentration of reactant species and current at steady-state condition is obtained using the analytical methods. The attained analytical result is compared with the numerical simulation (MATLAB) result. A satisfactory result is obtained between the series of solutions. The influence of parameters on the concentration of species and current is investigated and presented graphically.

Data-driven reconstruction of chaotic dynamical equations: The Hénon–Heiles type system

In this study, the classical two-dimensional potential VN=12mω2r2+1NrNsin(Nθ), N∈Z+, is considered. At N=1,2, the system is superintegrable and integrable, respectively, whereas for N>2 it exhibits a richer chaotic dynamics. For instance, at N=3 it coincides with the Hénon–Heiles system. The periodic, quasi-periodic and chaotic motions are systematically characterized employing time series, Poincaré sections, symmetry lines and the largest Lyapunov exponent as a function of the energy E and the parameter N. Concrete results for the lowest cases N=3,4 are presented in complete detail. This model is used as a benchmark system to estimate the accuracy of the Sparse Identification of Nonlinear Dynamical Systems (SINDy) method, a data-driven algorithm which reconstructs the underlying governing dynamical equations. We pay special attention at the transition from regular motion to chaos and how this influences the precision of the algorithm. In particular, it is shown that SINDy is a robust and stable tool possessing the ability to generate non-trivial approximate analytical expressions for periodic trajectories as well.

Interactions between different birds of prey as a random point process

The two-dimensional (2D) Coulomb gas is a one-parameter family of random point processes, depending on the inverse temperature β. Based on previous work, it is proposed as a simple statistical measure to quantify the intra- and interspecies repulsion among three different highly territorial birds of prey. Using data from the area of the Teutoburger Wald over 20 years, we fit the nearest-neighbour and next-to-nearest neighbour spacing distributions between the respective nests of the goshawk, eagle owl and the previously examined common buzzard to β of the Coulomb gas. Within each species, the repulsion measured in this way deviates significantly from the Poisson process of independent points in the plane. In contrast, the repulsion amongst each of two species is found to be considerably lower and closer to Poisson. Methodologically, we investigate the influence of the terrain, of a shorter interaction range given by the 2D Yukawa interaction, and the statistical independence of the time moving average we use for the yearly ensembles of occupied nests. We also check that an artificial random displacement of the original nest positions of the order of the mean level spacing quickly destroys the repulsion measured by β > 0. A simple, approximate analytical expression for the nearest-neighbour spacing distribution derived from non-Hermitian random matrix theory proves to be very useful.

Analysis of a detailed multi-stage model of stochastic gene expression using queueing theory and model reduction

We introduce a biologically detailed, stochastic model of gene expression describing the multiple rate-limiting steps of transcription, nuclear pre-mRNA processing, nuclear mRNA export, cytoplasmic mRNA degradation and translation of mRNA into protein. The processes in sub-cellular compartments are described by an arbitrary number of processing stages, thus accounting for a significantly finer molecular description of gene expression than conventional models such as the telegraph, two-stage and three-stage models of gene expression. We use two distinct tools, queueing theory and model reduction using the slow-scale linear-noise approximation, to derive exact or approximate analytic expressions for the moments or distributions of nuclear mRNA, cytoplasmic mRNA and protein fluctuations, as well as lower bounds for their Fano factors in steady-state conditions. We use these to study the phase diagram of the stochastic model; in particular we derive parametric conditions determining three types of transitions in the properties of mRNA fluctuations: from sub-Poissonian to super-Poissonian noise, from high noise in the nucleus to high noise in the cytoplasm, and from a monotonic increase to a monotonic decrease of the Fano factor with the number of processing stages. In contrast, protein fluctuations are always super-Poissonian and show weak dependence on the number of mRNA processing stages. Our results delineate the region of parameter space where conventional models give qualitatively incorrect results and provide insight into how the number of processing stages, e.g. the number of rate-limiting steps in initiation, splicing and mRNA degradation, shape stochastic gene expression by modulation of molecular memory.

Analytical Confirmation of the Calculated Dependencies for Heat Exchange in a Plate Recuperator when Humidifying the Auxiliary Flow

The relevance of the study is related to the need to ensure maximum reduction of energy consumption while ensuring the calculated parameters of the indoor climate in buildings under the Law of the Russian Federation “On Energy Saving...” and the updated version of SP 131. The purpose of the study is to obtain an approximate analytical expression of this dependence, which makes it possible to additionally confirm the results of field and numerical experiments previously performed by the authors for this flow treatment scheme. The objective of the study is to build a simplified mathematical model of the processes of changing the heat and humidity state of the air in the recuperator, identify the main factors affecting the increasing multiplier to the coefficient of temperature efficiency of the device and obtain the necessary numerical coefficients in formulas linking the desired and initial parameters. The general equation of thermal balance and heat transfer for the heat exchanger as a whole is used, including the integral characteristics of the apparatus in a dimensionless form, as well as standard techniques of algebraic transformations. An approximate analytical expression is obtained for an increasing multiplier to the coefficient of thermal efficiency of the recuperator taking into account the additional cooling of the inflow due to the heat consumption for evaporation. It is shown that the general structure of this ratio coincides with the one found by the authors earlier by processing the results of numerical modeling of a two-dimensional temperature field in a heat exchanger, taking into account experimental data on the amount of moisture carried away, which confirms their correctness and reliability.

Impact of strain on the SOT-driven dynamics of thin film Mn3Sn

Mn 3 Sn, a metallic antiferromagnet with an anti-chiral 120° spin structure, generates intriguing magneto-transport signatures such as a large anomalous Hall effect, spin-polarized current with novel symmetries, anomalous Nernst effect, and magneto-optic Kerr effect. When grown epitaxially as MgO(110)[001]∥Mn3Sn(01¯1¯0)[0001], Mn3Sn experiences a uniaxial tensile strain, which changes the bulk sixfold anisotropy to a twofold perpendicular magnetic anisotropy (PMA). Here, we investigate the field-assisted spin–orbit-torque (SOT)-driven dynamics in single-domain Mn3Sn with PMA. We find that for non-zero external magnetic fields, the magnetic octupole moment of Mn3Sn can be switched between the two stable states if the input current is between two field-dependent critical currents. Below the lower critical current, the magnetic octupole moment exhibits a stationary state in the vicinity of the initial stable state. On the other hand, above the higher critical current, the magnetic octupole moment shows oscillatory dynamics which could, in principle, be tuned from the 100s of megahertz to the terahertz range. We obtain approximate analytic expressions of the two critical currents that agree very well with the numerical simulations for experimentally relevant magnetic fields. We also obtain a unified functional form of the switching time vs the input current for different magnetic fields. Finally, we show that for lower values of Gilbert damping (α≲2×10−3), the critical currents and the final steady states depend significantly on α. The numerical and analytic results presented in our work can be used by both theorists and experimentalists to understand the SOT-driven order dynamics in PMA Mn3Sn and design future experiments and devices.

New results on the dynamics of critical collapse* *JQG is Supported by the Natural Science Foundation of Shandong Province, China (ZR2019MA068). YH, PPW and CGS are Supported by the National Natural Science Foundation of China (11925503)

We study the dynamics of the critical collapse of a spherically symmetric scalar field. Approximate analytic expressions for the metric functions and matter field in the large-radius region are obtained. In the central region, owing to the boundary conditions, the equation of motion for the scalar field is reduced to the flat-spacetime form.

Stochastic modeling of integrated order fulfillment processes with delivery time promise: Order picking, batching, and last-mile delivery

To guarantee high customer service and short and accurate lead times, many e-commerce retailers have started to home deliver their customer orders within a few hours or even minutes, also known as quick-commerce order fulfillment. Quick-commerce order fulfillment consists of three main processes: order picking in the warehouse, order batching for delivery, and last-mile delivery. The ultimate delivery performance depends on managing all three processes, which are highly stochastic, and interdependent. We capture this stochasticity and interdependency in an integrated analytical framework and derive approximate analytical expressions for the mean and variance of the total order fulfillment time. We validate the analytical expressions with both in-house detailed process simulations and external-party output measures. We then analyze the delivery cost-service quality trade-offs using an optimization model that minimizes the expected order fulfillment cost with a delivery probability (DP) constraint, focusing on meeting delivery time deadlines. The optimization model determines the number of pickers, the optimal delivery batch size, and the number of vehicles required to deliver the customer orders. Achieving a high delivery reliability comes at a cost. In comparison to the model with DP constraints, we observe that the expected order fulfillment cost averaged over all data parameter settings obtained from the model without DP constraints is 8.9% lower; however, the mean and standard deviation of order fulfillment time increase by 44.1% and 18.6%, respectively, which results in low delivery reliability. We further demonstrate that an integrated analysis of the order fulfillment process is essential to set reliable fulfillment due times.

Dimensionality effects on trap-assisted recombination: the Sommerfeld parameter

In the context of condensed matter physics, the Sommerfeld parameter describes the enhancement or suppression of free-carrier charge density in the vicinity of a charged center. The Sommerfeld parameter is known for three-dimensional systems and is integral to the description of trap-assisted recombination in solids. Here we derive the Sommerfeld parameter in one and two dimensions and compare with the results in three dimensions. We provide an approximate analytical expression for the Sommerfeld parameter in two dimensions. Our results indicate that the effect of the Sommerfeld parameter is to suppress trap-assisted recombination in decreased dimensionality.

A Kinetic Non-Steady-State Analysis of Immobilized Enzyme Systems Without External Mass Transfer Resistance

In this paper, a non-steady-state non-linear reaction diffusion in immobilized enzyme on the nonporous medium is considered for its mathematical analysis. The non-linear terms in this model are related to the Michaelis-Menten kinetics. For the considered model, the approximate analytical expressions of the substrate concentration and the effectiveness factor for the various geometric profiles of immobilized enzyme pellets are obtained using homotopy perturbation method (HPM). The obtained approximate analytical expressions proved to be fit for all values of parameters. Numerical solutions are also provided using the MATLAB software. When comparing the analytical and the numerical solutions, satisfactory results are noted. The effects of Thiele modulus and Michaelis-Menten kinetic constants on the effectiveness factor are also analyzed.