General Equation Research Articles

Ultraviolet supercontinuum generation using a differentially-pumped integrated glass chip

We investigate the generation of ultrabroadband femtosecond ultraviolet (UV) radiation via third-order harmonic generation in highly confined gas media. A dual-stage differential-pumping scheme integrated into a glass microfluidic chip provides an exceptional gas confinement up to several bar and allows the apparatus to be operated under high-vacuum environment. UV pulses are generated both in argon and neon with up to ∼0.8 μJ energy and 0.2% conversion efficiency for spectra that cover the UVB and UVC regions between 200 and 325 nm. Numerical simulations based on the unidirectional pulse propagation equation reveal that ionization plays a critical role for extending the spectral bandwidth of the generated third-harmonic pulse beyond the tripled 800 nm driving laser pulse bandwidth. By delivering UV supercontinua supporting Fourier transform limits below 2 fs, as well as comparable pulse energies with respect to capillary-based techniques that typically provide high spectral tunability but produce narrower bandwidths, our compact device makes a step forward towards the production and application of sub-fs UV pulses for the investigation of electron dynamics in neutral molecules.

Modeling of fluctuations in dynamical optoelectronic device simulations within a Maxwell-density matrix Langevin approach

We present a full-wave Maxwell-density matrix simulation tool including c-number stochastic noise terms for the modeling of the spatiotemporal dynamics in active photonic devices, such as quantum cascade lasers and quantum dot structures. The coherent light–matter interaction in such devices plays an important role in the generation of frequency combs and other nonlinear and non-classical optical phenomena. Since the emergence of nonlinear and non-classical features is directly linked to the noise properties, detailed simulations of the noise characteristics are required for the development of low-noise quantum optoelectronic sources. Our semiclassical simulation framework is based on the Lindblad equation for the electron dynamics, coupled with Maxwell’s equations for optical propagation in the laser waveguide. Fluctuations arising from the interactions of the optical field and quantum system with their reservoirs are treated within the quantum Langevin theory. Here, the fluctuations are included by adding stochastic c-number terms to the Maxwell-density matrix equations. The implementation in the mbsolve dynamic simulation framework is publicly available.

An efficient solution of the multi-term multi-harmonic electron Boltzmann equation for use in global models

Solving the electron Boltzmann equation is an essential but costly step in simulating low-temperature plasma kinetics. This work addresses the problem by introducing a solution of the general multi-term multi-harmonic Boltzmann equation (MTMH-BE) optimized for electrons in time-dependent non-equilibrium gases and electric fields. This is accomplished by configuring the numerical Jacobian as the product of a time-independent sparse matrix and an array of time-dependent coefficients. As this approach requires a fixed energy grid, the MTMH-BE is discretized using a finite volume scheme with support for non-uniform cell size, enabling robust solutions across a wide range of reduced field strengths. The new MTMH-BE solver is tested against a range of benchmarks, demonstrating excellent agreement with existing methods. Performance with time-dependent electron energy is evaluated using LoKI-B's pulsed electric field model, with the new solver demonstrating improved wall time scaling and an improvement in grid convergence of over three orders of magnitude. Alternatively, performance with quasi-stationary electrons in an evolving non-equilibrium gas is tested using a nitrogen cross-section set with up to 58 vibrational states and 3,570 processes. Here, the MTMH-BE solver achieves a wall time reduction of up to 99.6% over MultiBolt for error-controlled simulations in the two-term limit, reducing the average solution time from 1-2 seconds to less than 10 ms. These results indicate that the new MTMH-BE solver can improve accuracy and performance in state-to-state global models without the use of grid refinement, cross-section pruning, and other time-saving procedures.

A new solution of the nonlinear fractional logistic differential equations utilizing efficient techniques

The formulation of models and solutions for various physical problems are the primary goals of scientific achievements in engineering and physics. Our paper focuses on using the Caputo fractional derivative operator to solve nonlinear fractional logistic differential equations. In order to solve general nonlinear fractional differential equations, we first introduce a novel numerical methodology termed the Homotopy perturbation transform method. The perturbation approach and the Yang transform method are combined to create the suggested strategy. Second, we introduce a new hybrid method that uses the time-fractional Caputo derivative to approximate and analytically solve nonlinear fractional logistic differential equations. This method combines the Yang transform with the decomposition method. To validate the analysis, we offer three numerical cases of nonlinear fractional logistic differential equations employing the Caputo fractional derivative operator. The resulting solutions exhibit rapid convergence and are presented in series form. In order to verify the efficacy and relevance of the suggested methodologies, the investigated issues were assessed through the implementation of different fractional orders. We examine and show that, under the specified initial conditions, the solution approaches under evaluation are accurate and effective. Graphs in two and three dimensions show the results that were obtained. Numerical simulations are presented to confirm the efficacy of the strategies. The numerical results show that an accurate, reliable, and efficient approximation can be obtained with a minimal number of terms. The results obtained demonstrate that the new analytical solution method is easy to apply and very successful in solving difficult fractional problems that occur in relevant engineering and scientific domains.

Non-metricity with boundary terms: 𝖿(𝖰,𝖢) gravity and cosmology

We formulate f(Q,C) gravity and cosmology. Such a construction is based on the symmetric teleparallel geometry, but apart form the non-metricity scalar Q we incorporate in the Lagrangian the boundary term C of its difference from the standard Levi-Civita Ricci scalar R̊. We extract the general metric and affine connection field equations, we apply them at a cosmological framework, and adopting three different types of symmetric teleparallel affine connections we obtain the modified Friedmann equations. As we show, we acquire an effective dark-energy sector of geometrical origin, which can lead to interesting cosmological phenomenology. Additionally, we may obtain an effective interaction between matter and dark energy. Finally, examining a specific model, we show that we can obtain the usual thermal history of the universe, with the sequence of matter and dark-energy epochs, while the effective dark-energy equation-of-state parameter can be quintessence-like, phantom-like, or cross the phantom-divide during evolution.

Insights on equidistributed real spectral values in second-order delay systems: Perspectives in partial pole placement

In this work, we show that the coexistence of the maximal number of real spectral values of generic single-delay retarded second-order differential equations guarantees the realness of the rightmost spectral value. From a control theory standpoint, this entails that a delayed proportional-derivative (PD) controller can stabilize a delayed second-order differential equation. By assigning the maximum number of negative roots to the corresponding characteristic function (a quasipolynomial), we establish the conditions for asymptotic stability. If the assigned real spectral values are uniformly distributed, we specify a necessary and sufficient condition for the rightmost root to be negative, thus guaranteeing the exponential decay rate of the system’s solutions. We illustrate the proposed design methodology in the delayed PD control of the damped harmonic oscillator. It is worth mentioning that this work represents a natural continuation of Amrane et al. (2018) and Bedouhene et al. (2020), addressing the problem of coexisting real spectral values for linear dynamical systems including delays in their models.

The importance of the inertial coupling in the two-fluid model of two-phase flow

The new flux representation of the two-fluid model of two-phase flow, where the mixture is described in terms of the volumetric and drift fluxes, is currently the most consistent formulation to treat the inertial coupling between phases. In this representation, the dynamics of the relative motion between phases is revealed as a non-linear wave propagation equation. It is shown that the character and stability of this equation is determined by the balance between the inertial coupling and the interfacial drag. A novel stability criterion is derived that can be used to assess the interfacial closure laws and as a tool to determine the conditions under which a drift-flux correlation is stable. A family of inertial coupling functions for vertical two-phase flow, based on topologies of bubble's vortical wakes, is derived and the corresponding coupling parameters are assessed using available experimental data. The resulting stability maps reveal the occurrence of an unstable region at intermediate void fractions bound by a bistable condition at low and high void fractions, which can be associated with the slug flow-pattern regime.

Deep Learning of Partially Labeled Data for Quality Prediction Based on Stacked Target-Related Laplacian Autoencoder.

Partially labeled data, which is common in industrial processes due to the low sampling rate of quality variables, remains an important challenge in soft sensor applications. In order to exploit the information from partially labeled data, a target-related Laplacian autoencoder (TLapAE) is proposed in this work. In TLapAE, a novel target-related Laplacian regularizer is developed, which aims to extract structure-preserving and quality-related features by preserving the feature-target mapping according to the local geometrical structure of the data. In addition, stacked TLapAE (STLapAE) is further constructed to extract deep feature representations of the data by hierarchically stacking TLapAE blocks. For model training, backward propagation equations are derived based on matrix calculus techniques to update the model parameters of the proposed TLapAE. The effectiveness of the proposed STLapAE is evaluated using the butane content prediction case in a debutanizer column, the silicon content prediction case in a blast furnace (BF) ironmaking process, and the ethane concentration prediction case in an ethylene fractionator. The results show that the proposed TLapAE model has significantly improved prediction accuracy compared to soft sensors using only labeled data and other partially labeled data modeling methods.

On a Carnot working continuum with non-equilibrium state parameters

We explain that a full description of how the non-equilibrium state of the system evolves in time requires the consideration and solution of its general equation of motion. In the case of the Carnot medium, as a general equation of motion, there must be taken two balances of: nonequilibrium specific volume and non-equilibrium specific entropy. Instead of taking the classical approach where the balance of entropy is postponed to more advanced and theoretical treatments, we focus on the analysis of two, most general, volume and entropy fluxes. These fluxes of motion are universal features of thermodynamics. It has been shown that the Carnot working continuum mathematical model is captured by the two general nonmathematical statements valid for all systems that we call the first law and the second law of thermodynamics.

Excitation Equations for Longitudinally-Azimutally Irregular Waveguides Taking into Account the Finite of the Walls Conductivity

Excitation equations for longitudinal-azimuthally irregular waveguides are formulated taking into account losses in the walls. The inner surface of the waveguide walls is given by an arbitrary smooth function b(ϕ, z). The coordinate transformation method replaces the original cylindrical coordinate system r, ϕ, z with a new one ρ, ϕ, z, where ρ = r/(b(ϕ, z)). The new system defines the waveguide boundary as ρ = 1 = const, i. e. the waveguide geometry transforms as a regular cylinder. Taking these functions into account, the standard procedure of the incomplete Galerkin method is used to determine the amplitudes of partial waves. The resulting general equations can be used in the calculation and optimization of both microwave and EHF electronic devices of various types, as well as passive microwave devices of various applications.

Comparison of subcritical growth parameters of a Y-TZP obtained via cyclic or dynamic fatigue tests

ObjectiveThe objective of this study was to 1) compare the stress corrosion coefficient (n) of a Y-TZP obtained by two fatigue tests: cyclic and dynamic and 2) evaluate the effect of frequency in the characteristic lifetime and the existence of interaction between the cyclic fatigue and slow crack growth. MethodsA total of 145 Y-TZP specimens were produced in accordance with the manufacturer's instructions. These specimens, measuring 4.0 × 3.0 × 25.0 mm, were used for dynamic (n = 70) and cyclic fatigue tests (n = 75). The specimens were obtained from CAD/CAM blocks, sectioned, and sintered in a furnace at 1530 °C with a heating rate of 25 °C/min. They were tested in their "as-sintered" form without any additional surface treatment. The fatigue tests were conducted using a four-point bending to obtain the slow crack growth parameters (n). The cyclic fatigue test was also conducted in two frequencies (2 and 10 Hz), using stress levels between 350 and 600 MPa. Data from these tests were analyzed using ASTM C 1368–00 formulas and Weibull statistics. Scanning electron microscope (SEM) was used for fracture surface analysis to identify the origin of the fracture. Critical defect size was measured and used, along with flexural strength values, to estimate fracture toughness. Dynamic fatigue test data were used to obtain subcritical crack growth (SCG) parameters and perform Weibull statistical analysis. The cyclic fatigue data were used in the General Log-linear Model equation using the ALTA PRO software. Data were analyzed using one-way ANOVA followed by Tukey post-hoc tests and Student's t-test at a significance level of p ≤ 0.05. ResultsIn the dynamic fatigue test, the values obtained for σfo and n were 667 and 54, respectively. This parameter indicates how the strength of the material diminishes over time due to internal cracks. The Weibull parameters obtained from the same test results were m = 7.9, σ0 = 968, 9 and σ5% = 767, which indicates the reliability of the material. The Weibull parameters obtained by cyclic fatigue were statistically similar for the two frequencies used, the m* was 0.17 (2 Hz) and 0.21 (10 Hz); characteristic lifetimes (η) were 1.93 × 106 and 40,768, respectively. The n values obtained by cyclic fatigue were 48 and 40 at frequencies of 2 and 10 Hz, respectively. There was no effect of the frequency, the stress level or the interaction of the two in the Y-TZP lifetime, when analysed by General Log Linear Model. Significancethe n values obtained by cyclic and dynamic fatigue tests showed no statistically significant difference and the effect of frequency in the characteristic lifetime and the existence of interaction between the cyclic fatigue and subcritical growth were not observed in the tested specimens.

Single-stage approach for estimating optical parameters in spectral quantitative photoacoustictomography.

In quantitative photoacoustic tomography, the optical parameters of a target, most importantly the concentrations of chromophores such as deoxygenated and oxygenated hemoglobin, are estimated from photoacoustic data measured on the boundary of the target. In this work, a numerical approximation of a forward model for spectral quantitative photoacoustic tomography is constructed by utilizing the diffusion approximation for light propagation, the acoustic wave equation for ultrasound propagation, and spectral models of optical absorption and scattering to describe the wavelength dependence of the optical parameters. The related inverse problem is approached in the framework of Bayesian inverse problems. Concentrations of four chromophores (deoxygenated and oxygenated hemoglobin, water, and lipid), two scattering parameters (reference scattering and scattering power), and the Grüneisen parameter are estimated in a single-stage from photoacoustic data. The methodology is evaluated using numerical simulations in different full-view and limited-view imaging settings. The results show that, utilizing spectral data and models, the spectral optical parameters and the Grüneisen parameter can be simultaneously estimated. Furthermore, the approach can also be utilized in limited-view imaging situations.

Deducing solute differential heat capacity from experimental solubilities. An exemplified treatment of ascorbic acid to improve solubility prediction

ContextSolubility prediction based on the general solubility equation (GSE) rests on reliable values for the isobaric heat capacity difference ΔCp,1 of the solid solute. Usually, this value is estimated with either zero or the melting entropy ΔS1(Tm,1) or, in few cases, is extrapolated from data of thermally stable melts of the solute. This causes uncertainties in the prediction. ObjectiveTo improve prediction accuracy a simple regression method is proposed that determines ΔCp,1 from measured solubilities. Materials and methodsPublished experimental solubilities in neat organic solvents at 298 K of a model compound (L-(+)-ascorbic acid (LAA)) have been regressed using the GSE together with the Hansen parameter model for the activity coefficient. Results and discussionRegression yielded ΔCp,1 = 238 J∙mol−1∙K−1 which agrees well with cross-validation results and is consistent with estimates from various group contribution methods. It was found that prediction accuracy improved in the order of increasing ΔCp,1, that is, from 0, via 91 (=ΔS1(Tm,1)) to 238 J∙mol−1∙K−1. It could be shown that mole fraction solubility of LAA can be forecast this way with an accuracy within current inter-laboratory variation. ConclusionThe proposed method shows a general way to improve prediction accuracy of activity coefficient based solubility models by determining ΔCp,1 without resorting to common assumptions. The method is universally applicable and easy to implement.

HHL algorithm with mapping function and enhanced sampling for model predictive control in microgrids

This paper presents a refined quantum Harrow Hassidim Lloyd (HHL) algorithm for microgrid control. The first novelty of the developed method is that a mapping shift function enables the original HHL algorithm to handle general linear equations with non-singular and indefinite matrix. Second, a method of Matrix Extension for Amplifying Sampling Probabilities of Intended Solution (ME-ASPI) is proposed to design the reformulated linear algebraic equations, allowing for improved sampling efficiency of the quantum tomography in the refined HHL algorithm. Then, we applied the method to solve the model predictive control (MPC) problem in nonlinear dynamical microgrids. Specifically, with the ME-ASPI method, the refined HHL algorithm can effectively obtain the intended partial optimal control inputs for MPC. The optimization of quadratic programming problem in each time step of MPC is transformed into a linear system problem, which is addressed by the proposed quantum solver through using only partial information, with the time complexity improved from O(N2.37286) classically to O(N2logN×plogp) in quantum. Numerical examples have validated the effectiveness of the refined HHL algorithm with the proposed mapping function and the ME-ASPI method. By leveraging quantum properties, the proposed method provides a hybrid quantum–classical framework for microgrid control. This generic method can also potentially tackle many other challenges in analyzing and controlling general complex engineered systems.

A General Equation of State for a Quantum Simulator

A General Equation of State for a Quantum Simulator

A Technique for Efficient Estimation of Dynamic Structural Equation Models: A Case Study

Dynamic structural equation models (DSEM) are designed for time series analysis of latent structures. Inherent to the application of DSEM is model parameter estimation, which has to be addressed in many applications by a single time series. In this context, however, the methods currently available either lack estimation quality or are computationally inefficient. Given the era of big data, the necessity for a trade-off between these properties may be detrimental to the applicability of DSEM. The paper is aimed at tackling this trade-off by proposing a novel estimator recursioning technique (ER technique) that facilitates the development of computationally efficient raw-data maximum likelihood estimation algorithms through data transformation, covariance matrix block decomposition, likelihood function reduction, and estimator recursioning steps. The ER technique is introduced by applying it to a special case of the general dynamic structural equation model that encompasses a noisy Wiener-process-type structure with input from the factor-analytic model. The resulting algorithm has been verified through a number of numerical experiments as well as implemented in a brand new R package EMLI, which is available on CRAN.

Maximizing Four-Wave Mixing in Four-Subband Semiconductor Quantum Wells with Optimal-Shortcut Spatially Varying Control Fields

In the present article, we derive optimal spatially varying control fields, which maximize the four-wave mixing efficiency in a four-subband semiconductor asymmetric double quantum well, following analogous works in atomic systems. The control fields coherently prepare the medium, where a weak probe pulse is propagated and eventually converted to a signal pulse at the output. The optimal fields, which maximize the conversion efficiency for a given propagation length, are obtained by applying optimal control theory to a simplified form of propagation equations but are tested with numerical simulations using the full set of Maxwell–Schrödinger equations, which accurately describe the propagation of light pulses in the medium. For short propagation distances, the proposed optimal scheme outperforms a simpler spatially changing control protocol that we recently studied, while for larger distances, the efficiency of both protocols approaches unity. The present work is expected to find application in frequency conversion between light beams, conversion between light beams carrying orbital angular momentum, and nonlinear optical amplification.

Comparison of the symmetric hyperbolic thermodynamically compatible framework with Hamiltonian mechanics of binary mixtures

How to properly describe continuum thermodynamics of binary mixtures where each constituent has its own momentum? The Symmetric Hyperbolic Thermodynamically Consistent (SHTC) framework and Hamiltonian mechanics in the form of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) provide two answers, which are similar but not identical, and are compared in this article. They are compared both analytically and numerically on several levels of description, varying in the amount of detail. Namely, a reduction to a more common one-momentum setting is shown, where the effects of the second momentum translate into diffusive fluxes. Both SHTC and GENERIC can thus be interpreted as a method specifying diffusive flux in standard theory. The GENERIC equations, stemming from the Liouville equation, contain terms expressing self-advection of the relative velocity by itself, which lead to a vorticity-dependent diffusion matrix after the reduction. The SHTC equations, on the other hand, do not contain such terms. We also discuss the possibility to formulate a theory of mixtures with two momenta and only one temperature that is compatible with the Liouville equation and possesses the Hamiltonian structure, including Jacobi identity.

Thermal management of square light emitting diode arrays: modeling and parametric analysis

PurposeThis study investigates the impact of three parameters such as: number of LED chips, pitch and LED power on the junction temperature of LEDs using a best heat sink configuration selected according to a lower temperature. This study provides valuable insights into how to design LED arrays with lower junction temperatures.Design/methodology/approachTo determine the best configuration of a heat sink, a numerical study was conducted in Comsol Multiphysics on 10 different configurations. The configuration with the lowest junction temperature was selected for further analysis. The number of LED chips, pitch and LED power were then varied to determine the optimal configuration for this heat sink. A general equation for the average LED temperature as a function of these three factors was derived using Minitab software.FindingsAmong 10 configurations of the rectangular heat sink, we deduce that the best configuration corresponds to the first design having 1 mm of width, 0.5 mm of height and 45 mm of length. The average temperature for this design is 50.5 C. For the power of LED equal to 50 W–200 W, the average temperature of this LED drops when the number of LED chips reduces and the pitch size decreases. Indeed, the best array-LED corresponds to 64 LED chips and a pitch size of 0.5 mm. In addition, a generalization equation for average temperature is determined as a function of the number of LED chips, pitch and power of LED which are key factors for reducing the Junction temperature.Originality/valueThe study is original in its focus on three factors that have not been studied together in previous research. A numerical simulation method is used to investigate the impact of the three factors, which is more accurate and reliable than experimental methods. The study considers a wide range of values for the three factors, which allows for a more comprehensive understanding of their impact. It derives a general equation for the average temperature of the LED, which can be used to design LED arrays with desired junction temperatures.

On fully nonlinear Loewner-Nirenberg problem of Ricci curvature

We prove the existence of a smooth complete conformal metric with prescribed kth elementary symmetric function of negative Ricci curvature under certain condition on general domain in Euclidean space. We then formulate this problem for more general equations.