Approximation Theory Research Articles

On determinant representations of Hermite–Padé polynomials

In this work we introduce new concepts: weakly normal index, weakly perfect system of functions. With these concepts for an arbitrary system of power series we formulate and prove criteria for the uniqueness of solutions to two Hermite–Padé problems, and obtain explicit determinant representations of Hermite–Padé types 1 and 2 polynomials. Proven statements complement well-known results in Hermite–Padé approximation theory.

Irreversible Boltzmann samplers in dense liquids: Weak-coupling approximation and mode-coupling theory.

Exerting a nonequilibrium drive on an otherwise equilibrium Langevin process brings the dynamics out of equilibrium but can also speed up the approach to the Boltzmann steady state. Transverse forces are a minimal framework to achieve dynamical acceleration of the Boltzmann sampling. We consider a simple liquid in three space dimensions subjected to additional transverse pairwise forces, and quantify the extent to which transverse forces accelerate the dynamics. We first explore the dynamics of a tracer in a weak coupling regime describing high temperatures. The resulting acceleration is correlated with a monotonous increase of the magnitude of odd transport coefficients (mobility and diffusivity) with the amplitude of the transverse drive. We then develop a nonequilibrium version of the mode-coupling theory able to capture the effect of transverse forces, and more generally of forces created by additional degrees of freedom. Based on an analysis of transport coefficients, both odd and longitudinal, both for the collective modes and for a tracer particle, we find a systematic acceleration of the dynamics. Quantitatively, the gain, which is guaranteed throughout the ergodic phase, turns out to be a decreasing function of temperature beyond a temperature crossover, in particular as the glass transition is approached. Our theoretical results are in good agreement with available numerical results.

Insight on thermodynamic and thermoelectric properties of Ge based perovskites AGeO3 (A = Mg, Cd) for energy harvesting applications: A DFT approach

In this manuscript, transport properties like thermoelectric and thermodynamic properties of AGeO3 (A = Mg, Cd) perovskites have been performed in detail along with effect of chemical potential on various thermoelectric parameters. Furthermore, we present the electronic effects and thermodynamic stability of these materials. The MgGeO3 and CdGeO3 alloys have a gap of 3.368 and 2.555 eV, respectively, making them indirect band gap semiconductors. Furthermore, using a constant relaxation time approximation and semiclassical Boltzmann theory, the thermoelectric (TE) properties of both materials have been studied. The lattice thermal conductivity magnitudes for MGeO3 and CdGeO3 are 7.47 W/mK and 21.55 W/mK, respectively, at 300 K. Power Factor (PF) has been measured at approximately 0 chemical potential, or approximately 14 (1011 W/K2 ms) at 1200 K, 7.5 (1011 W/K2 ms) at 700 K, and 3.5 (1011 W/K2 ms) at 300 K for both MgGeO3 and CdGeO3. At room temperature, the electrical conductivities of MgGeO3 and CdGeO3 were measured to be 1.43 × 1015W/mK and 1.08 × 1015W/mK, respectively. Over a broad temperature range (0–1200 K), the thermodynamic parameters, such as heat capacity, entropy, thermal expansion, and Debye temperature, were also computed and discussed.

Approximating Choquet integral in generalized measure theory: Choquet-midpoint rule

In generalized measure theory, Choquet integral is a generalization of Lebesgue integral and mathematical expectation. Approximating Choquet integral in the continuous case on real line is not very easy. So, we need mainly to estimate Choquet integral with respect to non-additive measures. There are few studies on the approximating Choquet integral in the continuous case on real line. In approximation theory, there are many interesting properties of midpoint rule. As a subject for research, there are no results on the midpoint rule for Choquet integral. The main objective of this paper is to propose some applications of midpoint rule for approximating continuous Choquet integral. The choquet-midpoint rule helps us to numerically solve Choquet integrals, in particular, the singular and unbounded integrals. Several numerical examples are considered to illustrate the application of our proposed methodology.

Time–energy distribution of photoelectron from atomic states with different magnetic quantum numbers in elliptically polarized laser fields

Abstract A Wigner-distribution-like (WDL) function based on the strong-field approximation (SFA) theory is used to investigate the ionization time of the photoelectron emitted from the initial states with different magnetic quantum number m in elliptically polarized electric fields. The saddle-point method is adopted for comparisons. For different m states, a discrepancy exists in the WDL distributions of the photoelectrons emitted in a direction close to the major axis of the laser field ellipse. Based on the saddle-point analysis, this discrepancy can be ascribed to the interference between electrons ionized from two tunneling instants. Our results show that the relationships between the tunneling instants and kinetic energy of photoelectrons are the same for different m initial states when the Coulomb potential is not considered. Our work sheds some light on the ionization-time information of electrons from different magnetic quantum states.

Variable viscosity characteristics of two-layered isothermal calendering phenomena with non-Newtonian fluids

Two-layer calendering using non-Newtonian fluids is used frequently in different industries to improve the working and the quality of the products. In this article, two-layer calendering of incompressible non-Newtonian fluids, that is, couple stress fluid and viscoplastic fluid in the upper and lower layer respectively is investigated. For the simplified governing equations, Lubrication approximation theory is applied. By the use of no-slip condition and non-dimensional variables, the exact solution for pressure gradient, pressure, and velocity profiles for both layers of fluid is calculated. The parameters are important from an engineering point of view like roll separating force, power delivered by rollers, torque acting by each roller, and adiabatic temperature are also calculated analytically. Additionally, the results of a couple stress parameter, viscoplastic Casson parameter, and viscosity ratios on all the parameters are also discussed and investigated in detail. Moreover, all the effects investigated are also verified with the literature results of single-layer calendering of non-Newtonian fluids by using large numeric values for the couple stress parameter and viscoplastic Casson parameter.

First-principles study of the C2H4 adsorption on the small Ag-Cu clusters

The adsorption behaviors of the C2H4 molecule on the surface of the 7-atom Ag-Cu clusters are systematically investigated based on generalized gradient approximate density functional theory. The top site adsorption of C2H4 on the Ag7 cluster is physical adsorption, and the top site adsorption on the other clusters is chemical adsorption. The bridge site adsorption of C2H4 on Ag7 and Cu7 clusters is physical adsorption, while the chemical adsorption of C2H4 on the other clusters undergoes the transformation from the bridge site to the top site adsorption on the Cu atoms. In particular, the original structures of Ag5Cu2 and Ag4Cu3 clusters deformed significantly when C2H4 was adsorbed on their bridge sites. The adsorption of C2H4 on the bridge site of the Ag5Cu2 cluster is more stable than that on the top site, and the adsorption of C2H4 on the top site of the remainder clusters is more stable than that on the bridge site. Overall, the Ag2Cu5 cluster is the most stable for C2H4 adsorption configuration among all the studied clusters.

Far-from-equilibrium attractors with full relativistic Boltzmann approach in boost-invariant and non-boost-invariant systems

We study the universal behavior associated with a Relativistic Boltzmann Transport (RBT) approach with the full collision integral in 0+1D conformal systems. We show that all momentum moments of the distribution function exhibit universal behavior. Furthermore, the RBT approach allows to calculate the full distribution function, showing that an attractor behavior is present in both the longitudinal and transverse momentum dependence. We compare our results to the far-from-equilibrium attractors determined with other approaches, such as kinetic theory in Relaxation Time Approximation (RTA) and relativistic hydrodynamic theories, both in their viscous (DNMR) an anisotropic (aHydro) formulations, finding a very similar evolution, but an even faster thermalization in RBT for higher order moments. For the first time, we extended this analysis also to study the attractor behavior under a temperature-dependent viscosity η/s(T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta /s(T)$$\\end{document}, accounting also for the rapid increase toward the hadronic phase. We find that a partial breaking of the scaling behavior with respect to τ/τeq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au /\ au _{eq}$$\\end{document} emerges only at T≈Tc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T \\approx T_c$$\\end{document} generating a transient deviation from attractors; interestingly this in realistic finite systems may occur around the freeze-out dynamics. Finally, we investigate for the first time results beyond the boost-invariant picture, finding that also in such a case the system evolves toward the universal attractor. In particular, we present the forward and pull-back attractors at different space-time rapidities including rapidity regions where initially the distribution function is even vanishing.

Two-dimensional quad-stable Gaussian potential stochastic resonance model for enhanced bearing fault diagnosis

In this study, a two-dimensional quad-stable Gaussian potential stochastic resonance model is explored for the first time. First, the structure of the proposed model is analyzed to have a broader potential field and verified to break through the severe output saturation inherent in the classical two-dimensional quad-stable stochastic resonance model. Then, we analyze the relationship between the structure and parameters of the model and derive the steady-state probability density and the mean first-passage time using adiabatic approximation theory to describe the specific process of the Brownian particle transitions. By combining the adiabatic approximation theory and the probability flow equation, the spectral amplification factor of the model is derived, and the effects of different parameters on the model performance are investigated. Further, a fourth-order Runge-Kutta algorithm was applied to evaluate the model performance in multiple dimensions. Finally, the model parameters were optimized using an adaptive genetic algorithm and applied to complex practical engineering detection. The experimental results show that the proposed model is superior and universal in fault diagnosis. Overall, this study provides important mathematical support for solving various engineering problems and demonstrates a wide range of practical applications.

A computational investigation for heat and fluid transport with electro-osmosis phenomenon in a scraped surface heat exchanger

Scraped surface heat exchangers (SSHEs) have prominently shown their eminence in various food, chemical and in pharmaceutical industries when the regular processing of fluids and pertinent materials is encountered. The heat transfer analysis of Williamson fluid flow with electro osmotic effects in a narrow-gap SSHE is designed by implementing constant temperature difference between the gaps of the rotor and the stator. A fluid model is established by using lubrication approximation theory and Debye-Huckel simplifications. The computational solution is elaborated by implementing renowned BVP4C technique in MATLAB. Exact solution of Poisson-Boltzmann equation for different scenarios of SSHE is gathered, whereas solution for velocity, stream functions and temperature in various parts of SSHE are explored computationally. Impact of various crucial physical flow parameters like, Weissenberg number, mobility of the medium number, electro osmotic parameter on the velocity profile, stream function, temperature profile, and pressure gradient has been explored graphically. The numerical solution is compared with artificial neural network (ANN) and an absolute alignment of BVP4C solution is observed with ANN solution. It is reported that fluid flow is lifted with enhancing the viscoelastic behavior and exhibits dual behavior for electroosmosis phenomenon and mobility of medium parameter. Also, it is seen that temperature of the fluid is significantly declines with Weissenberg number and is surges with enhancing mobility of medium and Brinkman number. By increasing the Brinkman number conduction phenomenon is slow down and heat produced due to viscous dissipation, raises the temperature of the fluid. The viscoelastic effect could be of enormous significance in applications, like Polymer processing in certain heat exchangers, and in mixing and blending foodstuffs etc.

Exact hydrodynamic manifolds for the linear Boltzmann BGK equation I: spectral theory

We perform a complete spectral analysis of the linear three-dimensional Boltzmann BGK operator resulting in an explicit transcendental equation for the eigenvalues. Using the theory of finite-rank perturbations, we confirm the existence of a critical wave number kcrit\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_{\ extrm{crit}}$$\\end{document} which limits the number of hydrodynamic modes in the frequency space. This implies that there are only finitely many isolated eigenvalues above the essential spectrum at each wave number, thus showing the existence of a finite-dimensional, well-separated linear hydrodynamic manifold as a combination of invariant eigenspaces. The obtained results can serve as a benchmark for validating approximate theories of hydrodynamic closures and moment methods and provides the basis for the spectral closure operator.

Coupled piecewise tri-stable stochastic resonance system driven by dual inputs and joint EMD/VMD analysis

A novel piecewise tri-stable stochastic resonance (NPTSR) system is proposed to address the issue of output saturation caused by high-order terms limitation in this paper. Building upon this, the exploration is extended to a coupled piecewise tri-stable stochastic resonance system driven by dual inputs (DCPTSR). First, we uncover the influence of dual input interaction on output quality, finding that when the fRequencies of the two input signals are consistent, varying the amplitude of the driving signal can effectively enhance the output performance of the target signal. Secondly, by utilizing the adiabatic approximation theory, the steady-state probability density (SPD) and signal-to-noise ratio (SNR) of the DCPTSR system are derived, which allows us to analyze the effects of various parameters on both SPD and SNR. Next, three combined denoising systems, namely EMD-DCPTSR, VMD-DCPTSR, and SDCPTSR, are constructed by utilizing empirical mode decomposition (EMD), variational mode decomposition (VMD), and the stochastic resonance (SR). Through numerical simulations, we demonstrate that the combined denoising system outperforms the stand-alone SR system, and we analyze the stochastic resonance phenomenon of the DCPTSR system using the spectral amplification (SA) coefficient as an evaluation index. Finally, to assess practical applicability, these systems are deployed for bearing fault detection. The experimental results exhibit notable signal-to-noise gain improvements for the DCPTSR system compared to standalone SR systems by 0.7699 ∼ 9.4541 dB. The EMD-DCPTSR system shows signal-to-noise gain improvements of 0.3245 ∼ 1.1709 dB compared to the VMD-DCPTSR and SDCPTSR systems. Moreover, all three combined denoising systems outperform the standalone SR system in terms of signal processing capabilities. In conclusion, this paper extensively investigates the interaction between the two input signals in a dual-input system and studies the output performance of using EMD, VMD, and SR as preprocessing methods for the SR system. Through numerical simulations and practical engineering applications, we highlight the substantial advantages of combined denoising systems. These findings offer essential theoretical insights and promising prospects for engineering applications.

Magnetodielectric and Rheological Effects in Magnetorheological Suspensions Based on Lard, Gelatin and Carbonyl Iron Microparticles.

This study aims to develop low-cost, eco-friendly, and circular economy-compliant composite materials by creating three types of magnetorheological suspensions (MRSs) utilizing lard, carbonyl iron (CI) microparticles, and varying quantities of gelatin particles (GP). These MRSs serve as dielectric materials in cylindrical cells used to fabricate electric capacitors. The equivalent electrical capacitance (C) of these capacitors is measured under different magnetic flux densities (B≤160 mT) superimposed on a medium-frequency electric field (f = 1 kHz) over a period of 120 s. The results indicate that at high values of B, increasing the GP content to 20 vol.% decreases the capacitance C up to about one order of magnitude compared to MRS without GP. From the measured data, the average values of capacitance Cm are derived, enabling the calculation of relative dielectric permittivities (ϵr') and the dynamic viscosities (η) of the MRSs. It is demonstrated that ϵr' and η can be adjusted by modifying the MRS composition and fine-tuned through the magnetic flux density B. A theoretical model based on the theory of dipolar approximations is used to show that ϵr', η, and the magnetodielectric effect can be coarsely adjusted through the composition of MRSs and finely adjusted through the values B of the magnetic flux density. The ability to fine-tune these properties highlights the versatility of these materials, making them suitable for applications in various industries, including electronics, automotive, and aerospace.

Revisit Birkhoff’s Theorem: The Post-Newtonian Metric of a Self-Gravitating and Collapsing Thin Spherical Shell

We calculate the metric of a self-gravitating and collapsing infinitely thin spherical shell in the weak-field and slow-motion limits, and we demonstrate that Birkhoff’s theorem is not consistent with the theory of the post-Newtonian approximation. More importantly, it is illustrated that performing a coordinate transformation in solving Einstein field equations may change the matter energy-momentum tensor, making the resultant solution not correspond to the original problem.

Non-isothermal electroosmotic flow of a viscoelastic fluid through a porous medium in a microchannel

The influence of the temperature-dependent viscosity on the electroosmotic flow through a porous medium in a microchannel of parallel flat plates. At the end, an imposed electric field induces Joule heating and temperature gradients, modifying the viscosity of the fluid. The Phan-Thien–Tanner and Darcy–Forchheimer models describe the viscoelastic fluid through the porous media. Consequently, the electric double layer (EDL) overlaps at the center plane of the microchannel, causing an electrically charged porous matrix and modifying the zeta potential along the microchannel walls. Therefore, we define a zeta potential ratio of the porous matrix and that at the microchannel walls. Unlike similar investigations, the boundary condition for the zeta potential varies locally due to the temperature field, modifying the interaction among hydrodynamics, electrical, viscoelastic, thermal and porosity effects. A modified Péclet number is defined as a function of the porosity and thermal conductivity of the fluid and an effective charge density is determined based on the Poisson–Boltzmann equation, accounting for the pore size and the zeta potential ratio. The set of equations are solved numerically considering the lubrication approximation theory. We found that the influence of the temperature-dependent viscosity increases the shear stresses at the EDL region, particularly where the temperature rises to maximum values. Also, the overlapping of the EDL occurs mainly for low values of permeability. Finally, the pressure gradient involved in the Darcy law can be altered through the zeta potential ratio.

Adomian Modification Methods via Orthogonal Polynomials: A Comparative Study

The present manuscript proposes different modification procedures for the standard Adomian Decomposition Method (ADM). These procedures are based on the application of orthogonal polynomials that play vital parts in approximation theories. Moreover, the study also scrutinizes four test nonlinear inhomogeneous initial-value problems, and distinctively examines their respective absolute error differences. Remarkably, different computational benefits of the proposed modification are noted with regard high-level of accuracy and fewer computational steps.

Tensor approximation of functional differential equations.

Functional differential equations(FDEs) play a fundamental role in many areas of mathematical physics, including fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equations), and statistical physics. Despite their significance, computing solutions to FDEs remains a longstanding challenge in mathematical physics. In this paper we address this challenge by introducing approximation theory and high-performance computational algorithms designed for solving FDEs on tensor manifolds. Our approach involves approximating FDEs using high-dimensional partial differential equations(PDEs), and then solving such high-dimensional PDEs on a low-rank tensor manifold leveraging high-performance (parallel) tensor algorithms. The effectiveness of the proposed approach is demonstrated through its application to the Burgers-Hopf FDE, which governs the characteristic functional of the stochastic solution to the Burgers equationevolving from a random initial state.

On the accuracy of interpolation based on single-layer artificial neural networks with a focus on defeating the Runge phenomenon

Artificial Neural Networks (ANNs) are a tool in approximation theory widely used to solve interpolation problems. In fact, ANNs can be assimilated to functions since they take an input and return an output. The structure of the specifically adopted network determines the underlying approximation space, while the form of the function is selected by fixing the parameters of the network. In the present paper, we consider one-hidden layer ANNs with a feedforward architecture, also referred to as shallow or two-layer networks, so that the structure is determined by the number and types of neurons. The determination of the parameters that define the function, called training, is done via the resolution of the approximation problem, so by imposing the interpolation through a set of specific nodes. We present the case where the parameters are trained using a procedure that is referred to as Extreme Learning Machine (ELM) that leads to a linear interpolation problem. In such hypotheses, the existence of an ANN interpolating function is guaranteed. Given that the ANN is interpolating, the error incurred occurs outside the sampling interpolation nodes provided by the user. In this study, various choices of nodes are analyzed: equispaced, Chebychev, and randomly selected ones. Then, the focus is on regular target functions, for which it is known that interpolation can lead to spurious oscillations, a phenomenon that in the ANN literature is referred to as overfitting. We obtain good accuracy of the ANN interpolating function in all tested cases using these different types of interpolating nodes and different types of neurons. The following study is conducted starting from the well-known bell-shaped Runge example, which makes it clear that the construction of a global interpolating polynomial is accurate only if trained on suitably chosen nodes, ad example the Chebychev ones. In order to evaluate the behavior when the number of interpolation nodes increases, we increase the number of neurons in our network and compare it with the interpolating polynomial. We test using Runge’s function and other well-known examples with different regularities. As expected, the accuracy of the approximation with a global polynomial increases only if the Chebychev nodes are considered. Instead, the error for the ANN interpolating function always decays, and in most cases we observe that the convergence follows what is observed in the polynomial case on Chebychev nodes, despite the set of nodes used for training. Then we can conclude that the use of such an ANN defeats the Runge phenomenon. Our results show the power of ANNs to achieve excellent approximations when interpolating regular functions also starting from uniform and random nodes, particularly for Runge’s function.

Monotone path-connected sets in geometric approximation theory and their applications

Monotone path-connected sets in geometric approximation theory and their applications

A Hyper-Transformer model for Controllable Pareto Front Learning with Split Feasibility Constraints

Controllable Pareto front learning (CPFL) approximates the Pareto optimal solution set and then locates a non-dominated point with respect to a given reference vector. However, decision-maker objectives were limited to a constraint region in practice, so instead of training on the entire decision space, we only trained on the constraint region. Controllable Pareto front learning with Split Feasibility Constraints (SFC) is a way to find the best Pareto solutions to a split multi-objective optimization problem that meets certain constraints. In the previous study, CPFL used a Hypernetwork model comprising multi-layer perceptron (Hyper-MLP) blocks. Transformer can be more effective than previous architectures on numerous modern deep learning tasks in certain situations due to their distinctive advantages. Therefore, we have developed a hyper-transformer (Hyper-Trans) model for CPFL with SFC. We use the theory of universal approximation for the sequence-to-sequence function to show that the Hyper-Trans model makes MED errors smaller in computational experiments than the Hyper-MLP model.