Approximation Theory Research Articles

Weighted twisted inhomogeneous diophantine approximation

Abstract We prove a multidimensional weighted analogue of the well-known theorem of Kurzweil (1955) in the metric theory of inhomogeneous Diophantine approximation. Let ∑ i = 1 m α i = m and | ⋅ | α = max 1 ⩽ i ⩽ m | ⋅ | 1 / α i . Given an n-tuple of monotonically decreasing functions Ψ = ( ψ 1 , … , ψ n ) with ψ i : R + → R + such that each ψ i ( r ) → 0 as r → ∞ and fixed A ∈ R n × m define W A ( Ψ ) := { b ∈ [ 0 , 1 ] n : | A i ⋅ q − b i − p i | < ψ i ( | q | α ) ( 1 ⩽ i ⩽ n ) , for infinitely many ( p , q ) ∈ Z n × ( Z m ∖ { 0 } ) } . We prove that the set W A ( Ψ ) has zero-full Lebesgue measure under convergent–divergent sum conditions with some mild assumptions on A and the approximating functions Ψ. We also prove the Hausdorff dimension results for this set. Along with some geometric arguments, the main ingredients are the weighted ubiquity and weighted mass transference principle introduced recently by Kleinbock & Wang (2023 Adv. Math. 428 109154), and Wang & Wu (2021 Math. Ann. 381 243–317) respectively.

Absence of Additional Stretching-Induced Electron Scattering in Highly Conductive Cross-linked Nanocomposites with Negligible Tunneling Barrier Height and Width.

The intrinsic resistance of stretchable materials is dependent on strain, following Ohm's law. Here the invariable resistance of highly conductive cross-linked nanocomposites over 53% strain is reported, where additional electron scattering is absent with stretching. The in situ generated uniformly dispersed small silver nanosatellite particles (diameter=3.6nm) realize a short tunneling barrier width of 4.1nm in cross-linked silicone rubber matrix. Furthermore, the barrier height can be precisely controlled by the gap state energy level modulation in silicone rubber using cross-linkers. The negligible barrier height (0.01eV) and short barrier width, achieved by the silver nanosatellite particles in cross-linked silicone rubber, dramatically increase the electrical conductivity (51710Scm-1) by more than 4 orders of magnitude. The high conductance is also maintained over 53% strain. The quantum tunneling behavior is observed when the barrier height is increased, following the Simmons approximation theory. The transport becomes diffusive, following Ohm's law, when the barrier width is increased beyond 10.3nm. This study provides a novel strain-invariant resistance mechanism in highly conductive cross-linked nanocomposites.

Manufacturing Anti-Reflective Subwavelength Structures on ZnS Using Femtosecond Laser Bessel Beam with Burst Mode

Increasing the transmittance of zinc sulfide (ZnS) infrared windows can effectively improve the imaging quality of infrared detection. In this study, an anti-reflective subwavelength structure (ASS) was manufactured on ZnS using a femtosecond burst Bessel laser with the goal of achieving high transmittance in the mid-infrared range. The period and depth parameters of the ASS were initially determined using the effective medium approximation (EMA) theory and subsequently optimized using the rigorous coupled-wave analysis (RCWA) method to eliminate surface Fresnel anti-reflections. The depth of the ASS increases with the number of bursts, while the structure profile transitions from Gaussian to conical. In addition, the ASS achieves 86% transmittance in the 7–10 µm range, and the average transmittance improves by 10% in the 5–12 µm range. Moreover, the wide-angle ASS with the hydrophobicity (contact angle 160°) is achieved on the ZnS window. Ultimately, the ASS on ZnS enhances the clarity of the infrared image.

Modelling of planetary accretion and core-mantle structure formation

Abstract We advance a thermodynamically consistent model of self-gravitational accretion and differentiation in planets. The system is modeled in actual variables as a compressible thermoviscoelastic fluid in a fixed, sufficiently large domain. The supply of material to the accreting and differentiating system is described as a bulk source of mass, volume, impulse, and energy localized in some border region of the domain. Mass, momentum, and energy conservation, along with constitutive relations, result in an extended compressible Navier–Stokes-Fourier-Poisson system. The centrifugal and Coriolis forces are also considered. After studying some single-component setting, we consider a two-component situation, where metals and silicates mix and differentiate under gravity, eventually forming a core-mantle structure. The energetics of the models are elucidated. Moreover, we prove that the models are stable, in that self-gravitational collapse is excluded. Eventually, we comment on the prospects of devising a rigorous mathematical approximation and existence theory.

A-priori and a-posteriori error estimates for discontinuous Galerkin method of the Maxwell eigenvalue problem

This paper is devoted to a-priori and a-posteriori error analysis of discontinuous Galerkin (DG) method for the Maxwell eigenvalue problem. The discrete compactness of DG space is proved so that the Babuška and Osborn spectral approximation theory can be applicable in the a-priori error analysis. Then we prove the optimal error estimates for DG eigenfunctions in mesh-dependent norm and DG eigenvalues. A special contribution of this work is to prove that the error in L2-norm for smooth eigenfunctions is of higher order than that in mesh-dependent norm, so that the DG eigenvalues can approximate the true eigenvalues from upper. Another contribution of this work is to provide a-posteriori error analysis for the DG method. A reliable a-posteriori error estimator is analyzed. The upper bound property of DG eigenvalues and the robustness of adaptive methods are verified through numerical experiments.

Fabrication and evaluation of eicosane/poly(styrene-co-butylacrylate) microencapsulated phase change materials through ultrasonicated mini-emulsion technique

This study reports the synthesis and characterization of Microencapsulated Phase Change Materials for the pertinence of latent heat storage. Eicosane/ Poly(Styrene-co-Butylacrylate) [ESE/Poly(Sty-co-BA)] microcapsules were synthesized by encapsulating n-Eicosane (n-ESE) with a shell made of Poly(Styrene-co-Butylacrylate) [Poly(Sty-co-BA)] via employing ultrasonic-assistant mini-emulsion polymerization technique. The properties of the synthesized ESE/Poly(Sty-co-BA) microcapsules were analyzed by Differential scanning calorimetry (DSC),Thermogravimetric analysis (TGA), Transmission electron microscopy (TEM), X-ray diffraction (XRD), and Fourier transform infrared (FTIR)spectroscopy. DSC results indicated that ESE/Poly(Sty-co-BA) exhibited a melting and crystallizing temperature of 35.72 and 32.27 °C, and latent heat of 220.53 and 218.57 J/g, respectively. The ESE/Poly(Sty-co-BA) microcapsules prepared in this work unveiled an effective encapsulation ratio (91.93 %), encapsulation efficiency (91.59 %) and displayed excellent thermal storage capacity (99.63 %). Various models such as Kissinger’s, Augis and Bennett’s approximation, Mahadevan and Arvami theory were employed to determine the effective activation energy value alongside the crystal growth of n-ESE and ESE/Poly(Sty-co-BA). TGA analysis revealed that ESE/Poly(Sty-co-BA) demonstrated three step degradation and exhibited good thermal stability. The findings of the work manifested a novel technique for producing the phase transition materials for TES that can be applied to a wide range of instances, including smart textiles, thermo-sensitized functional coatings, passive space cooling or heating.

A First-Principles Study of the Structural and Thermo-Mechanical Properties of Tungsten-Based Plasma-Facing Materials

Tungsten (W) and tungsten alloys are being considered as leading candidates for structural and functional materials in future fusion energy devices. The most attractive properties of tungsten for the design of magnetic and inertial fusion energy reactors are its high melting point, high thermal conductivity, low sputtering yield, and low long-term disposal radioactive footprint. Despite these relevant features, there is a lack of understanding of how the structural and mechanical properties of W-based alloys are affected by the temperature in fusion power plants. In this work, we present a study on the thermo-mechanical properties of five W-based plasma-facing materials. First-principles density functional theory (DFT) calculations are combined with the quasi-harmonic approximation (QHA) theory to investigate the electronic, structural, mechanical, and thermal properties of these W-based alloys as a function of temperature. The coefficient of thermal expansion, temperature-dependent elastic constants, and several elastic parameters, including bulk and Young’s modulus, are calculated. Our work advances the understanding of the structural and thermo-mechanical behavior of W-based materials, thus providing insights into the design and selection of candidate plasma-facing materials in fusion energy devices.

Finite element simulations of quartz crystal microbalances (QCM): from Sauerbrey to fractional viscoelasticity under water

2D finite element simulations are performed on QCM working in the thickness-shear mode and loaded with different homogeneous films. They include a purely elastic film, a viscoelastic Maxwellian liquid, viscoelastic-Voigt solid, and the fractional viscoelastic (power-law) version of each case. Unlike single-relaxation kind models, fractional viscoelasticity considers the relaxation-time spectrum often found in polymeric materials. The films are tested in air or covered with liquids of different viscosities. Two substrate thicknesses are tested: 100 nm and 500 nm, the latter being close to the condition that promotes the resonance of the adsorbed film. In all cases the simulations are compared with small-load approximation theory (SLA). The 100 nm films follow the theory closely, although significant deviations of the SLA are observed as the overtone number n increases, even in purely elastic films. We also show that it is possible to identify the viscoelastic ‘fingerprint’ of the 100 nm films in air using raw data and Sauerbrey’s equivalent thickness obtained with the QCM in the 3 < n < 13 range. These numerical data are validated by experimental measurements of crosslinked polydimethylsiloxane films with thicknesses ∼150 nm. In contrast, the 500 nm films deviate notoriously from the SLA, for all viscoelastic models and overtones, with the largest deviation observed in the elastic film. When a liquid layer covers the QCM without an adsorbed film, the only overtone that numerically reproduces the theoretical value is the fundamental, n = 1. For n > 1, strong coupling between the solid and liquid is detected, and the original vibration modes of the crystal are altered by the presence of the liquid. Finally, the numerical simulations suggest that it is possible to detect whether a viscoelastic film is formed under a liquid layer using only the information from n = 1. In these film/liquid systems we also observe the so-called missing-mass effect, although the theory and simulations exhibit different levels of impact of such effect when the liquid viscosity is high.

On the rate of convergence for the q-Durrmeyer polynomials in complex domains

Abstract The q-Durrmeyer polynomials are one of the popular q-versions of the classical operators of approximation theory. They have been studied from different points of view by a number of researchers. The aim of this work is to estimate the rate of convergence for the sequence of the q-Durrmeyer polynomials in the case 0 &lt; q &lt; 1. It is proved that for any compact set 𝓓 ⊂ ℂ, the rate of convergence is O(qn ) as n → ∞. The sharpness of the obtained result is demonstrated.

Effective medium theory for viscoelasticity of soft jammed solids

Abstract The viscoelastic properties of soft jammed solids, such as foams, emulsions, and soft colloids, have been extensively studied in experiments. A particular focus has been placed on the phenomenon of anomalous viscous loss, characterized by a storage modulus $G' \propto \omega^0$ and a loss modulus $G'' \propto \sqrt{\omega}$, where $\omega$ represents the frequency of the applied strain. In this work, we aim to develop a microscopic theory that explains these experimental observations. Our approach is based on effective medium theory (EMT), also referred to as coherent potential approximation theory. By incorporating the effects of contact damping, a key characteristic of soft jammed solids, into the EMT, we offer new insights into the viscoelastic behavior of these materials. The theory not only explains the observed viscoelastic properties but also links the anomalous viscous loss to the marginal stability inherent in amorphous systems. This research lays the groundwork for a microscopic theory that effectively describes the impact of damping on soft jammed solids and their characteristic viscoelastic behaviors.

Characteristic Equations of Chebyshev Polynomials of Third and Fourth Kinds and Their Generating Matrices

The main goal of the article is to obtain matrix representation for the third and fourth kinds of Chebyshev polynomials by using a tridiagonal matrix. We present a connection between the determinant of the tridiagonal matrix and the third and fourth kinds of Chebyshev polynomials. We also determine the characteristic equations for the third and fourth kinds of the Chebyshev polynomials up to degree three. We also prove some properties relating to matrix representation. We obtain a connection between the second, third kind and fourth kinds of Chebyshev polynomials and matrix power. It elaborates the theorem to validate through examples. The applications of the Chebyshev polynomials is also discussed. The practical application of the Chebyshev polynomials of the third kind in approximation theory is also detailed.

Radial Kernels Collocation Method for the Solution of Volterra Integro-Differential Equations

Radial kernel interpolation is an advanced method in approximation theory for the construction of higher order accurate interpolants for scattered data up to higher dimensional spaces. In this manuscript, we formulate a radial kernel collocation approach for solving problems involving the Volterra integro-differential equations using two radial kernels: The Generalized Multi-quadrics and the linear Laguerre-Gaussians. This was achieved by simplifying the Volterra integral problem's solution to an algebraic system of equations. The impact of the shape parameter present in every kernel on the method's accuracy is examined and found to be significant. Two examples were used to illustrate the process; the numerical results are shown as tables and graphs. MATLAB 2018a was employed in the process.

New Results on Differential Subordination and Superordination for Multivalent Functions Involving New Symmetric Operator

This article aims to significantly advance geometric function theory by providing a valuable contribution to analytic and multivalent functions. It focuses on differential subordination and superordination, which characterize the interactions between analytic functions. To achieve our goal, we employ a method that relies on the characteristics of differential subordination and superordination. As one of the latest advancements in this field, this technique is able to derive several results about differential subordination and superordination for multivalent functions defined by the new operator within the open unit disk . Additionally, by employing the technique, the differential sandwich outcome is achieved. Therefore, this work presents crucial exceptional instances that follow the results. The findings of this paper can be applied to a wide range of mathematical and engineering problems, including system identification, orthogonal polynomials, fluid dynamics, signal processing, antenna technology, and approximation theory. Furthermore, this work significantly advances the knowledge and understanding of the analytical functions of the unit and its interactive higher relations. The characteristics and consequences of differential subordination theory are symmetric to those of differential superordination theory. By combining them, sandwich-type theorems can be derived.

A robust force identification method based on L1+2-norm approximation theory

Obtaining the real force of the structure is the basic condition for efficient vibration reduction design. However, the uneasiness and instability of the inverse problem make the force identification still challenging. On the basis of L2-norm regularization, we propose a new method for force identification based on L1+2-norm approximation theory. By introducing the L1-norm penalty operator, a joint norm approximation optimization model is constructed to limit the non-zero force and punish the force value which approaching to zero, so that it can adapt to the identification of various types of forces. Then the optimal force solution is obtained by using the Primal-dual interior point method. Finally, the effectiveness and accuracy of the proposed method are verified by numerical simulation and experiment of asymmetric plate and double-layer plate structures. Different from previous studies, the L1+2-norm method in this work has high robustness, considers the characteristics of sparse regularization and L2-norm regularization, and can adapt to the identification of continuous forces and sparse forces at the same time, which has potential engineering application value.

Systems of fixpoint equations: Abstraction, games, up-to techniques and local algorithms

Systems of fixpoint equations over complete lattices, consisting of (mixed) least and greatest fixpoint equations, allow one to express many verification tasks such as model-checking of various kinds of specification logics or the check of coinductive behavioural equivalences. In this paper we develop a theory of approximation for systems of fixpoint equations in the style of abstract interpretation: a system over some concrete domain is abstracted to a system in a suitable abstract domain, with conditions ensuring that the abstract solution represents a sound/complete overapproximation of the concrete solution. Interestingly, up-to techniques, a classical approach used in coinductive settings to obtain easier or feasible proofs, can be interpreted as abstractions in a way that they naturally fit into our framework and extend to systems of equations. Additionally, relying on the approximation theory, we can characterise the solution of systems of fixpoint equations over complete lattices in terms of a suitable parity game, generalising some recent work that was restricted to continuous lattices. The game view opens the way for the development of local algorithms for characterising the solution of such equation systems. We describe a local algorithm for checking the winner on specific game positions. This corresponds to answering the associated verification question (i.e., for model checking, whether a state satisfies a formula or, for equivalence checking, whether two states are behaviourally equivalent). The algorithm can be combined with abstraction and up-to techniques, thus providing ways of speeding up the computation.

A computation analysis with heat and mass transfer for micropolar nanofluid in ciliated microchannel: With application in the ductus efferentes

Inherited heat enhancement capabilities and their significance in the field of medical sciences and industry make nanofluids the focus of research nowadays. Furthermore, due to the remarkable advancements in bionanotechnology and its significance in biomedical fields such as drug delivery systems, cancer tumor therapy, bioimaging, and many others, it has emerged as a key research area. Contribution of cilia for the flow in ductus efferentes of human male reproductive tract is elaborated. A novel mathematical scheme is presented for the heat and mass transfer of MHD micropolar nanofluid transport in an asymmetric channel lined with cilia. The pertinent equations of nanofluid transport are exposed to lubrication approximation theory and solution for the physical problem is examined with efficient bvp4c technique in MATLAB. Fluid rheology is explored with the variations of different transport parameters like Hartmann number, Grashof number, Brownian motion, buoyancy, thermophoresis and Darcy number. It is reported that nanofluid transport is affected with rise in the Lorentz force and show reverse behavior with rising permeability. The temperature of the nanofluid in ciliated microchannel is raised with enhanced value of Hartmann number, Grashof number, Prandtl number, and Darcy number while diffusion phenomenon of nanofluid is slowed down with these parameters. Spinning motion of the nanofluid is enhanced with Grashof number and slow down with nanoparticle Grashof number and different behavior is recorded for Darcy and viscosity parameters in different flow regime. Reported investigation presents crucial findings for ciliary transport of micropolar nanofluid and tackled with appropriate selection of micropolar parameter, Brownian motion parameter, thermophoresis and Grashof number. Moreover, this investigation will be handful for cilia-based actuators which work as micro-mixers in controlling the flow in minute bio-sensors and may prove their worth in micro-pumps employed in various drug-delivery systems.

Nonlinear coupled asymmetric stochastic resonance for weak signal detection based on intelligent algorithm optimization

Stochastic resonance has been extensively studied for detecting weak signals. To improve the diagnostic ability of weak signals, a novel nonlinear coupled asymmetric stochastic resonance (NCASR) system is investigated in this paper. Firstly, the NCASR system is established by coupling the asymmetric bistable system with the monostable system. Next, the expressions for the steady-state probability density (SPD) function, the mean first passage time (MFPT) and the signal-to-noise ratio (SNR) of the proposed system are derived based on the adiabatic approximation theory. Furthermore, the impact of system parameters on the SPD, the MFPT and the SNR is analyzed. Then, by simulation experiments, we verify the effectiveness of detecting weak signals for the NCASR system with Lévy noise. Finally, the NCASR system optimized by Adaptive Weighted Particle Swarm Optimization (AWPSO) algorithm is applied to detect the bearing fault signal. Compared with the optimized classical bistable stochastic resonance (CBSR) system, it is found that the detection performance of the NCASR system is superior to the CBSR system in detecting bearing fault signals.

On the use of fast projection methods with unsteady velocity boundary conditions

This article aims at clarifying and amending a previously published result on the use of pseudo-pressure approximations for fast incompressible Navier-Stokes solvers with time-dependent boundary conditions. Fast projection methods were proposed to speed up high-order incompressible Navier-Stokes solvers by replacing pressure projections with simple approximations. A generalized approximation theory was proposed in [1], in which the time-dependent boundary conditions were ignored. In this comment, we investigate the effect of unsteady boundary conditions on the order of accuracy and demonstrate that the pressure approximations derived in [1] hold for all RK2 integrators, while RK3 approximations are only third order for certain sets of coefficients. We show that third order is lost for other cases and shed light on why order is broken. This work serves as a reference for the treatment of unsteady boundary conditions for fast-projection methods. Numerical validation is performed on a well established channel flow problem with a variety of unsteady inlet conditions for a wide range of explicit RK2 and RK3 integrators.

Cilia-assisted flow of electrically conducting micropolar fluid in a heated curved channel under Soret and Dufour effects

Cilia flow plays a crucial role in biological systems such as the movement of mucus in the respiratory tract, circulation of cerebrospinal fluid in the brain, and propulsion of sperm cells. Understanding the cilia flow of viscous fluids is essential for elucidating the biomechanics of these processes and their implications for health and disease. Motivated by such numerous biomedical applications, this article aims to exhibit the influence of heat and mass transfer on the ciliary flow of electrical conducting micropolar fluid in a curved channel. The energy and concentration equations are modulated with viscous dissipation, Soret, and Dufour effects. The constitutive equations are simplified by the hypothesis of lubrication approximation theory and then solved numerically using the implicit finite difference method (FDM). Results for velocity, pumping phenomenon, concentration, thermal field, rate of heat transfer, streamlines, skin friction coefficient, Nusselt number, and Sherwood number are analyzed subject to pertinent parameters. The study reveals that velocity is enhanced via both the micropolar parameter and curvature parameter. Temperature enhances for larger values of Dufour and Brickman numbers. Furthermore, the radial magnetic field plays a resistive role in the trapped bolus. The findings presented in this study should prove beneficial for researchers in the domains of medicine, engineering, science, and fluid mechanics. Further, it is found that the micropolar fluid model is more suitable for biofluids like blood.

Approximation theory of wavelet frame based image restoration

In this paper, we analyze the error estimate of a wavelet frame based image restoration method from degraded and incomplete measurements. We present the error between the underlying original discrete image and the approximate solution which has the minimal ℓ1-norm of the canonical wavelet frame coefficients among all possible solutions. Then we further connect the error estimate for the discrete model to the approximation to the underlying function from which the underlying image comes.