Approximation Theory Research Articles

Increasing mathematics students' involvement in research activities at teacher education universities

The paper explores the problem of creating interest in mathematics students' research endeavors at teacher preparation institutions. The purpose of the study is to determine whether it is possible to organize students' scientific investigation using the 5E Instructional Model during a workshop on approximation theory and the Fourier series. The study takes into account the survey responses from the students who assisted in assessing the workshop participants' emotional states. The Differential Emotion Scale, a tool for expressing evaluations of positive and negative emotionality, was used to gather the data. The benefits of the atmosphere created by the 5E Instructional Model on students' emotional states and how it shapes their motivation in scientific inquiry when setting up workshop classes are covered in the article. We have reason to believe that running workshops based on the 5E Instructional Model is effective. The students' involvement throughout the session and enthusiasm in research activities increased as a result of the index reduction of their negative feelings.

Virtual element method for the Helmholtz transmission eigenvalue problem of anisotropic media

In this paper, we propose a conforming virtual element method for the Helmholtz transmission eigenvalue problem of anisotropic media. By using [Formula: see text]-coercivity theory, the spectral approximation theory of compact operator and the projection and interpolation error estimates, we prove the spectral convergence of the discrete scheme and the optimal a priori error estimates for the discrete eigenvalues and eigenfunctions. The virtual element method has great flexibility in handling polygonal meshes, which motivates us to construct a fully computable a posteriori error estimator for the virtual element method. Then the upper bound of the approximation error is derived from the residual equation and the inf-sup condition. In turn, the related lower bound is established by using the bubble function strategy. Finally, we provide numerical examples to verify the theoretical results, including the optimal convergence of the virtual element scheme on uniformly refined meshes and the efficiency of the estimator on adaptively refined meshes.

General closed-form expressions for the three-dimensional vibrations of elastic bodies using the Ritz method

A novel Ritz-β approach is proposed for accurately determining the vibrational frequencies of a body based on its three-dimensional elasticity. In the conventional Ritz method, when admissible 3D displacement fields are expressed as either simple algebraic, Chebyshev, or Legendre polynomials, the volumetric integrals are transformed into beta integrals through a change of variables and then furthered simplified to beta functions and gamma functions; the resulting stiffness and mass matrices are expressed in a closed-form and the computational time is significantly decreased. The proposed approach does not involve the intensive computational procedures employed in the classical finite element, finite difference, and spectral approximation theories. Furthermore, it does not involve the kinematic constraints employed in classical beam, plate, and shell theories. The correctness and accuracy of the proposed approach were confirmed through comprehensive convergence studies and comparisons with existing results.

Blindfolded Attackers Still Threatening: Strict Black-Box Adversarial Attacks on Graphs

Adversarial attacks on graphs have attracted considerable research interests. Existing works assume the attacker is either (partly) aware of the victim model, or able to send queries to it. These assumptions are, however, unrealistic. To bridge the gap between theoretical graph attacks and real-world scenarios, in this work, we propose a novel and more realistic setting: strict black-box graph attack, in which the attacker has no knowledge about the victim model at all and is not allowed to send any queries. To design such an attack strategy, we first propose a generic graph filter to unify different families of graph-based models. The strength of attacks can then be quantified by the change in the graph filter before and after attack. By maximizing this change, we are able to find an effective attack strategy, regardless of the underlying model. To solve this optimization problem, we also propose a relaxation technique and approximation theories to reduce the difficulty as well as the computational expense. Experiments demonstrate that, even with no exposure to the model, the Macro-F1 drops 6.4% in node classification and 29.5% in graph classification, which is a significant result compared with existent works.

A Theoretical Study of Reverse Roll Coating for a Non-Isothermal Third-Grade Fluid under Lubrication Approximation Theory

The flow of non-Newtonian fluids is extremely important in a variety of industrial applications and processes, including painting, printing, and coating. The goal of this paper is to develop a mathematical model for an incompressible, non-isothermal third-grade fluid as it travels through a minor gap between two heated rolls that are revolving in the opposite direction. Using appropriate dimensionless parameters, the dimensionless version of the governing equations is constructed. To simplify the nondimensional form of the governing equations, the LAT (lubrication approximation theory) is applied. It is mentioned that the perturbation approach should be used. It is a very important tool in modern mathematical physics and practical fluid mechanics, and it is used a lot. Through perturbation techniques, exact solutions for velocity distribution, flow rate, temperature distribution, and pressure gradient are offered. To observe the flow pattern, streamlines are also drawn. Numerical outcomes of some substantial engineering quantities like flow rate, roll separation force, power input, and pressure distribution are found. Some outcomes are displayed graphically although others are presented in tabular form. We have observed that the third-grade parameters give a way to control velocity, pressure gradients, flow rate, and coating thickness. Further, increasing the velocities ratio values lowers the coating thickness on the web. Furthermore, the Brickman number has a significant influence on the temperature profile and has increased with the increase in the Brickman number. Both the model and the numerical calculations are in agreement with the viscous fluid model that is already present in the literature.

Measuring the phase of ambient noise cross correlations: anisotropic Rayleigh and Love wave tomography across the Oman Mountains

SUMMARY Ambient seismic noise tomography has, over the last two decades, developed into a well-established tool for imaging seismic properties of the Earth’s crust. Fundamental mode Rayleigh and Love wave phase velocity dispersion curves can be measured from ambient noise cross-correlation functions (CCF) either using a high-frequency approximation theory, or by fitting the spectrum of the CCF to a Bessel function. Here, we advance the latter approach and present an automated algorithm that fits the phase of the Hankel function to the phase of the causal symmetric part of the CCF in order to determine phase velocity curves as continuous functions of frequency. Synthetic tests verify the reliability of the proposed method in the presence of low signal-to-noise ratio (SNR). Moreover, usage of the phase allows for robust phase velocity measurements at longer periods than when using the zero crossings of the Bessel function only and is, therefore, particularly useful at short inter-station distances. In the frequency domain, acceptable bandwidths of smooth phase velocity curves are obtained in an automated procedure using a set of fine-tuned quality criteria. We apply the method to 2.5 yr of continuous waveform data recorded by 58 temporary and permanent broad-band seismic stations in northern Oman. We obtain 1072 and 670 phase velocity curves for Rayleigh and Love waves, respectively, in the period range of 2–40 s. The data are inverted for isotropic and azimuthally anisotropic period-dependent phase velocity maps. Synthetic reconstruction tests show that the phase velocity maps have a lateral resolution of ∼30 km. The results suggest distinctly different middle to lower crustal architecture between the northern and eastern Oman Mountains. Azimuthal anisotropy shows contrasting fast propagation orientations in the shallow and deep crust, which we attribute to stress-induced and structural anisotropy in the upper crust and to lattice-preferred orientation in the lower crust.

A hybrid data-driven method for rapid prediction of lithium-ion battery capacity

The quick and accurate prediction of future capacity is a challenging issue in the field of battery health management. To solve this problem, the paper proposes a data-driven model based on advanced machine-learning techniques. First, wavelet transform is used to denoise the initial features, and the relatively important features are selected with the help of random forest. Second, the paper uses the approximate entropy theory to construct multi-time scale sliding window data to solve the problem that the sliding window size can not be determined in traditional methods. Third, in order to overcome the disadvantages of traditional methods, such as the comparative inefficiency in converging and using temporal contextual information, the paper proposes a novel mogrifier long short-term memory network denoted as QGA-ASM-LSTM, which is based on attention mechanism and similarity judgment mechanism, using quantum genetic algorithm to optimize its parameters. Finally, the proposed data-driven method is used to model the degradation of lithium-ion batteries and finish the fast capacity prediction. The results of experiments on different public datasets demonstrate that the performance of QGA-ASM-LSTM exceeds the state-of-the-art models in terms of prediction accuracy and training time.

Systematically Improvable Generalization of Self-Interaction Corrected Density Functional Theory.

Perdew-Zunger self-interaction correction (PZSIC) reintroduces an exact constraint to approximate density functional theory (DFT). However, PZSIC can paradoxically degrade performance, and standard DFT approximations (with or without PZSIC) are not systematically improvable. We use the adiabatic projection formalism (Janesko, B. G. J. Chem. Phys. 2022, 156, 014111, https://doi.org/10.1063/5.0076144) to derive PZSIC in terms of a reference system experiencing only electron self-interaction. Generalization to a "self-and-some-others" interaction introduces correlation into the reference system, systematically bridging from PZSIC to exact wave function theory without the double counting of correlation. Minimal active spaces accurately treat nearly one electron, near-equilibrium, and strongly correlated model systems at modest computational expense.

Tailoring Three-Dimensional copper framework in polyamide 6 for High-Performance thermal management applications

Polymer plays an essential role in efficient thermal management of high-performance electronic devices, but the cooling efficiency of polymer composites is limited by some technical difficulties, such as the lack of thermal transport pathways and high cost of nano filler. Herein, we introduced solid-state shear milling (S3M) technology to prepare copper flakes (Cu-f) and constructed an enhanced Cu flakes network through hot-pressing of Cu and polyamide 6 (PA6) particles with different melt temperature. The low melt temperature PA6 coating on Cu flake surface acted as a glue to tightly connect Cu flakes and high melt temperature PA6 particles, thus generating an enhanced filler network. This brings the PA6/Cu-f network composites with a record high thermal conductivity at 6.3 Wm−1Κ−1, which is 3 times higher than that of traditional PA6/Cu-s composites (1.8 Wm−1Κ−1) at 80 wt% Cu loading. Basing on Foygel’s model from effective medium approximation theory, the PA6/Cu-f network composites had a lower thermal contact resistance (Rc) of 9.8 × 103 KW−1 and interfacial thermal resistance (ITR) of 6.3 × 10−8 W−1m2K. In LED lamps cooling performance test, PA6/Cu-f network composite showed 15 °C lower than traditional PA6/Cu-s composite in operating temperature. We believed that with the excellent thermal management ability and low cost, the PA6/Cu-f network composites open up opportunities for efficient thermal management in high power density devices.

Measurement of Forest Ecological Benefits Based on Big Data

Socioeconomic development combined with the development and utilization of forest resources have resulted in the forest ecological environment becoming more and more valuable, and its impact on the national economy has also increased. Firstly, based on a modern statistical model, the dependent variable and independent variable set of standardized forest ecological benefits were determined, and the seemingly unrelated forest ecological benefit model was established. According to the alternative market method and market approximation theory, the first and second monetary models of forest ecological benefit value were constructed using the market approximation coefficient, the effective area coefficient, and the physical quantity conversion coefficient. Next, the concept of “whole diffusion” was introduced into the establishment of the forest ecological benefit model, and the whole diffusion model was established to estimate the physical amount of forest ecological benefits. The concepts of classical forest ecological benefit and generalized forest ecological benefit were proposed. Forest management survey data were used to measure forest ecological benefit, and classic ecological benefit and generalized forest benefit were integrated into one system for the first time. The economic value is 94.31 million RMB ¥.

Neural network interpolation operators optimized by Lagrange polynomial

In this paper, we introduce a new type of interpolation operators by using Lagrange polynomials of degree r, which can be regarded as feedforward neural networks with four layers. The approximation rate of the new operators can be estimated by the (r+1)-th modulus of smoothness of the objective functions. By adding some smooth assumptions on the activation function, we establish two important inequalities of the derivatives of the operators. With these two inequalities, by using the K-functional and Berens–Lorentz lemma in approximation theory, we establish the converse theorem of approximation. We also give the Voronovskaja-type asymptotic estimation of the operators for smooth functions. Furthermore, we extend our operators to the multivariate case, and investigate their approximation properties for multivariate functions. Finally, some numerical examples are given to demonstrate the validity of the theoretical results obtained and the superiority of the operators.

A Piecewise Nonlinear Tri-Stable Stochastic Resonance System and its Application in Bearing Fault Diagnosis

An improved Piecewise Tri-stable Stochastic Resonance (PTSR) system is investigated to enhance the low performance of Classical Tri-stable Stochastic Resonance (CTSR) system for weak signal extraction. Firstly, the proposed PTSR system is theoretically compared with CTSR system. Secondly, based on adiabatic approximation theory, the analytical expressions for the Mean First Pass Time (MFPT) and the output Signal-to-Noise Ratio (SNR) of the system are derived. Afterwards, a PTSR-EMD system that combines PTSR system with empirical mode decomposition (EMD) is proposed. Finally, the above three systems are applied to the detection of harmonic signals and actual fault signals, and the superior performance of PTSR-EMD system over PTSR system and CTSR system are confirmed.

The diagonal of the operahedra

The primary goal of this article is to set up a general theory of coherent cellular approximations of the diagonal for families of polytopes by developing the method introduced by N. Masuda, A. Tonks, H. Thomas and B. Vallette. We apply this theory to the study of the operahedra, a family of polytopes ranging from the associahedra to the permutahedra, and which encodes homotopy operads. After defining Loday realizations of the operahedra, we make a coherent choice of cellular approximations of the diagonal, which leads to a compatible topological cellular operad structure on them. This gives a model for topological and algebraic homotopy operads and an explicit functorial formula for their tensor product.

Statistical Dynamics of Mean Flows Interacting with Rossby Waves, Turbulence, and Topography

Abridged statistical dynamical closures, for the interaction of two-dimensional inhomogeneous turbulent flows with topography and Rossby waves on a beta–plane, are formulated from the Quasi-diagonal Direct Interaction Approximation (QDIA) theory, at various levels of simplification. An abridged QDIA is obtained by replacing the mean field trajectory, from initial-time to current-time, in the time history integrals of the non-Markovian closure by the current-time mean field. Three variants of Markovian Inhomogeneous Closures (MICs) are formulated from the abridged QDIA by using the current-time, prior-time, and correlation fluctuation dissipation theorems. The abridged MICs have auxiliary prognostic equations for relaxation functions that approximate the information in the time history integrals of the QDIA. The abridged MICs are more efficient than the QDIA for long integrations with just two relaxation functions required. The efficacy of the closures is studied in 10-day simulations with an easterly large-scale flow impinging on a conical mountain to generate rapidly growing Rossby waves in a turbulent environment. The abridged closures closely agree with the statistics of large ensembles of direct numerical simulations for the mean and transients. An Eddy Damped Markovian Inhomogeneous Closure (EDMIC), with analytical relaxation functions, which generalizes the Eddy Dampened Quasi Normal Markovian (EDQNM) to inhomogeneous flows, is formulated and shown to be realizable under the same circumstances as the homogeneous EDQNM.

Transient response of a thin linearly elastic plate with Norton or Tresca friction

By using a nonlinear version of Trotter’s theory of approximation of semi-groups acting on variable Hilbert spaces, we propose an asymptotic modeling for the behavior of a linearly elastic plate in bilateral contact with a rigid body along part of its lateral boundary with Norton or Tresca friction.

On Fractional Inequalities Using Generalized Proportional Hadamard Fractional Integral Operator

The main objective of this paper is to use the generalized proportional Hadamard fractional integral operator to establish some new fractional integral inequalities for extended Chebyshev functionals. In addition, we investigate some fractional integral inequalities for positive continuous functions by employing a generalized proportional Hadamard fractional integral operator. The findings of this study are theoretical but have the potential to help solve additional practical problems in mathematical physics, statistics, and approximation theory.

Dynamic mode decomposition with core sketch

With the increase in collected data volumes, either from experimental measurements or high fidelity simulations, there is an ever-growing need to develop computationally efficient tools to process, analyze, and interpret these datasets. Modal analysis techniques have gained great interest due to their ability to identify patterns in the data and extract valuable information about the system being considered. Dynamic mode decomposition (DMD) relies on elements of the Koopman approximation theory to compute a set of modes, each associated with a fixed oscillation frequency and a decay/growth rate. Extracting these details from large datasets can be computationally expensive due to the need to implement singular value decomposition of the input data matrix. Sketching algorithms have become popular in numerical linear algebra where statistical theoretic approaches are utilized to reduce the cost of major operations. A sketch of a matrix is another matrix, which is significantly smaller, but still sufficiently approximates the original system. We put forth an efficient DMD framework, SketchyDMD, based on a core sketching algorithm that captures information about the range and corange (their mutual relationship) of input data. The proposed sketching-based framework can accelerate various portions of the DMD routines, compared to classical methods that operate directly on the raw input data. We conduct numerical experiments using the spherical shallow water equations as a prototypical model in the context of geophysical flows. We show that the proposed SketchyDMD is superior to existing randomized DMD methods that are based on capturing only the range of the input data.

AJ-Statistical Approximation of Continuous Functions by Sequence of Convolution Operators

In this paper, following the concept of AI-statistical convergence for real sequences introduced by Savas et al. [22], we deal with Korovkin type approximation theory for a sequence of positive convolution operators defined on C[a, b], the space of all real valued continuous functions on [a, b], in the line of Duman [6]. In the Section 3, we study the rate of AI-statistical convergence.

(Retracted) Field weed recognition algorithm based on machine learning

Currently, large-scale spraying of herbicides pollutes the environment, destroys the quality of cultivated land, and increases the cost of agricultural production. For this reason, “precision farming,” in which machine learning technology is used to detect weeds in crops and soil and then herbicides are sprayed at specific locations according to the distribution of weeds, is proposed. Machine learning is a multidisciplinary interdisciplinary major, covering knowledge of probability theory, statistics, approximate theory and complex algorithms. Machine learning uses computers as tools and is committed to simulating human learning methods in real time, and divides existing content into knowledge structures to effectively improve learning efficiency. We aim to use machine learning technology to study weed detection algorithms and provide feasible solutions for accurate weed classification and variable spraying of herbicides. Based on the research of four kinds of germinated corn and wheat weeds by Chinese and foreign experts, we use machine learning technology as the method of extracting and recognizing the combination of color features, shape features, and texture features. Plant development and software are used to process weeds to classify and identify wheat and corn. To better describe the gray distribution of each pixel and neighboring pixels, a more effective weed image classification function is required. The experimental results of this paper show that the machine learning-based weed image recognition classifier can describe the grayscale distribution of each pixel and adjacent pixels well, and it is beneficial to identify the characteristics of weeds, in which the correct rate of weed recognition exceeds 97.5%.

Study on the Oil Film Force with Unlimited Length Oil Film of the Sliding Bearing and the ERSFD with Bosses

The research background is the elastic ring squeeze film damper. Four contact pressure models were established by analyzing the structural characteristics and movement, combined with the sliding bearing theory, including structural parameters and eccentricities. Multi-structure and multi-interval dynamic boundary conditions were selected by analyzing actual structures. Simpson, polynomial, and integrated parameters methods extended Booker formulas. By combining existences and forms of the solution and mean-value theories, approximate analytical solutions of the finite length bearing were obtained under different contacts. Combined with the short and long bearing, general structures and expressions of analytical solutions of oil film pressures and forces under three approximation theories were obtained. The oil film characteristics of the dynamic equilibrium state were obtained, and the correctness was verified by theoretical comparison. Numerical simulations analyzed the relationship among relevant parameters. It provided a theoretical basis upon which to study the geometric form, motion state, and the approximate analytical solution of the ERSFD dynamic model, and increased its research ability.