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The density-fitted equation-of-motion (EOM) orbital-optimized second-order perturbation theory (DF-EOM-OMP2) method is presented for the first time. In addition, κ-DF-EOM-MP2 and κ-DF-EOM-OMP2 methods are implemented with the addition of κ-regularization. The accuracy of the DF-EOM-OMP2, κ-DF-EOM-MP2, and κ-DF-EOM-OMP2 methods are compared to the density-fitted EOM-MP2 (DF-EOM-MP2), EOM coupled-cluster (CC) singles and doubles (DF-EOM-CCSD), and EOM-CCSD with the triples excitation correction model [EOM-CCSD(fT)] for excitation energies of many closed- and open-shell chemical systems. The excitation energies computed using different test cases and methods were compared to the EOM-CCSD(fT) method and mean absolute errors (MAEs) are presented. The MAE values of closed- and open-shell cases (closed-shell organic chromophores set, open-shell set, peptide radicals set, and radical set) according to the EOM-CCSD(fT) method show that the κ-regularization technique yields highly accurate results for the first excited states. Our results indicate that the κ-DF-EOM-MP2 and κ-DF-EOM-OMP2 methods perform noticeably better than the DF-EOM-MP2 and DF-EOM-OMP2 methods. They approach the EOM-CCSD quality, at a significantly reduced cost, for the computation of excitation energies. Especially, the κ-DF-EOM-MP2 method provides outstanding results for most test cases considered. Overall, we conclude that the κ-versions of DF-EOM-MP2 and DF-EOM-OMP2 emerge as a useful computational tool for the study of excited-state molecular properties. [ABSTRACT FROM AUTHOR]

Currently, the largest ( r a max = 19.9 Å) and by far the most complicated (234 atoms, C1 symmetry, 696 independent geometrical parameters, and 27 261 interatomic terms) experimental molecular structure of a cage-type Si6Tip6 (Tip = 2,4,6-iPr3C6H2) isomer has been investigated in the gas phase by the electron diffraction method (Tav = 645 K) supplemented with theoretical simulations. A detailed analysis of the current possibilities for experimentally investigating large molecular structures is performed. A series of density functional theory approximations and the role of dispersion interactions have been benchmarked using the obtained data. Based on the refined geometry of Si6Tip6, various quantum-chemical methods have been applied for the investigation of the electronic structure of its Si6 core. In particular, natural bond orbital, quantum theory of atoms in molecules, interacting quantum atoms, fractional occupation number weighted density, and complete active space self-consistent field (CASSCF) methods were utilized. The diradical character of the molecule has been assessed by the UHF and CASSCF approximations. The problem of bonding between the hemispheroidal silicon atoms has been investigated. [ABSTRACT FROM AUTHOR]

Currently, the largest ( r a max = 19.9 Å) and by far the most complicated (234 atoms, C1 symmetry, 696 independent geometrical parameters, and 27 261 interatomic terms) experimental molecular structure of a cage-type Si6Tip6 (Tip = 2,4,6-iPr3C6H2) isomer has been investigated in the gas phase by the electron diffraction method (Tav = 645 K) supplemented with theoretical simulations. A detailed analysis of the current possibilities for experimentally investigating large molecular structures is performed. A series of density functional theory approximations and the role of dispersion interactions have been benchmarked using the obtained data. Based on the refined geometry of Si6Tip6, various quantum-chemical methods have been applied for the investigation of the electronic structure of its Si6 core. In particular, natural bond orbital, quantum theory of atoms in molecules, interacting quantum atoms, fractional occupation number weighted density, and complete active space self-consistent field (CASSCF) methods were utilized. The diradical character of the molecule has been assessed by the UHF and CASSCF approximations. The problem of bonding between the hemispheroidal silicon atoms has been investigated. [ABSTRACT FROM AUTHOR]

The accuracy of orbital-free density functional theory depends on the approximations made for a Kinetic Energy (KE) functional. Until now, the most accurate KEDFs are based on non-local kernels constructed from the linear response theory of homogeneous electron gas. In this work, we explore beyond the HEG by employing a more general kernel based on the jellium-with-gap model (JGM). The proposed functional incorporates several new features, such as (i) having the correct low momentum(q) limit of the response function for metals and semiconductors without any modeling term, (ii) the underlying kernel is density-independent, and most importantly, (iii) parameter-free. The accuracy and efficiency of the proposed JGM NL-KEDF have been demonstrated for several semiconductors and metals. The encouraging results indicate the utility and predictive power of the JGM kernel for NL KEDF developments. This approach is also physically appealing and practically useful as we have presented a general formalism to incorporate the gap kernel in all existing Lindhard-based functionals. [ABSTRACT FROM AUTHOR]

A major difficulty in quantum simulation is the adequate treatment of a large collection of entangled particles, synonymous with electron correlation in electronic structure theory, with coupled cluster (CC) theory being the leading framework for dealing with this problem. Augmenting computationally affordable low-rank approximations in CC theory with a perturbative account of higher-rank excitations is a tractable and effective way of accounting for the missing electron correlation in those approximations. This is perhaps best exemplified by the "gold standard" CCSD(T) method, which bolsters the baseline CCSD with the effects of triple excitations using considerations from many-body perturbation theory (MBPT). Despite this established success, such a synergy between MBPT and the unitary analog of CC theory (UCC) has not been explored. In this work, we propose a similar approach wherein converged UCCSD amplitudes are leveraged to evaluate energy corrections associated with triple excitations, leading to the UCCSD[T] method. In terms of quantum computing, this correction represents an entirely classical post-processing step that improves the energy estimate by accounting for triple excitation effects without necessitating new quantum algorithm developments or increasing demand for quantum resources. The rationale behind this choice is shown to be rigorous by studying the properties of finite-order UCC energy functionals, and our efforts do not support the addition of the fifth-order contributions as in the (T) correction. We assess the performance of these approaches on a collection of small molecules and demonstrate the benefits of harnessing the inherent synergy between MBPT and UCC theories. [ABSTRACT FROM AUTHOR]

Modeling excited state processes in molecular crystals is relevant for several applications. A popular approach for studying excited state molecular crystals is to use cluster models embedded in point charges. In this paper, we compare the performance of several embedding models in predicting excited states and S1–S0 optical gaps for a set of crystals from the X23 molecular crystal database. The performance of atomic charges based on ground or excited states was examined for cluster models, Ewald embedding, and self-consistent approaches. We investigated the impact of various factors, such as the level of theory, basis sets, embedding models, and the level of localization of the excitation. We consider different levels of theory, including time-dependent density functional theory and Tamm–Dancoff approximation (TDA) (DFT functionals: ωB97X-D and PBE0), CC2, complete active space self-consistent field, and CASPT2. We also explore the impact of selection of the QM region, charge leakage, and level of theory for the description of different kinds of excited states. We implemented three schemes based on distance thresholds to overcome overpolarization and charge leakage in molecular crystals. Our findings are compared against experimental data, G0W0-BSE, periodic TDA, and optimally tuned screened range-separated functionals. [ABSTRACT FROM AUTHOR]

Gaussian Approximation Potentials (GAPs) are a class of Machine Learned Interatomic Potentials routinely used to model materials and molecular systems on the atomic scale. The software implementation provides the means for both fitting models using ab initio data and using the resulting potentials in atomic simulations. Details of the GAP theory, algorithms and software are presented, together with detailed usage examples to help new and existing users. We review some recent developments to the GAP framework, including Message Passing Interface parallelisation of the fitting code enabling its use on thousands of central processing unit cores and compression of descriptors to eliminate the poor scaling with the number of different chemical elements. [ABSTRACT FROM AUTHOR]

Describing analytically the transport properties of electrolytes, such as their conductivity or the self-diffusion of the ions, has been a central challenge of chemical physics for almost a century. In recent years, this question has regained some interest in light of Stochastic Density Field Theory (SDFT) – an analytical framework that allows the approximate determination of density correlations in fluctuating systems. In spite of the success of this theory to describe dilute electrolytes, its extension to concentrated solutions raises a number of technical difficulties, and requires simplified descriptions of the short-range repulsion between the ions. In this article, we discuss recent approximations that were proposed to compute the conductivity of electrolytes, in particular truncations of Coulomb interactions at short distances. We extend them to another observable (the self-diffusion coefficient of the ions) and compare them to earlier analytical approaches, such as the mean spherical approximation and mode-coupling theory. We show how the treatment of hydrodynamic effects in SDFT can be improved, that the choice of the modified Coulomb interactions significantly affects the determination of the properties of the electrolytes, and that comparison with other theories provides a guide to extend SDFT approaches in this context. [ABSTRACT FROM AUTHOR]

The concept of high‐entropy materials has been introduced based on the idea that multiple principal components can be mixed through the increase in configurational entropy. Implicit in this idea is that the vibrational entropy, the other component of the mixing entropy, is small compared to the configurational entropy. To explore this relationship, we examined the mixing enthalpy, configurational entropy, and vibrational entropy of two binary ceramic systems—the transition metal carbides and transition metal diborides. We computed the vibrational entropy directly using the dynamical matrices obtained from density functional theory and the quasiharmonic approximation. The mixing vibrational entropy of the mixed diborides is at least as large as the configurational entropy while it is smaller for the carbides. Utilizing the phonon density of states, we further demonstrate the origin of the high mixing vibrational entropy arises because of a large number of new low‐frequency modes that appear in the diborides. Similar modes occur in the carbides but occur at larger frequencies. These differences ultimately arise because of the structural differences where metal atoms share nearest neighbors in the diborides, while they do not in the carbides. This increased vibrational mixing entropy dramatically enhances the mixing of the diborides and demonstrates that this type of entropy cannot be neglected when considering what stabilizes mixtures and provides a new perspective on what is considered high entropy. [ABSTRACT FROM AUTHOR]

A sharp step on a chaotic potential can enhance primordial curvature fluctuations on smaller scales to the O (10 − 2) to form primordial black holes (PBHs). The present study discusses an inflationary potential with a sharp step that results in the formation of PBHs in four distinct mass ranges. Also this inflationary model allows the separate consideration of observable parameters ns and r on the CMB scale from the physics at small scales, where PBHs formation occur. In this work we computed the fractional abundance of PBHs ( f PBH ) using the GLMS approximation of peak theory and also the Press–Schechter (PS) formalism. In the two typical mass windows, 10 − 13 M ⊙ and 10 − 11 M ⊙ , f PBH calculated using the GLMS approximation is nearly equal to 1 and that calculated via PS is of 10−3. In the other two mass windows 1 M ⊙ and 6 M ⊙ , f PBH obtained using GLMS approximation is 0.01 and 0.001 respectively, while f PBH calculated via PS formalism yields 10−5 and 10−6. The results obtained via GLMS approximation are found to be consistent with observational constraints. A comparative analysis of f PBH obtained using the GLMS perspective and the PS formalism is also included. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The fractal function approximation is an important branch of fractal theory. Compared with the ordinary function approximation, its advantage is that it can not only improve the accuracy of the approximation, but also maintain the fractional-dimensional characteristics of the objective function. However, in the process of studying the fractal function approximation, it is inevitable to calculate the fractal dimension of the sum function after adding multiple functions. This paper systematically sorts out and computes the fractal dimension of sum functions for different cases, so as to make a good theoretical foundation for fractal function approximation. Finally, considering that the change of aircraft aerodynamic value has fractional dimension characteristics, the use of the fractional function to approximate the aerodynamic numerical function has an inherent advantage, and the use of fractional calculus for aerodynamic modeling can more accurately measure the maximum angle of attack. Therefore, the application of fractional calculus and fractal function approximation theory in aerodynamic numerical modeling has been discussed in detail at the end of this paper, and the corresponding fractional aerodynamic modeling process is also demonstrated. [ABSTRACT FROM AUTHOR]

Summary: Methods from mainstream texts in calculus provide resources and opportunities to fit realistic curves to data from epidemics, such as Kermack and McKendrick's approximation, which is a shifted Verhulst curve. [ABSTRACT FROM AUTHOR]

Reverse roll coating is a high-speed coating process used in many applications. The single largest application for reverse roll coating is applying architectural paint to coils of sheet metal; however, in the converting industry reverse roll coating is used for some thermal coatings, adhesives, color laser paper receivers, wax coatings, magnetic coatings, vinyl dispersions for wallcovering, some specialty light sensitive applications, and many specialty products that require a high precision application process. This work presents a mathematical model for the steady flow of non-isothermal, incompressible, micropolar fluid in the metering nip of a reverse roll coater with velocity slip applied at both roll's surfaces. Lubrication approximation theory (LAT) is utilized to reduce the complexity of the non-dimensional equations. Exact solutions for pressure gradient, velocity, and temperature are achieved. At the same time, numerical techniques (Simpson's 3/8 rule with Newton Raphson method) are used to compute the pressure distribution, coating thickness, flow rate, and separation points. Effects of coupling number, slip parameter, microrotation, and velocity ratio on velocity, pressure, temperature, and pressure gradient are presented in graphs. It is observed from the current analysis that the intrinsic rotations of the fluid particles prove to be the controlling parameter of the pressure gradient, which significantly varies the coating thickness. Also, as a result of velocity slip, the flow rate declines and the coating thickness reduces as the fluid moves faster along the boundary walls. [ABSTRACT FROM AUTHOR]

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 < q < 1. It is proved that for any compact set 퓓 ⊂ ℂ, the rate of convergence is O(qn) as n → ∞. The sharpness of the obtained result is demonstrated. [ABSTRACT FROM AUTHOR]

The theory of fractal surfaces, in its basic setting, asserts the existence of a bivariate continuous function defined on a triangular domain. The extant literature on the construction of fractal surfaces over triangular domains use certain assumptions for the construction and deal primarily with the interpolation aspects. Working in the framework of fractal surfaces over triangular domains, this note has a two-fold target. Firstly, to revamp the existing constructions of fractal surfaces over triangular domains, and secondly to connect the idea of fractal surfaces further with the theory of approximation. To this end, in the same spirit of the so-called α -fractal functions on intervals and hyperrectangles, we study a salient subclass of fractal surfaces, which provides a parameterized family of bivariate fractal functions corresponding to a fixed continuous function defined on a triangular domain. Some elementary properties of the single-valued (linear and nonlinear) and multi-valued fractal operators associated with the α -fractal function formalism of the bivariate fractal functions on a triangular domain are recorded. A fractal approximation process, that is a sequence of single-valued fractal operators converging strongly to the identity operator on C (Δ , R) , the space of all real-valued continuous functions defined on a triangular domain Δ , is obtained. An approximation class of fractal functions, referred to as the fractal polynomials, is hinted at. The notion of α -fractal function and associated single-valued fractal operator in conjunction with appropriate stability results for Schauder bases provide Schauder bases consisting of self-referential functions for C (Δ , R) . [ABSTRACT FROM AUTHOR]

The Boussinesq equation and its related types are able to provide a significant explanation for a variety of different physical processes that are relevant to plasma physics, ocean engineering, and fluid flow. Within the framework of shallow water waves, the aim of this research is to find solutions for solitary waves using newly developed nonlocal models of Boussinesq’s equations. The extraction of bright and dark solitary wave solutions along with bright–dark hybrid solitary wave solutions is accomplished through the implementation of two integration algorithms. The general projective Riccati equations method and the enhanced Kudryashov technique are the ones that have been implemented as techniques. The enhanced Kudryashov method combines the benefits of both the original Kudryashov method and the newly developed Kudryashov method, which may generate bright, dark, and singular solitons. The Projective Riccati structure is determined by two functions that provide distinct types of hybrid solitons. The solutions get increasingly diverse as these functions are combined. The techniques that were applied are straightforward and efficient enough to provide an approximation of the solutions discovered in the research. Furthermore, these techniques can be utilized to solve various kinds of nonlinear partial differential equations in mathematical physics and engineering. In addition, plots of the selected solutions in three dimensions, two dimensions, and contour form are provided. [ABSTRACT FROM AUTHOR]

In this paper, the Fuzzy Sumudu Transform Iterative method (FSTIM) was applied to find the exact fuzzy solution of the fuzzy space-time fractional telegraph equations using the Fuzzy Caputo Fractional Derivative operator. The Telegraph partial differential equation is a hyperbolic equation representing the reaction-diffusion process in various fields. It has applications in engineering, biology, and physics. The FSTIM provides a reliable and efficient approach for obtaining approximate solutions to these complex equations improving accuracy and allowing for fine-tuning and optimization for better approximation results. The work introduces a fuzzy logic-based approach to Sumudu transform iterative methods, offering flexibility and adaptability in handling complex equations. This innovative methodology considers uncertainty and imprecision, providing comprehensive and accurate solutions, and advancing numerical methods. Solving the fuzzy space-time fractional telegraph equation used a fusion of the Fuzzy Sumudu transform and iterative approach. Solution of fuzzy fractional telegraph equation finding analytically and interpreting its results graphically. Throughout the article, whenever we draw graphs, we use Mathematica Software. We successfully employed FSTIM, which is elegant and fast to convergence. [ABSTRACT FROM AUTHOR]

This paper proposes a novel segmented underdamped asymmetric tristable stochastic resonance (SUATSR) system. The research involves deriving the equivalent potential function for the underdamped background and analyzing the system’s output saturation. Results demonstrate that the SUATSR system effectively mitigates output saturation issues, enhancing the output amplitude. The study also employs adiabatic approximation theory to derive theoretical expressions for mean first passage time (MFPT) and spectral amplification (SA). The paper discusses the impact of varying system parameters on these performance indicators. Numerical simulations of the classical tristable stochastic resonance (CTSR), segmented underdamped tristable stochastic resonance (SUTSR) and SUATSR systems validate the theoretical derivations. Furthermore, the three systems are applied to different bearing fault tests and optimized using the adaptive genetic algorithm (AGA). The results demonstrate that the SUATSR system outperforms the other two systems, exhibiting higher SA and signal-to-noise ratio (SNR) values at the fault frequency. This highlights the SUATSR system’s superior noise resistance and its effectiveness in detecting and enhancing fault signals. [ABSTRACT FROM AUTHOR]

This paper introduced a new definition of rough fuzzy sets based on a fuzzy ideal l defined on a fuzzy approximation space (X, R). This definition considered the degree r ∈ [0, 1] in defining the after sets and the before sets of the arbitrary fuzzy relation R on the finite set of objects X. It is shown that this new type of roughness is a generalization to many of previous definitions in the fuzzy case and also in the ordinary case. As an application, it is given a rough fuzzy‐connected set notion depending on some degree r ∈ [0, 1]. [ABSTRACT FROM AUTHOR]

This work investigates piecewise Vand der Pol oscillator under the arbitrary order, piecewise derivatives, and power nonlinearities to present a novel idea of piecewise systems using the classical‐power‐law randomness and classical Mittag–Leffler‐law‐randomness. We are able to study the dynamics of piecewise Van der Pol oscillators because the oscillator is one of the prototype models of the systems. It has become famous for examining the dynamics of the systems. To obtain the approximate solution for the aforementioned system, the famous Adams–Bashforth method is applied. The algorithm for obtaining numerical solutions is presented as well as the obtained results were presented graphically along with their different behavior. This oscillator is successfully applied to the model, a diversity of physical processes like nonlinear electronic circuits. Despite the simplicity of this oscillator, the behavior of the models is extremely prosperous and research is still ongoing. [ABSTRACT FROM AUTHOR]

Blade coating is a process that applies a fluid to a surface using a fixed blade. Among various coating technologies, blade coating offers significant economic advantages. It is commonly employed in the production of paper, information preservation, the application of coloring agents, and the manufacture of photographic films and magnetic storage devices. The novelty of this work lies in the investigation of the blade coating process for an electrically conducting Oldroyd 4‐constant liquid, incorporating velocity slippage at the blade surface in an area previously underexplored. The mathematical equations are modeled with the use of Lubrication Approximation Theory (LAT) and the normalized equations of the Oldroyd 4‐constant fluid are numerically solved by the Matlab built‐in function bvp4c using Regula‐Falsi Method. The impact of sundry parameters on physical quantities is examined through graphical representation. It is noted from the theoretical results that for the fixed value of the MHD parameter (M = 2.5), the coating thickness and blade load increased by 31% and 1648% respectively, for plane coater. For the exponential coater, these values increased by 29% and 1618% from their Newtonian value. These findings offer new insights into optimizing the blade coating process for complex fluid systems. [ABSTRACT FROM AUTHOR]

Roll coating is a popular method for applying a thin coating film to a continuous substrate, such as paper or foil. This study aims to investigate analytically the fluid flow through counter rotating rolls using a non-Newtonian model. The governing equation for the constitutive of the Oldroyd-B as a non-Newtonian fluid with momentum equation has been discovered. The nondimensionalizing of the fluid flow equations is done by using suitable dimensionless parameters. The equations are then simplified by using Lubrication Approximation Theory (LAT) to obtain the solution for various quantities. Analytical solutions for the velocity distribution, pressure and pressure gradients have been obtained by regular perturbation method. Numerical integration has been used to calculate separation point, coating thickness, separation force and power input of the fluid. The effect of involved parameter such as Weissenberg number We and viscosity on velocities, pressure gradient and pressure is presented in a variety of graphs and several other results are shown in tables. It has been observed that as the modified capillary number N C a 2 increases, the separation point moves toward the nip region, whereas the coating thickness declines. It has been witnessed that the We serves as the controlling parameter for flow rate, pressure distribution and power input, whereas for the coating thickness, the separation force increases, which are important for coating industries. [ABSTRACT FROM AUTHOR]

This paper is devoted to the approximation of a function by a trigonometric polynomial based on its inaccurate values at selected points. Two methods of observation are considered. The first method is to make observations at points evenly distributed on the segment where the function is specified. The second method is to take observations at the points of division into a finite number of equal parts of the neighbourhoods of the selected points. Upper estimates of the standard deviation of the function from trigonometric polynomials are constructed and the rate of their convergence is estimated. Differences were found in the computational complexity of these approximations and in the number of observations of the function values at the selected points. Thus, the problem of approximating a function from inaccurate observations of their values at selected points is a multi-criteria one and its solution depends on the choice of observation points. [ABSTRACT FROM AUTHOR]

We aim to explore the bubble nature of the exotic nucleus 20 N within the microscopic antisymmetrized molecular dynamics (AMD) approach. Constraining its structural parameters, we analyse its static properties. Subsequently, we use the AMD infused finite range distorted wave Born approximation theory to calculate the Coulomb breakup of 20 N as an indirect approach to estimate the 19 N (n , γ) 20 N radiative capture rate. [ABSTRACT FROM AUTHOR]

The authors present the results of the research combining the methods of approximation theory and optimal decision theory. Namely, a solution to the optimization problem for the biharmonic Poisson integral in the upper half-plane is considered one of the most optimal solutions to the biharmonic equation in Cartesian coordinates. The approximate properties of the biharmonic Poisson operator in the upper half-plane on the classes of quasi-smooth functions are obtained in the form of an exact equality for the deviation of quasi-smooth functions from the positive operator under consideration. [ABSTRACT FROM AUTHOR]

The problem of machining complex surfaces with non-ball-end cutters by strip-width-maximization machining is formulated as a kind of surface fitting problem in which the tool surface envelope feature line approximates the design surface under the movement transform. The theory of surface envelope−approximation is proposed as a general method for optimizing tool movement in single-contact strip-width-maximization machining of sculptured surfaces with non-ball-end cutters. Based on the surface moving frame, the velocity equations and transformation matrices for the tool motion relative to the workpiece, described by the motion-invariant parameters of the tool surface and design surface, are derived. A functional extremum model for optimizing the tool position ensures continuous and symmetrical motion relative to the workpiece to achieve the highest machining efficiency and accuracy. Finally, a Matlab-based simulation example verifies the machining efficiency and accuracy of the envelope approximation theory. [ABSTRACT FROM AUTHOR]

Teachers' ability to respond to student thinking is important for effective teaching. Considering the importance of the decisions and steps to be taken by teachers in responding to student thinking in teaching environments where conceptual understanding that brings meaningful learning is aimed at, it is understood that prospective teachers' skills in responding to student thinking should be improved. One of the current approximation of practices used in the development of prospective teachers' instructional skills is scripting task practices. The aim of this study is to reveal the alignment between theory and approximation of practice in the context of conceptual and procedural understanding of prospective mathematics teachers' ability to respond to student thinking through scripting tasks. Being one of the qualitative research methods, in the present study, we used a case study to examine how prospective teachers responded to students' ideas. For this purpose, we asked 38 prospective mathematics teachers to write how they would take an instructional step on a problem related to student thinking and then justify it by completing a scenario prepared on the same problem. We subjected the data to content analysis and open coding. The findings revealed that nearly half of the prospective teachers aspired to teach for conceptual understanding and were able to demonstrate this in their scripting task practices. There were prospective teachers who aimed for conceptual teaching in theory, but could not reflect this focus in their approximation of practices. There were also prospective teachers who pursued only a procedural focus, both in theory and in approximation of practice. There were no prospective teachers who pursued a procedural focus in theory and realized conceptual teaching as an approximation of practice. We discuss the findings in the context of the relevant literature. [ABSTRACT FROM AUTHOR]

In this paper, we collect some recent results on the approximation properties of generalized sampling operators and Kantorovich operators, focusing on pointwise and uniform convergence, rate of convergence, and Voronovskaya-type theorems in weighted spaces of functions. In the second part of the paper, we introduce a new generalization of sampling Durrmeyer operators including a special function ρ which satisfies certain assumptions. For the family of newly constructed operators, we obtain pointwise convergence, uniform convergence and rate of convergence for functions belonging to weighted spaces of functions. [ABSTRACT FROM AUTHOR]

The paper presents research results combining the methods of approximation theory and optimal decision theory. Namely, the optimization problem for the biharmonic Poisson integral in the upper half-plane is considered as one of the most optimal solutions to the biharmonic equation in Cartesian coordinates. The approximate properties of the biharmonic Poisson operator in the upper half-plane on the classes of quasi-smooth functions are obtained in the form of an exact equality for the deviation of quasi-smooth functions from the positive operator under consideration. [ABSTRACT FROM AUTHOR]

In this work we present a formalism based on scalar Green's functions to deal with electromagnetic scattering problems. Although the formulations of the Mie theory and Born approximations in terms of electromagnetic scattering are well known and relevant, they have certain disadvantages: complexity, computational time, few symmetries, etc. Therefore, the study with scalar Green's functions allows dealing with these problems with greater simplicity and efficiency. However, the information provided by the vector formulation is sacrificed. Nevertheless, different cases of electromagnetic scattering of dielectric media with different dimensions, geometries and refractive indices will be presented. Thus, we will be able to verify the capacity of this scalar method in predicting light-scattering problems. [ABSTRACT FROM AUTHOR]

Motivated by kernel methods in machine learning theory, we study the uniform approximation of functions from reproducing kernel Hilbert spaces by Bernstein operators. Rates of approximation are provided in terms of the function norm in the reproducing kernel Hilbert space. A case study of contracting properties of the Bernstein operators is also presented. [ABSTRACT FROM AUTHOR]

Roll coating plays a significant role in various coating industries such as magnetic records, wallpapers, wrapping, adhesive tapes, books and magazines, photographic and plastic films. The thin layer coating of a magnetohydrodynamic (MHD) viscous hybrid nanofluid by passing through the space between two co-rotating rolls has been studied in an isothermal and incompressible analysis. The governing equation of mass and momentum are obtained then dimensionless using lubrication approximation theory (LAT). The velocity, pressure gradient, and pressure distribution are determined by the exact solution. Using Simpson's (3/8) rule for numerical integration, the complex integral is examined. Important engineering parameters including power and roll separating force delivered by the rolls to the fluid are also estimated numerically. Raising the volume fraction of nanoparticles raises the pressure distribution and pressure gradient while having little effect on the velocity profile. It seems that the magnetic field and hybrid nanofluid, both seem very advantageous for the efficient roll coating process, controlling the separation force, power input, and distance between the attachment and separation point. [ABSTRACT FROM AUTHOR]

In several applications, ranging from computational geometry and finite element analysis to computer graphics, there is a need to approximate functions defined on triangular domains rather than rectangular ones. For this purpose, frequently used interpolation methods include barycentric interpolation, piecewise linear interpolation, and polynomial interpolation. However, the use of polynomial interpolation methods may suffer from the Runge phenomenon, affecting the accuracy of the approximation in the presence of equidistributed data. In these situations, the constrained mock-Chebyshev least squares approximation on rectangular domains was shown to be a successful approximation tool. In this paper, we extend it to triangular domains, by using both Waldron and discrete Leja points. This paper is dedicated to Len Bos on the occasion of his retirement. Len, for us, is a master of mathematics and also a big friend. He introduced us to the fascinating world of "finding good interpolation nodes and effective interpolation strategies", mostly in the multivariate setting. The set of points we are using in this paper, Waldron and Leja, have been introduced to us by him and we hope that this note can be of some interest for him and all people working on approximation theory. [ABSTRACT FROM AUTHOR]

In approximation theory, classical discrete operators, like generalized sampling, Szász‐Mirak'jan, Baskakov, and Bernstein operators, have been extensively studied for scalar functions. In this paper, we look at the approximation of curves by a class of discrete operators, and we exhibit graphical examples concerning several cases. The topic has useful implications about the computer graphics and the image processing: We discuss applications on the approximation and the reconstruction of curves in images. [ABSTRACT FROM AUTHOR]

In this article, we calculate the value-at-risk (VaR) for large portfolios in intensity models, where the idiosyncratic and systematic risk exposures as well as the impact of past default losses are subsumed into the intensity processes. The adopted method is based on the theory of saddlepoint approximation for continuous-time Markov processes whose transition densities and distribution functions can be approximated in closed forms. A simple example with theoretical and numerical results is presented in the end. [ABSTRACT FROM AUTHOR]

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 C m are derived, enabling the calculation of relative dielectric permittivities ( ϵ r ′ ) and the dynamic viscosities (η) of the MRSs. It is demonstrated that ϵ r ′ 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 ϵ r ′ , 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. [ABSTRACT FROM AUTHOR]

The article is devoted to the problem of approximation of classes of periodic functions by rectangular linear means of Fourier series. Asymptotic equalities are found for upper bounds of deviations in the uniform metric of rectangular Fejér means on classes of periodic functions of several variables generated by sequences that tend to zero at the rate of geometric progression. In one-dimensional cases, these classes consist of Poisson integrals, namely functions that can be regularly extended in the fixed strip of a complex plane. [ABSTRACT FROM AUTHOR]

On the real line, we study the generalized Dunkl harmonic analysis depending on a parameter . The case of corresponds to the usual Dunkl harmonic analysis. All constructions depend on the parameter . The differences and moduli of smoothness are defined using a generalized translation operator. The Sobolev space and the -functional are defined using a differential-difference operator. An approximate Jackson-type inequality is proved. The equivalence of the -functional and the modulus of smoothness is established. [ABSTRACT FROM AUTHOR]

In this paper, periodic structure theory with the wave approximation is used to present a simple approximate solution technique to characterize wave motions propagating in periodic line-supported cylinders in the circumferential direction. To develop displacement functions that adhere to Floquet's concept, a combination of simple beam functions of the bounding modes(BM)of propagation bands(PB) of a periodic beam are formulated. This study is developed for the motion type known as a plane wave. Consequently, only waves that are simply propagating without attenuation are taken into account. The circumferential modes of a single periodic curved panel (unit cell) have been defined in terms of classical beam functions that satisfy Floquet's wave principle, but the axial modes are thought to be sinusoidal waves. Displacement functions are used to strain energy and kinetic energy expressions. The Rayleigh- Ritz technique is then used to generate the stiffness and mass matrices of the periodic unit cell. By solving the eigenvalue equation, phasefrequency relation is obtained. It has also been possible to predict the bounding frequencies of the PB for various axial modes of a cylindrical shell with a certain circumferential phase constant. The findings are then put through comparison with those outlined in the literature. Further, the bounding frequency results for the optimum periodic curved panel which gives lowest frequency for a given cylindrical shell geometry are also found out. It has been found that the current beam function with a periodic structure (PS) wave approach can find the bounding frequencies (BF) and bounding modes (BM) with reasonable accuracy. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the asymptotic convergence of the sequences. Concretely, we will give the asymptotically statistical convergence of sequences. The asymptotically statistical convergence is studied in many papers including papers [3], [5], [23], [26], [29], [39], [41]. In this paper, we introduce and investigate the concepts of asymptotically deferred statistical equivalent functions of order $ \alpha $ and strong asymptotically deferred statistical equivalent functions of order $ \alpha $ defined on discrete countable amenable semigroups, which generalizes some of the results known in the literature. This is achieved by introducing the Folner sequences and sequences of non-negative integers. Based on this concept we apply it to the approximation theory, and we have proved the Korovkin type theorem and study the rate of convergence for positive linear operators which are deferred statistically convergent of order $ \alpha $. [ABSTRACT FROM AUTHOR]

In the realm of specialized functions, the allure of q -calculus beckons to many scholars, captivating them with its prowess in shaping models of quantum computing, noncommutative probability, combinatorics, functional analysis, mathematical physics, approximation theory, and beyond. This study explores a new idea called the multidimensional q -Hermite polynomials, using different q -calculus techniques. Numerous properties and novel findings regarding these polynomials are derived, encompassing their generating function, series representations, recurrence relations, q -differential formula, and operational principles. Further, we proved that these polynomials are quasi-monomial in q -aspect. As the applications, these findings are subsequently employed to address connection between the multidimensional q -Hermite polynomials and the two-variable q -Legendre polynomials for the first time. Various characterizations are examined, as well the graphical representations of the two-variable q -Legendre polynomials are provided by the surface plots and graphs of distribution of zeros for the q -Legendre polynomials with some specific set of parameters are shown using Mathematica. Our investigations shed light on the intricate nature of these polynomials, elucidating their behaviour and facilitating deeper understanding within the realm of q -calculus. [ABSTRACT FROM AUTHOR]

With the progress in Global Navigation Satellite Systems (GNSS) technology, accurate geoid modelling has started to play an essential role in geodetic applications such as establishing height datum as a continuous surface model and related vertical control for infrastructure projects. Thus, numerous geoid modelling methods have been offered since 1990's, each of them has its own algorithm and approximation theories. Classical Stokes-Helmert is one of the most well-known methods all over the world by geodetic communities. However, a user-friendly software package of the method is not publicly accessible on the Internet. Therefore, a compact and user-friendly software package "CSHSOFT" is developed and presented for scholars in this field. A fractionated programming strategy has been treated to build individual components striving high accuracy and computational efficiency for geoid heights. Subsequently, the CSHSOFT is simply tested to construct a geoid model in the mountainous area in Auvergne test-bed where several geoid modelling techniques are implemented. Afterward, the new geoid model of the region is externally evaluated by GNSS-levelling data in terms of rigorous orthometric heights. The fitting statistics of 2.75 cm and 0.36 ppm in absolute and relative height differences fairly indicate that the CSHSOFT is a vigorous tool for gravimetric geoid modelling, and can be comfortably employed for geoscientific and technical studies. [ABSTRACT FROM AUTHOR]

This paper deals with global well-posedness of the solution to the complex short pulse equation. We first use regularized technology and the approximation argument to prove the local existence and uniqueness of this equation. Then, based on conserved quantities and energy analysis, we show that the solution can be extended globally in time for suitably small initial data. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We give investigations on a kernel function approximation problem arising from learning theory and show the convergence rate from the view of classical Fourier analysis. First, we provide the general definition for a modulus of smoothness and a K -functional, and show that they are equivalent. In particular, we give explicit representation for some moduli of smoothness. Second, we establish some Jackson-type inequalities for the approximation error associated with some non-radial kernels. Also we apply these results to some concrete classical kernel function spaces and give Jackson-type inequalities for some concrete RKHS approximation problems. Finally, we apply these discussions to learning theory and describe the learning rates with the moduli of smoothness. The tools we used are Fourier analysis and the semigroup operator. The results show that the Jackson-type inequalities of approximation by some radial kernel functions on compact set with nonempty interiors cannot be expressed with the classical moduli of smoothness. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let H be a separable Hilbert space and let { x n } be a sequence in H that does not contain any zero elements. We say that { x n } is a Bessel-normalizable or frame-normalizable sequence if the normalized sequence { x n ‖ x n ‖ } is a Bessel sequence or a frame for H, respectively. In this paper, several necessary and sufficient conditions for sequences to be frame-normalizable and not frame-normalizable are proved. Perturbation theorems for frame-normalizable sequences are also proved. As applications, we show that the Balazs–Stoeva conjecture holds for Bessel-normalizable sequences. Finally, we apply our results to partially answer the open question raised by Aldroubi et al. as to whether the iterative system { A n x ‖ A n x ‖ } n ≥ 0 , x ∈ S associated with a normal operator A : H → H and a countable subset S of H, is a frame for H. In particular, if S is finite, then we are able to show that { A n x ‖ A n x ‖ } n ≥ 0 , x ∈ S is not a frame for H whenever { A n x } n ≥ 0 , x ∈ S is a frame for H. [ABSTRACT FROM AUTHOR]

In this paper, we describe an algorithm for fitting an analytic and bandlimited closed or open curve to interpolate an arbitrary collection of points in R 2 . The main idea is to smooth the parametrization of the curve by iteratively filtering the Fourier or Chebyshev coefficients of both the derivative of the arc-length function and the tangential angle of the curve and applying smooth perturbations, after each filtering step, until the curve is represented by a reasonably small number of coefficients. The algorithm produces a curve passing through the set of points to an accuracy of machine precision, after a limited number of iterations. It costs O(N log N) operations at each iteration, provided that the number of discretization nodes is N. The resulting curves are smooth, affine invariant, and visually appealing and do not exhibit any ringing artifacts. The bandwidths of the constructed curves are much smaller than those of curves constructed by previous methods. We demonstrate the performance of our algorithm with several numerical experiments. [ABSTRACT FROM AUTHOR]

We present a scheme for finding all roots of an analytic function in a square domain in the complex plane. The scheme can be viewed as a generalization of the classical approach to finding roots of a function on the real line, by first approximating it by a polynomial in the Chebyshev basis, followed by diagonalizing the so-called “colleague matrices.” Our extension of the classical approach is based on several observations that enable the construction of polynomial bases in compact domains that satisfy three-term recurrences and are reasonably well-conditioned. This class of polynomial bases gives rise to “generalized colleague matrices,” whose eigenvalues are roots of functions expressed in these bases. In this paper, we also introduce a special-purpose QR algorithm for finding the eigenvalues of generalized colleague matrices, which is a straightforward extension of the recently introduced structured stable QR algorithm for the classical cases (see Serkh and Rokhlin 2021). The performance of the schemes is illustrated with several numerical examples. [ABSTRACT FROM AUTHOR]