Approximation Theory Research Articles

Generalization Analysis of CNNs for Classification on Spheres.

Deep learning based on deep convolutional neural networks (CNNs) is extremely efficient in solving classification problems in speech recognition, computer vision, and many other fields. But there is no enough theoretical understanding about this topic, especially the generalization ability of the induced CNN algorithms. In this article, we develop some generalization analysis of a deep CNN algorithm for binary classification with data on spheres. An essential property of the classification problem is the lack of continuity or high smoothness of the target function associated with a convex loss function such as the hinge loss. This motivates us to consider the approximation of functions in the Lp space with 1≤ p ≤ ∞. We provide rates of Lp -approximation when the approximated function lies in a Sobolev space and then present generalization bounds and learning rates for the excess misclassification error of the deep CNN classification algorithm. Our novel analysis is based on efficient cubature formulae on spheres and other tools from spherical analysis and approximation theory.

Switched Energy and Neural Network-Based Approach for Swing-Up and Tracking Control of Double Inverted Pendulum

The double inverted pendulum (DIP) system is a benchmark underactuated mechanical system. The control problem of the DIP system is challenging since it not only has high nonlinearity, but also has underactuation degree equal to two. In this article, a switched energy-based swing-up and neural network (NN) controller is proposed to solve the swing-up and tracking control problem of the DIP system when the desired position is time varying. The implementation of the proposed controller can be divided into three steps. In step one, the energy-based control method is adopted to drive the first pendulum from the downward position to the neighborhood of the upright position. In step two, the sliding mode control method is adopted to stabilize the first pendulum. Meanwhile, a method combining energy-based control and “equivalent cart” is adopted to swing up the second pendulum. In step three, on the basis of approximate nonlinear output regulation theory, the NN controller is used for achieving satisfactory position tracking performance. Finally, the effectiveness of our design is verified by experimental results.

Norm-Attainable Polynomials: Characterizations and Properties in Orthogonal Polynomial Families

Norm-attainable polynomials play a crucial role in various mathematical contexts. This research paper investigates the characterizations and properties of norm-attainable polynomials in different orthogonal polynomial families. We explore their behavior, convexity, and positive definiteness. Additionally, we establish their norm-attainability in specific intervals for various weight functions. The findings contribute to a deeper understanding of norm-attainable polynomials and their applications in approximation theory and mathematical modeling.

Further investigation on fixed point theorems via C-class functions in extended b-metric spaces

The purpose of this research project is to develop new theories, discuss, and extend some recent common fixed point results established when the underlying ambient space is an extended b-metric space and the contraction condition involves a new class of ψ-φ-C-contraction type mappings where ψ is the altering distance function and φ is the ultra-altering distance function. The unique fixed point theorems for such mappings in the setting of ψ-φ-complete metric spaces are proven. We also prove the fixed point theorem in partially ordered metric spaces. Moreover, some examples supporting the main results are given. Our results extend and generalize corresponding results in the literature. The start of the development of the theory of fixed points is tied to the end of the 19th century. The method of successive approximations is used in order to prove the solution's existence and uniqueness at the beginning of differential and integral equations. This branch of nonlinear analysis has been developed through various classes of spaces, such as metric spaces, topological spaces, probabilistic metric spaces, fuzzy metric spaces, and others. In developing the theory of fixed points, achievements are applied in various sciences, such as optimization, economics, and approximation theory. A very important step in the development of fixed point theory was taken by A.H. Ansari through the introduction of a C-class function. Using C-class functions, we generalize some known fixed point results, and Kamran et al. introduced a new intuitive concept of distance measure to extend the notion of b-metric space by further weakening the triangle inequality.

Finite coating thickness of Prandtl fluid in non-isothermal reverse roll coating process

PurposeThe coating process is broadly employed in the manufacturing of wallpapers, adhesive tapes, wrapping, protection of fabrics and metals, X-ray and photographic films, beautification, books and magazines, film foils, magnetic records, coated paper, etc.Design/methodology/approachIn this study, an incompressible flow of non-Newtonian fluid is modeled to inspect the rheological behavior of finite coating thickness in the reverse roll coating process. With the assistance of lubrication approximation theory (LAT), the dimensionless form of governing expressions is simplified. Exact solutions for distributions for velocity, flow rate, temperature and pressure gradient attained utilizing perturbation technique and their variation is presented as well as discussed in graphs. Meanwhile, some important factors from an engineering perspective including coating thickness and transition point were calculated mathematically and are displayed in a tabular manner. Also, streamlines are drawn to observe the flow pattern.FindingsPrandtl fluid parameters provide a controlling factor to regulate the flow rate, velocity, coating thickness, and pressure gradient leading to an efficient coating process. Moreover, the Brinkman number and Prandtl fluid parameters significantly improve the temperature distribution.Originality/valueIn the literature, this study fills a gap in the theoretical prediction of coating thickness rheologically influenced by Prandtl fluid in reverse roll coating process.

Enhanced magneto-optical properties of Fe3O4@Au nanoparticles and its reverse core-shell nanostructure embedded in host matrix SiO2

We studied the effect of electric and magnetic polarizability of Fe3O4@Au and the reverse structure Au@Fe3O4 core-shell spherical nanostructures embedded in SiO2. Employing electrostatic approximation and Maxwell-Garnett effective medium theory, we evaluate the electric and magnetic polarizability, the refractive index and absorbance as a function of radiation energy. The modified Drude-Lorentz form and Lorentz model used to evaluate ϵ(ω) and μ(ω), respectively. For fixefd r2 = 30 nm, β = 0.875, εh = 3.9 and f = 0.001 maximum absorbance obtained at E = 2.742 eV for Au@Fe3O4 core-shell and at E = 2.937 eV for the reverse structure with β = 0.578. The ratio of the maximum absorbance peak of Au@Fe3O4 core-shell to Fe3O4@Au core-shell gives 3.5. Both graphs of n2 and absorbance possess three sets of peaks. All sets of resonance peaks of Fe3O4@Au found in the visible region. For the revers structure the first two sets of resonance peaks found in the visible region and the third set of resonance peaks in UV region. The peaks arise due to the coupling between the surface plasmon of the gold metal with Fe3O4∕SiO2 interfaces and the interaction of incident radiation with magnetic dipole moment of the magnetic semiconducting nanoparticles. The findings may be used in applications that require the combined plasmonic and magnetic effects such as drug delivery.

Comparative numerical and analytical computations for magnetized pseudoplastic materialwith roll-coating applications

The phenomenon of wire coating occurs in various applications in manufacturing systems, cables, drawing of sheets, polymer industries etc. The current investigation deals with the study of the reverse coating phenomenon of magnetized non-Newtonian fluid. A pseudoplastic material with applications of magnetic force is considered and the rheological impact based on the non-Newtonian model is investigated. The thin layer of pseudoplastic material is coated when it travels due to the small gap conveying by two reverse coating rollers. The formulation of problem is based on lubrication approximation theory. Both analytical and numerical simulations are performed, and the corresponding results are compared. The numerical assessment is based on the shooting technique while analytical outcomes are obtained using perturbation technique. The effects of the governing parameters on the flow profiles are depicted. The main parameters influence the flow are the Weissenberg number and velocity ratio parameter.

Unveiling the Potential of Sheffer Polynomials: Exploring Approximation Features with Jakimovski–Leviatan Operators

In this article, we explore the construction of Jakimovski–Leviatan operators for Durrmeyer-type approximation using Sheffer polynomials. Constructing positive linear operators for Sheffer polynomials enables us to analyze their approximation properties, including weighted approximations and convergence rates. The application of approximation theory has earned significant attention from scholars globally, particularly in the fields of engineering and mathematics. The investigation of these approximation properties and their characteristics holds considerable importance in these disciplines.

Deep estimation for Q⁎ with minimax Bellman error minimization

In this paper we consider the estimation of optimal state-action value function Q⁎ with ReLU ResNet based on minimax Bellman error minimization. We construct the non-asymptotic error bounds for the minimax estimator and the estimated Q function induced by the estimated greedy policy. To bound the Bellman residual error, we guarantee the approximation errors based on deep approximation theory and the statistical ones by utilizing empirical processes taking into account the Markov decision process dependency. We provide a novel generalization bound with dependent data and an approximation bound in the Hölder class which are of independent interest. This bound depends on the sample size, the ambient dimension, the width and depth of the neural network, which can bring prior insights into tuning these hyper-parameters to achieve a desired convergence rate in practice. Furthermore, the bound circumvents the curse of dimensionality if the distribution of state-action pairs is assumed to be supported on a set of low intrinsic dimension.

TaBE: Decoupling spatial and spectral processing with Taylor’s unfolding method in the beamspace domain for multi-channel speech enhancement

In recent years, significant advancements have been made in neural beamforming, leveraging spectral and spatial cues to enhance their performance in multi-channel speech enhancement. When the frame-wise processing mechanism is required, there exists a trade-off between performance and algorithmic delay for existing all-neural beamformers. However, from the perspective of multi-source information fusion, the network is often encapsulated into a black box to entangle and fuse the spatial and spectral features into a non-linear feature space, which hinders our understanding of how they work collaboratively for target speech extraction. In this regard, this paper proposes to decouple the spatial and spectral domain processing inspired by Taylor’s approximation theory. Specifically, we reformulate the time-variant beamforming defined in the spatial domain into the adaptive weighting and mixing of different beam components in the beamspace domain. This reformulation enables us to model the recovery of target speech as a weighted sum operation in the beamspace domain, where each beam component is associated with an introduced unknown term for residual interference cancellation. By virtue of Taylor’s series expansion, the recovery process can be decomposed into the superimposition of the 0th-order non-derivative and high-order derivative terms, where the former acts as spatial filtering in the spatial domain, and the latter serves as a residual interference canceller in the spectral domain. We conduct extensive experiments on the spatialized LibriSpeech and L3DAS Challenge datasets. Experimental results show that, compared with existing advanced approaches, the proposed method not only achieves competitive performance in terms of multiple objective metrics but also provides feasible guidance in multi-channel speech enhancement pipeline design.

Developing machine-learned potentials to simultaneously capture the dynamics of excess protons and hydroxide ions in classical and path integral simulations.

The transport of excess protons and hydroxide ions in water underlies numerous important chemical and biological processes. Accurately simulating the associated transport mechanisms ideally requires utilizing abinitio molecular dynamics simulations to model the bond breaking and formation involved in proton transfer and path-integral simulations to model the nuclear quantum effects relevant to light hydrogen atoms. These requirements result in a prohibitive computational cost, especially at the time and length scales needed to converge proton transport properties. Here, we present machine-learned potentials (MLPs) that can model both excess protons and hydroxide ions at the generalized gradient approximation and hybrid density functional theory levels of accuracy and use them to perform multiple nanoseconds of both classical and path-integral proton defect simulations at a fraction of the cost of the corresponding abinitio simulations. We show that the MLPs are able to reproduce abinitio trends and converge properties such as the diffusion coefficients of both excess protons and hydroxide ions. We use our multi-nanosecond simulations, which allow us to monitor large numbers of proton transfer events, to analyze the role of hypercoordination in the transport mechanism of the hydroxide ion and provide further evidence for the asymmetry in diffusion between excess protons and hydroxide ions.

Propagation characteristics of a high-power beam in weakly relativistic-ponderomotive thermal quantum plasma

The present work explores the propagation characteristics of high-power beams in weakly relativistic-ponderomotive thermal quantum plasma. A q-Gaussian laser beam is taken in the present investigation. The quasi-optics equation obtained in the present study is solved through a well-established Wentzel–Kramers–Brillouin approximation and paraxial theory approach for obtaining the second-order differential equation describing the behavior of beam width of the laser beam. Further, a numerical simulation of this second-order differential equation is carried out for determining the behavior of the beam width with dimensionless distance for established laser–plasma parameters. The comparison of the present study is made with ordinary quantum plasma and classical relativistic plasma cases.

Mathematical Model of the Flow in a Nanofiber/Microfiber Mixed Aerosol Filter

A new mathematical model of an aerosol fibrous filter, composed of a variety of nano- and microfibers, is developed. The combination of nano- and microfibers in a mixed-type filter provides a higher overall quality factor compared with filters with monodisperse fibers. In this paper, we propose a mathematical model of the flow of an incompressible fluid in a porous region consisting of a set of cylinders of various diameters in the range of nano- and micrometers to describe a mixed-type aerosol filter. The flow domain is a rectangular periodic cell with one microfiber and many nanofibers. The motion of the carrier medium is described by the boundary value problem in Stokes flow approximation with the no-slip boundary condition for microfibers and the slip condition for nanofibers. The boundary element method taking into account the slip and non-slip conditions is developed. The calculated velocity field, streamlines, vorticity distribution, and drag of separate fibers and the entire periodic cell are presented. Numerical results for the drag force of the porous medium of a mixed-type filter for the various ratios of mass proportion of nano- and microfibers, porosity, and filtration velocity are presented. The obtained results are compared with the analytical formulas based on the approximate theory of filtration of bimodal filters and with known experimental data. It is shown that with an increase in the mass fraction of nanofibers, the total drag force of the cell increases, while the relative contribution of nanofibers to the total drag force tends toward the value that is less than unity. An approximate analytical formula for the drag coefficient of a mixed aerosol filter is derived. The developed flow model and analytical formulas allow for estimating the aerodynamic drag of a mixed filter composed by nano- and microfibers.

Electron impact fine-structure cross-section calculations of Cu and their applications in the diagnostics of laser-produced Cu-plasma through the collisional radiative model

We present the calculations of electron impact excitation cross-sections for the transitions from the ground state of Cu 3d104sS1/22 to the excited fine structure energy levels of the configurations 3d94s2,3d94s4p,3d104l,(l=p,d,f),3d105l,(l=s,p,d),3d10nl,(n=6,7;l=s,p) using the fully relativistic distorted wave theory. In addition, cross-sections for the transitions from the metastable states 3d94s2D5/2,3/22 to the upper excited levels are also presented. For calculating these cross-sections, we have obtained the atomic wave functions of copper using the multi-configuration Dirac–Fock approximation theory. To ensure the accuracy of these wave functions, we compared our calculated oscillator strengths of some dipole-allowed transitions of Cu with the available earlier reported values. We have utilized all the calculated electron impact fine-structure level excitation cross-sections from the ground as well as the metastable states of Cu to develop a fine structure-resolved collisional-radiative (CR) model for the diagnostics of laser-produced Cu plasma at atmospheric pressure. The model has been applied for diagnostics by coupling it with the spatially resolved optical emission spectroscopy (OES) measurements of Singh and Sharma [Phys. Plasmas 23 122104 (2016)] in the presence and absence of a static magnetic field. For diagnostics, the intensities of three OES observed strong atomic emission lines of Cu viz, 510.5, 515.3, and 521.8 nm have been utilized in the absence and presence of a magnetic field at different axial lengths (z = 0.5–6.5 mm) from the Cu target. Also, these intensities are corrected using the internal reference self-absorption method to overcome the effect of radiation trapping. The normalized intensities of the considered lines obtained from the CR model have been compared with their corresponding OES-measured values to extract the plasma parameters, i.e., electron temperature and electron density. Further, the obtained electron temperatures from the CR model are compared with the corresponding values reported in the experimental paper of Singh and Sharma [Phys. Plasmas 23 122104 (2016)] obtained using the Boltzmann plot method.

Radiative properties of non-spherical opacifiers doped in silica aerogels for high-temperature thermal insulation

Silica aerogel, a super-insulating material, has garnered significant attention due to its extensive thermal insulation applications. However, the high-temperature performance of silica aerogels is compromised due to their infrared transparency. This study tackles this issue by investigating the influence of opacifier shape on the insulation properties of silica aerogel composites. Employing the Discrete Dipole Approximation (DDA) theory, the radiative properties of opacifiers of varying shapes are investigated. The study also employs a theoretical model to ascertain the effective thermal conductivity of silica aerogels doped with non-spherical opacifiers. To counter oxidation at ultra-high temperatures, an innovative ellipsoidal [email protected] core–shell opacifier is introduced. Non-spherical opacifiers are further optimized to identify the ideal particle diameters and volume fractions for specific temperature ranges. The investigation uncovers that non-spherical carbon black opacifiers with low shape parameter ratios considerably enhance the insulating performance at elevated temperatures. [email protected] core–shell opacifiers, distinguished by high inner/outer radius ratios, offer superior thermal stability and great infrared radiation extinction. In summary, this study offers a pragmatic approach to fabricating aerogels with enhanced insulation properties at high temperatures.

Gravity from thermodynamics: Optimal transport and negative effective dimensions

We prove an equivalence between the classical equations of motion governing vacuum gravity compactifications (and more general warped-product spacetimes) and a concavity property of entropy under time evolution. This is obtained by linking the theory of optimal transport to the Raychaudhuri equation in the internal space, where the warp factor introduces effective notions of curvature and (negative) internal dimension. When the Reduced Energy Condition is satisfied, concavity can be characterized in terms of the cosmological constant \LambdaΛ; as a consequence, the masses of the spin-two Kaluza-Klein fields obey bounds in terms of \LambdaΛ alone. We show that some Cheeger bounds on the KK spectrum hold even without assuming synthetic Ricci lower bounds, in the large class of infinitesimally Hilbertian metric measure spaces, which includes D-brane and O-plane singularities. As an application, we show how some approximate string theory solutions in the literature achieve scale separation, and we construct a new explicit parametrically scale-separated AdS solution of M-theory supported by Casimir energy.

Hybrid Threshold Denoising Framework Using Singular Value Decomposition for Side-Channel Analysis Preprocessing.

The traces used in side-channel analysis are essential to breaking the key of encryption and the signal quality greatly affects the correct rate of key guessing. Therefore, the preprocessing of side-channel traces plays an important role in side-channel analysis. The process of side-channel leakage signal acquisition is usually affected by internal circuit noise, external environmental noise, and other factors, so the collected signal is often mixed with strong noise. In order to extract the feature information of side-channel signals from very low signal-to-noise ratio traces, a hybrid threshold denoising framework using singular value decomposition is proposed for side-channel analysis preprocessing. This framework is based on singular value decomposition and introduces low-rank matrix approximation theory to improve the rank selection methods of singular value decomposition. This paper combines the hard threshold method of truncated singular value decomposition with the soft threshold method of singular value shrinkage damping and proposes a hybrid threshold denoising framework using singular value decomposition for the data preprocessing step of side-channel analysis as a general preprocessing method for non-profiled side-channel analysis. The data used in the experimental evaluation are from the raw traces of the public database of DPA contest V2 and AES_HD. The success rate curve of non-profiled side-channel analysis further confirms the effectiveness of the proposed framework. Moreover, the signal-to-noise ratio of traces is significantly improved after preprocessing, and the correlation with the correct key is also significantly enhanced. Experimental results on DPA v2 and AES_HD show that the proposed noise reduction framework can be effectively applied to the side-channel analysis preprocessing step, and can successfully improve the signal-to-noise ratio of the traces and the attack efficiency.

Approximating smooth and sparse functions by deep neural networks: Optimal approximation rates and saturation

Constructing neural networks for function approximation is a classical and longstanding topic in approximation theory. In this paper, we aim at constructing deep neural networks with three hidden layers using a sigmoidal activation function to approximate smooth and sparse functions. Specifically, we prove that the constructed deep nets with controllable magnitude of free parameters can reach the optimal approximation rate in approximating both smooth and sparse functions. In particular, we prove that neural networks with three hidden layers can avoid the phenomenon of saturation, i.e., the phenomenon that for some neural network architectures, the approximation rate stops improving for functions of very high smoothness.

Analytical study of viscoelastic fluid during forward roll coating process under lubrication approximation theory

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 [Formula: see text] 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.

A Rough Estimation of Acoustics of the Cuboidal Room with Impedance Walls

The cuboidal room acoustics field is modelled with the Fourier method. A combination of uniform, impedance boundary conditions imposed on walls is assumed, and they are expressed by absorption coefficient values. The absorption coefficient, in the full range of its values in the discrete form, is considered. With above assumptions, the formula for a rough estimation of the cuboidal room acoustics is derived. This approximate formula expresses the mean sound pressure level as a function of the absorption coefficient, frequency, and volume of the room separately. It is derived based on the least-squares approximation theory and it is a novelty in the cuboidal room acoustics. Theoretical considerations are illustrated via numerical calculations performed for the 3D acoustic problem. Quantitative results received with the help of the approximate formula may be a point of reference to the numerical calculations.