Approximation Theory Research Articles

Best Approximation of Unbounded Functions by Modulus of Smoothness

In this paper, we study the approximation of unbounded functions in a weighted space by modulus of smoothness using various linear operators. We establish direct theorems for such approximations and analyze the properties of the modulus of smoothness within the same space. Specifically, we investigate the behavior of the modulus of smoothness under different types of linear operators, including the Bernstein-Durrmeyer operator, the Fejer operator, and the Jackson operator. We also provide a detailed analysis of the convergence rate of these operators. Furthermore, we discuss the relationship between the modulus of smoothness and the Lipschitz constant of a function. Our findings have important implications for the field of approximation theory and may help to inform future research in this area.

A manifold two-sample test study: integral probability metric with neural networks

Abstract Two-sample tests are important areas aiming to determine whether two collections of observations follow the same distribution or not. We propose two-sample tests based on integral probability metric (IPM) for high-dimensional samples supported on a low-dimensional manifold. We characterize the properties of proposed tests with respect to the number of samples $n$ and the structure of the manifold with intrinsic dimension $d$. When an atlas is given, we propose a two-step test to identify the difference between general distributions, which achieves the type-II risk in the order of $n^{-1/\max \{d,2\}}$. When an atlas is not given, we propose Hölder IPM test that applies for data distributions with $(s,\beta )$-Hölder densities, which achieves the type-II risk in the order of $n^{-(s+\beta )/d}$. To mitigate the heavy computation burden of evaluating the Hölder IPM, we approximate the Hölder function class using neural networks. Based on the approximation theory of neural networks, we show that the neural network IPM test has the type-II risk in the order of $n^{-(s+\beta )/d}$, which is in the same order of the type-II risk as the Hölder IPM test. Our proposed tests are adaptive to low-dimensional geometric structure because their performance crucially depends on the intrinsic dimension instead of the data dimension.

Heat transfer analysis of the non-Newtonian polymer in the calendering process with slip effects

In this paper, a non-isothermal study of the calendering processes is presented using Carreau–Yasuda model along with nonlinear slip condition introduced at the upper roll surface. The flow equations for the problem are developed and converted into dimensionless form with the help of dimensionless variables and then finally simplified by a well-known lubrication approximation theory. The final equations are solved numerically using “bvp4c” to find stream function and velocity profiles, while the hybrid numerical method which is the combination of shooting and finite difference methods is used to solve the energy equation. Graphs show the impact of the concerned material parameters on various quantities of interest. The pressure distribution decreases with the increasing values of the slip parameter and Weissenberg number. The mechanical variables show an increasing trend with the increasing values of the slip parameter and Weissenberg number. The temperature distribution increases with an increase in the Brinkman number, while temperature shows declining trend near the roll surface with the increasing values of the slip parameter. The force separating the two rollers, total power input into both rolls, increase with the increasing values of the Weissenberg number and slip parameters. The results show that the Newtonian model predicts higher pressure in the nip zone than the Carreau–Yasuda model. It is interesting to note that for the case of shear thinning, the Carreau–Yasuda model predicts 30% less pressure in the nip region when compared to the Newtonian model.

Fast Linear Canonical Transform for Nonequispaced Data

The investigations of the discrete and fast linear canonical transform (LCT) are becoming one of the hottest research topics in modern signal processing and optics. Among them, the fast calculation of LCT for non-uniform data is one of key problems. Focus on this problem, a new fast algorithm of the LCT has been proposed in this paper firstly by interpolation and approximation theory. The proposed algorithms can calculate quickly the LCT of the data, whether the input or output data is uniform. Secondly, the complexity and precision of derived algorithms have been analyzed for different situations. Finally, the experimental results are presented to verify the correctness of the obtained results.

JARNÍK TYPE THEOREMS ON MANIFOLDS

AbstractLet $\psi $ be a decreasing function. We prove zero-infinity Hausdorff measure criteria for the set of dual $\psi $ -approximable points and for the set of inhomogeneous multiplicative $\psi $ -approximable points on nondegenerate planar curves. Our results extend theorems of Huang [‘Hausdorff theory of dual approximation on planar curves’, J. reine angew. Math.740 (2018), 63–76] and Beresnevich and Velani [‘A note on three problems in metric Diophantine approximation’, in: Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, 631 (American Mathematical Society, Providence, RI, 2015), 211–229] from s-Hausdorff measure, where $s\in \mathbb R$ , to the more general g-Hausdorff measure, where g is a suitable class of dimension functions.

Resonance of ellipsoidal billiard trajectories and extremal rational functions

We study resonant billiard trajectories within quadrics in the d-dimensional Euclidean space. We relate them to the theory of approximation, in particular the extremal rational functions on the systems of d intervals on the real line. This fruitful link enables us to prove fundamental properties of the billiard dynamics and to provide a comprehensive study of a large class of non-periodic trajectories of integrable billiards. A key ingredient is a functional-polynomial relation of a generalized Pell type. Applying further these ideas and techniques to s-weak billiard trajectories, we come to a functional-polynomial relation of the same generalized Pell type.

Asphaltene precipitation described with a Yukawa SAFT-VR/MSA equation of state

We combine the Statistical Associating Fluid Theory for Potentials of Variable Range (SAFT-VR) and the Mean Spherical Approximation (MSA) to develop an equation of state to model asphaltene precipitation in petroleum fluids including the measured asphaltene onset pressures (AOP’s) of a reservoir live oil. In this model, dispersive solvent–solvent and asphaltene–solvent interactions in reservoir fluid mixtures are represented by Yukawa potentials. The approach is based in the modeling of polar and ionic interactions via Yukawa potentials, as originally proposed by the Generalized Mean Spherical Approximation Theory, and also in recent studies in molecular simulation of charged fluids using non-Ewald effective pair interactions, as in the Wolf method.

Mechanical behaviour of rubber blocks

This study investigates the behaviour of rubber blocks bonded between two plates under combined compression and shear loading, using experimental and numerical analyses, and also approximate analytical theories. First, experimental data from a series of compression and shear tests of rubber blocks with different aspect ratios are presented. Next, numerical simulations are carried out with three-dimensional finite element (FE) models, allowing insight to be gained into the stress and strain fields within the blocks. Existing analytical theories for blocks under compression and combined compressive and shear loading are then reviewed, and their accuracy is evaluated against test and numerical results. The study shows that those theories accounting for the effect of the axial shortening of the blocks provide a better description of the combined compression and shear behaviour, compared to theories, developed for laminated structural bearings with many thin rubber layers, that ignore this effect. An improved theory is also proposed, which better describes the effects of the bulging of the compressed blocks on their shear and flexural parameters and provides a better fit to experimental and numerical results.

Approximating constant potential DFT with canonical DFT and electrostatic corrections.

The complexity of electrochemical interfaces has led to the development of several approximate density functional theory (DFT)-based schemes to study reaction thermodynamics and kinetics as a function of electrode potential. While fixed electrode potential conditions can be simulated with grand canonical ensemble DFT (GCE-DFT), various electrostatic corrections on canonical, constant charge DFT are often applied instead. In this work, we present a systematic derivation and analysis of the different electrostatic corrections on canonical DFT to understand their physical validity, implicit assumptions, and scope of applicability. Our work highlights the need to carefully address the suitability of a given model for the problem under study, especially if physical or chemical insight in addition to reaction energetics is sought. In particular, we analytically show that the different corrections cannot differentiate between electrostatic interactions and covalent or charge-transfer interactions. By numerically testing different models for CO2 adsorption on a single-atom catalyst as a function of the electrode potential, we further show that computed capacitances, dipole moments, and the obtained physical insight depend sensitively on the chosen approximation. These features limit the scope, generality, and physical insight of these corrective schemes despite their proven practicality for specific systems and energetics. Finally, we suggest guidelines for choosing different electrostatic corrections and propose the use of conceptual DFT to develop more general approximations for electrochemical interfaces and reactions using canonical DFT.

Periodic DFT study on heavy metals Cu(II) and Pb(II) atoms adsorption on Na-montmorillonite (010) edge surface

Heavy metal (HM) pollution is of great concern currently because it has been recognized as a potential threat to air, water,and soil. Adsorption is one of the most effective strategies for removing heavy metals. Montmorillonite as the adsorbent with a low cost and high thermal stability has received much attention. The adsorption of heavy metal Copper [Cu(II)] and Plumbum [Pb(II)] atoms on Na-montmorillonite hydroxylated (010) edge surface was analyzed using the generalized gradient approximation and density functional theory in the supercell technique. The coverage dependence of the adsorption structure and energetics was systematically investigated for a wide range of coverage from 0 to 1.0 monolayer (ML). Among all the possible adsorption sites, the most stable adsorption site for Cu(II) atom was a three-fold hollow site while the most favorable adsorption site for Pb(II) atom was a three-fold hollow site and followed by a two-fold bridge site. The adsorption energy increased with increasing coverage of Pb(II) atoms, indicating higher stability of surface adsorption and a tendency for the formation of adsorbate island clusters with increasing coverage. However, the adsorption energy of Cu(II) atoms decreased with increasing coverage. In the coverage range of 0 < Θ ≤ 0.5 ML the recovery capacity of the Na-montmorillonite for the heavy metal atoms was in the order of Cu(II)>Pb(II), while the recovery capacity of the Na-montmorillonite in the coverage range of 0.5 < Θ ≤ 1 ML was in the order of Pb(II)>Cu(II). The other aspects of the Cu(II) and Pb(II)/Na-montmorillonite (010) systems, including adsorption geometry, differential charge distribution, and electronic density of states were also studied and discussed in detail.

The shrinking target problem for matrix transformations of tori: Revisiting the standard problem

Let T be a d×d matrix with real coefficients. Then T determines a self-map of the d-dimensional torus Td=Rd/Zd. Let {En}n∈N be a sequence of subsets of Td and let W(T,{En}) be the set of points x∈Td such that Tn(x)∈En for infinitely many n∈N. For a large class of subsets (namely, those satisfying the so called bounded property (B) which includes balls, rectangles, and hyperboloids) we show that the d-dimensional Lebesgue measure of the shrinking target set W(T,{En}) is zero (resp. one) if a natural volume sum converges (resp. diverges). In fact, we prove a quantitative form of this zero-one criteria that describes the asymptotic behaviour of the counting function R(x,N):=#{1≤n≤N:Tn(x)∈En}. The counting result makes use of a general quantitative statement that holds for a large class measure-preserving dynamical systems (namely, those satisfying the so called summable-mixing property). We next turn our attention to the Hausdorff dimension of W(T,{En}). In the case the subsets En are balls, rectangles or hyperboloids we obtain precise formulae for the dimension. These shapes correspond, respectively, to the simultaneous, weighted and multiplicative theories of classical Diophantine approximation. The dimension results for balls generalises those obtained in [25] for integer matrices to real matrices. In the final section, we discuss various problems that stem from the results proved in the paper.

Finite-temperature equilibrium density profiles of integrable systems in confining potentials.

We study the equilibrium density profile of particles in two one-dimensional classical integrable models, namely hard rods and the hyperbolic Calogero model, placed in confining potentials. For both of these models the interparticle repulsion is strong enough to prevent particle trajectories from intersecting. We use field theoretic techniques to compute the density profile and their scaling with system size and temperature, and we compare them with results from Monte Carlo simulations. In both cases we find good agreement between the field theory and simulations. We also consider the case of the Toda model in which interparticle repulsion is weak and particle trajectories can cross. In this case, we find that a field theoretic description is ill-suited and instead, in certain parameter regimes, we present an approximate Hessian theory to understand the density profile. Our work provides an analytical approach toward understanding the equilibrium properties for interacting integrable systems in confining traps.

Attenuation Characterization of Terahertz Waves in Foggy and Rainy Conditions at 0.1–1 THz Frequencies

Investigating the absorption and scattering effects of atmospheric particles, i.e., raindrops and fog droplets, is required to establish a comprehensive and accurate channel model. However, for long-distance communication in outdoor scenarios, research on the propagation characterization of fog and rain attenuation in the terahertz (THz) band is insufficient. In this study, fog and rain attenuation characterization with different conditions are characterized. First, fog attenuation at different temperatures and diverse visibility is explored using Rayleigh approximation theory and Mie theory. The results demonstrate that visibility and frequency have a stronger effect than temperature on fog attenuation. Then, rain attenuation as a function of rainfall rate is theoretically determined using Mie theory and the Joss, M-P, and Weibull distribution. The results show that rainfall rate and frequency have greater impact than raindrop distribution on rain attenuation. There are large differences in rainfall attenuation under diverse distributions. Accurate fog and rainfall attenuation information can be used to better estimate path loss and the link budget for terahertz communication in outdoor scenarios.

The joy of mathematical discovery: To the 120th anniversary of Academician A.N. Kolmogorov

This article is dedicated to the memory of the outstanding mathematician of the 20th century, Andrei Nikolaevich Kolmogorov, whose 120th birthday is celebrated this year. The author describes in detail the formation of Kolmogorov as a scientist, the significance of Moscow State University in the scientific life and pedagogical activity of the famous mathematician. Kolmogorov’s contribution to such branches of mathematics as classical analysis, topology, geometry, approximation theory, functional analysis, and probability theory is analyzed, and the contribution of his scientific school to Russian science, including not only mathematics and physics but also the humanities, is shown.

Possible Violation of the Spin-Hamiltonian Concept in Interpreting the Zero-Field Splitting.

The majority of experimental data in electron spin resonance and molecular magnetism are interpreted in terms of the spin-Hamiltonian (SH) formalism. However, this is an approximate theory that requires a proper testing. In the older variant, the multielectron terms are used as a basis in which the D-tensor components are evaluated by employing the second-order perturbation theory (PT) for nondegenerate states; here, the spin-orbit interaction expressed via the spin-orbit splitting parameter λ serves for the perturbation. The model space is restricted only to the fictitious spin functions |S, M⟩. In the case of the orbital (quasi) degeneracy of the ground term, the PT tends to diverge and the subtracted D, E, and g parameters are false. In the second variant working in the "complete active space" (CAS), the spin-orbit coupling operator is involved by the variation method resulting in the spin-orbit multiplets (energies and eigenvectors) The multiplets can be evaluated either by applying ab initio CASSCF + NEVPT2 + SOC calculations or by using semiempirical generalized crystal-field theory (with the one-electron SOC operator depending upon ξ). The resulting states can be projected onto the subspace of the spin-only kets in the way that the eigenvalues stay invariant. Such an effective Hamiltonian matrix can be reconstructed using six independent components of the symmetric D-tensor from which the D and E values are obtained by solving linear equations. The eigenvectors of the spin-orbit multiplets in the CAS allow determining the dominating composition of the spin projection─cumulative weights of |±M⟩. These are conceptually different from those generated by the SH alone. It is shown that in some cases, the SH theory works satisfactorily for a series of transition-metal complexes; however, sometimes it fails. The ab initio calculations on the SH parameters are compared with the approximate generalized crystal-field theory conducted at the experimental geometry of the chromophore. In total, 12 metal complexes have been analyzed. One of the criteria that assesses the validity of SH is the projection norm N for spin multiplets (this has not to be far from 1). Another criterion is the gap in the spectrum of the spin-orbit multiplets that separates the hypothetical (fictitious) spin-only manifold from the rest of the states.

Further reflections about the “Born” term used in thermodynamic models for electrolytes

This paper is a follow-up of previous work about the “Born” term, which must not be confused with the famous Born equation for ion solvation energy. The “Born” term is an empirical formula that is often introduced in explicit-solvent models for electrolytes in conjunction with a contribution for ion-ion interactions taken from a model with implicit solvent. An inconsistency is found for the expression commonly used in the literature. The “Born” term is supposed to describe the effect of ion–solvent interactions. The relevance of the “Born” term for that purpose is further investigated by considering various different situations: The transfer of an ion from pure water into a pure molten salt is examined; then, the nonprimitive mean spherical approximation theory for mixtures of ions and dipoles (MSA-ID) is used to get further insight in the case of a waterlike solvent and, besides, in the limit of low density and low coupling where the MSA becomes exact. The various situations considered in this work further indicate that this term is unsuitable. The possibility of deriving a formula in the spirit of the original Born equation is also considered.

Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves

We extend the pseudoholomorphic curve methods from Floer theory to infinite-dimensional phase spaces and use our results to prove the existence of a forced time-periodic solution to a general Hamiltonian PDE with regularizing nonlinearity. In particular, when the nonlinearity is sufficiently regularizing, bounded and time-periodic, we prove an infinite-dimensional version of Gromov-Floer compactness by using ideas from the theory of Diophantine approximations to overcome the small divisor problem. Furthermore, in the case when the infinite-dimensional phase space is a product of a finite-dimensional closed symplectic manifold with linear symplectic Hilbert space, we prove a cup-length estimate for the number of periodic solutions.

Virtual lattice resonance of a single nanoresonator in a metal nanoslit

We study the transmission through a subwavelength metallic slit loaded with a single nanoresonator. To gain physical insight into the problem a theoretical model combining the dipole approximation and image theory is developed. The model shows that the coupling between the single nanoresonator and the slit's cavity modes serves as a localized analog to an infinite nanoresonator array. This virtual array supports a surface image-lattice resonance due to the coherent self-scattering of the single nanoresonator. Thereby, it may lead to the ability to mimic many recently reported intriguing physical phenomena of real surface-lattice resonances in nanoresonator arrays, by a single-subwavelength system. We specifically show that it leads to enhanced light-matter interaction, and to the appearance of an extraordinary transmission window. The theoretical results are in good agreement with full-wave numerical simulations.

Revisiting the Mechanisms of Charge Transport in Solutions of Redox-Active Molecules Using Computer Simulations: When and Why Do Analytical Theories Fail?

Understanding charge transport is essential for the development of energy-storage applications. This work introduces a new theoretical methodology to model diffusive charge transport in solutions of redox-active molecules by combining Langevin dynamics for the spatial degrees of freedom and a master-equation formalism to describe the electron-hopping events between redox molecules. The model is used to analyze the effects of the concentration of the redox molecules and the strength of the intermolecular interactions on the charge-transport mechanism. In the past, the rate of charge transport has been modeled with the analytical Dahms-Ruff equation; however, this is a mean-field equation, whose range of validity has not been tested with less approximate theories. We show that the Dahms-Ruff equation fails to quantitatively predict the diffusion coefficient for charge transport for large concentrations of the redox species and high bimolecular electron-transfer rates, i.e., the most relevant conditions for energy-storage applications. Under these conditions, the diffusion coefficient for charge transport obtained from simulations is larger than that predicted from the Dahm-Ruff equation because of the formation of transient clusters of redox molecules. Also, intermolecular interactions, which are not taken into account by the Dahms-Ruff equation, play a central role in the charge transport of redox species. We show that the apparent diffusion coefficient experiences a maximum with respect to the strength of the intermolecular attractions. This maximum is traced back to the formation of clusters and their two opposite effects on the diffusion coefficient: electron hopping is fast within a cluster but inefficient between neighboring clusters.

Error-Correction Method for High-Throughput Sizing of Nanoscale Vesicles with Single-Molecule Localization Microscopy.

Single-molecule localization microscopy (SMLM) allows super-resolution imaging, mapping, counting, and sizing of biological nanostructures such as cell organelles and extracellular vesicles (EVs), but sizing structures smaller than ∼100 nm can be inaccurate due to single-molecule localization error caused by distortion of the point spread function and limited photon number. Here we demonstrate a method to correct localization error when sizing vesicles and other spherical nanoparticles with SMLM and compare sizing results using two vesicle labeling schemes. We use mean approximation theory to derive a simple equation using full width at half-maximum (FWHM) for correcting particle sizes measured by two-dimensional SMLM, validate the method by sizing streptavidin-coated polystyrene nanobeads with the SMLM technique dSTORM with and without error correction, using transmission electron microscopy (TEM) for comparison, and then apply the method to sizing small seminal EVs. Nanobead sizes measured by dSTORM became increasingly less accurate (larger than TEM values) for beads smaller than 50 nm. The error-correction method reduced the size difference versus TEM from 15% without error correction to 7% with error correction for 40 nm beads, from 44% to 9% for 30 nm beads, and from 66% to 15% for 20 nm beads. Seminal EVs were labeled with a lipophilic membrane dye (MemBright 700) and with an Alexa Fluor 488-anti-CD63 antibody conjugate, and were sized separately using both dyes by dSTORM. Error-corrected exosome diameters were smaller than uncorrected values: 72 nm vs 79 nm mean diameter with membrane dyes; 84 nm vs 97 nm with the antibody-conjugated dyes. The mean error-corrected diameter was 12 nm smaller when using the membrane dye than when using the antibody-conjugated dye likely due to the large size of the antibody. Thus, both the error-correction method and the compact membrane labeling scheme reduce overestimation of vesicle size by SMLM. This error-correction method has a low computational cost as it does not require correction of individual blinking events, and it is compatible with all SMLM techniques (e.g., PALM, STORM, and DNA-PAINT).