Approximation Theory Research Articles

Calculated ionization energies, orbital eigenvalues (HOMO), and related QSAR descriptors of organic molecules: a set of 61 experimental values enables elimination of systematic errors and provides realistic error estimates.

Ionization energies (IEs) of organic compounds come in different forms-adiabatic, vertical, as electrode potentials, or as orbital eigenvalues via Koopmans' theorem. They have been linked to the reactivity towards electrophiles and have been used to quantitatively describe electron transfer processes. The de novo prediction of IEs is only meaningful when an estimate of the prediction's uncertainty is included. Pulsed-field ionization (PFI) experiments have quantified adiabatic IEs with unprecedented precision. In this work, a new set of PFI-derived IEs is compiled from the literature as a benchmark for prediction methods. This set includes many common functional groups, a size range from diatomics to two aromatic rings, and IEs between 7 and 14 eV. The first-principles CCSD(T)/CBS protocol presently used reproduces these values within 0.05 eV. For adiabatic IEs and vertical IEs/orbital eigenvalues predicted using approximate density functional theory (DFT), linear regression models are proposed, so that IEs calculated using different methods can be directly compared on a physical scale. This elimination of systematic errors improves the error statistics and allows the performance of predicted IEs to be evaluated if used in quantitative structure-property or -activity relationships, as the latter implicitly correct a descriptor's bias. Owing to the structural scope of the test set, the minimum and maximum deviations from experiment should correspond to those expected for common organic molecules. Deviations from reference values found for orbital eigenvalues but also for IEs calculated explicitly with HF or semi-empirical MO methods were as large as 0.5 eV to 2.0 eV. Such large errors could also propagate into quantitative structure-property models, as shown in illustrative examples of oxidation rate constants in solution.

A generalization of modified $$\alpha $$-Bernstein operators and its related estimations and errors

AbstractIn the present article, we introduce a novel generalization of modified Bernstein operators which is again a positive linear operator. We show the necessary and sufficient condition for the convergence of these operators. We also study some other approximation properties of these operators using standard tools of approximation theory.

Existence of weak solutions for nonlocal Dirichlet problems via Young measure theory

This article investigates the existence of weak solutions for a class of nonlocal problems with Dirichlet boundary conditions. The proof of the existence result relies on Galerkin's approximation and Young's measure theory. For more information see https://ejde.math.txstate.edu/Volumes/2024/73/abstr.html

Fast and Efficient Lunar Finite Element Gravity Model

In this paper, the finite element method (FEM) is integrated with orthogonal polynomial approximation in high-dimensional spaces to innovatively model the Moon’s surface gravity anomaly. The aim is to approximate solutions to Laplace’s classical differential equations of gravity, employing classical Chebyshev polynomials as basis functions. Using classical Chebyshev polynomials as basis functions, the least-squares approximation was used to approximate discrete samples of the approximation function. These test functions provide an understanding of errors in approximation and corresponding errors due to differentiation and integration. These test functions provide an understanding of errors in approximation and corresponding errors due to differentiation and integration. The first application of this project is to substitute the globally valid classical spherical harmonic series of approximations with locally valid series of orthogonal polynomial approximations (i.e., using the FEM approach). With an error tolerance set at 10−9ms−2, this method is used to adapt the gravity model radially upwards from the lunar surface. The results showcase a need for a higher degree of approximation on and near the lunar surface, with the necessity decreasing as the radius increases. Notably, this method achieves a computational speedup of five orders of magnitude when applying the method to radial adaptation. More intrinsically, the second application involves using the methodology as an effective tool in solving boundary value problems. Specifically, this approach is implemented to solve classical differential equations involved with high-precision, long-term orbit propagation. This application provides a four-order-of-magnitude speedup in computational time while maintaining an error within the 10−10ms−2 error range for various orbit propagation tests. Alongside the advancements in orthogonal approximation theory, the FEM enables revolutionary speedups in orbit propagation without compromising accuracy.

Interpretability as Approximation: Understanding Black-Box Models by Decision Boundary

Currently, interpretability methods focus more on less objective human-understandable semantics. To objectify and standardize interpretability research, in this study, we provide notions of interpretability based on approximation theory. We first define explainable models in terms of explicitness and then use completeness to define interpretability, thereby turning interpretability into the process of approximating black-box models with interpretable models. In particular, we think that the decision boundary of a classification model is equivalent to its interpretability. Next, we implement this approximation interpretation on multilayer perceptrons (MLPs) and then propose to use the MLP as a universal interpreter to explain other complex black-box models. Compared to the LIME method, which can only extract local linear features, our method is global and therefore termed as GIME. Extensive experiments demonstrate the effectiveness of our approaches.

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

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

On the Lanczos Method for Computing Some Matrix Functions

The study of matrix functions is highly significant and has important applications in control theory, quantum mechanics, signal processing, and machine learning. Previous work has mainly focused on how to use the Krylov-type method to efficiently calculate matrix functions f(A)β and βTf(A)β when A is symmetric. In this paper, we mainly illustrate the convergence using the polynomial approximation theory for the case where A is symmetric positive definite. Numerical results illustrate the effectiveness of our theoretical results.

Accumulation points of normalized approximations

Building on classical aspects of the theory of Diophantine approximation, we consider the collection of all accumulation points of normalized integer vector translates of points qα with α∈Rd and q∈Z. In the first part of the paper we derive measure theoretic and Hausdorff dimension results about the set of α whose accumulation points are all of Rd. In the second part we focus primarily on the case when the coordinates of α together with 1 form a basis for an algebraic number field K. Here we show that, under the correct normalization, the set of accumulation points displays an ordered geometric structure which reflects algebraic properties of the underlying number field. For example, when d=2, this collection of accumulation points can be described as a countable union of dilates (by norms of elements of an order in K) of a single ellipse, or of a pair of hyperbolas, depending on whether or not K has a non-trivial embedding into C.

Effective medium theory for viscoelasticity of soft jammed solids

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

Borel Summation Can Be Controlled by Critical Indices

We consider application of the self-similarity principle in approximation theory under the conditions of asymptotic scale-invariance. For the effective summation of the asymptotic series methods, an iterative Borel summation with self-similar iterated roots is applied. The approximants follow from the self-similarity considerations and behave asymptotically as a power-law satisfying the asymptotic scale invariance. Optimal conditions on convergence of the sequence of approximants are imposed through the critical indices defined from the approximants. The indices are understood as control parameters for the optimal convergence of the asymptotic series. Such interpretation of the indices leads to an overall improvement of accuracy in calculations of the indices. The statement is supported by fifteen examples from condensed matter physics, quantum mechanics and field theory.

Hydrodynamization and resummed viscous hydrodynamics

In this paper, I review our current understanding of the applicability of hydrodynamics to modeling the quark–gluon plasma (QGP), focusing on the question of hydrodynamization/thermalization of the QGP and the anisotropic hydrodynamics (aHydro) far-from-equilibrium hydrodynamic framework. I discuss the existence of far-from-equilibrium hydrodynamic attractors and methods for determining attractors within different hydrodynamical frameworks. I also discuss the determination of attractors from exact solutions to the Boltzmann equation in relaxation time approximation and effective kinetic field theory applied to quantum chromodynamics. I then present comparisons of the kinetic attractors with the attractors obtained in standard second-viscous hydrodynamics frameworks and aHydro. I demonstrate that, due to the resummation of terms to all orders in the inverse Reynolds number, the aHydro framework can describe both the weak- and strong-interaction limits. I then review the phenomenological application of aHydro to relativistic heavy-ion collisions using both quasiparticle aHydro and second-order viscous aHydro. The phenomenological results indicate that aHydro provides a controlled extension of dissipative relativistic hydrodynamics to the early-time far-from-equilibrium stage of heavy-ion collisions. This allows one to better describe the data and to extract the temperature dependence of transport coefficients at much higher temperatures than linearized second-order viscous hydrodynamics.

Online adaptive critic designs with tensor product B-splines and incremental model techniques

As an effective optimal control scheme in the field of reinforcement learning, adaptive dynamic programming (ADP) has attracted extensive attention in recent decades. Neural networks (NNs) are commonly employed in ADP algorithms to realize nonlinear function approximation. However, the learning process with NN approximation typically requires a substantial amount of training data and places a heavy computational burden. To improve learning efficiency and alleviate computational load, this paper develops a novel model-free ADP approach based on multivariate splines and incremental control techniques, aimed at achieving online learning of nonlinear control. By utilizing input and output data, an incremental linear approximation model is identified online without any prior knowledge of system dynamics. To improve nonlinear approximation capabilities and lessen computational demands, tensor product B-splines, instead of NNs, are integrated into the critic module to approximate the value function, and the recursive least squares temporal difference (RLS-TD) algorithm is employed to update the weights of spline bases. Convergence analysis of the proposed control scheme is conducted based on the dual-timescale stochastic approximation theory. To illustrate the effectiveness and feasibility of the proposed control scheme, numerical simulations are performed in solving the online nonlinear control problem within the context of the inverted pendulum system.

Weighted twisted inhomogeneous diophantine approximation

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

Motility-Induced Pinning in Flocking System with Discrete Symmetry.

We report a motility-induced pinning transition in the active Ising model for a self-propelled particle system with discrete symmetry. This model was known to exhibit a liquid-gas type flocking phase transition, but a recent study reveals that the polar order is metastable due to droplet excitation. Using extensive MonteCarlo simulations, we demonstrate that, for an intermediate alignment interaction strength, the steady state is characterized by traveling local domains, which renders the polar order short-ranged in both space and time. We further demonstrate that interfaces between colliding domains become pinned as the alignment interaction strength increases. A resonating back-and-forth motion of individual self-propelled particles across interfaces is identified as a mechanism for the pinning. We present a numerical phase diagram for the motility-induced pinning transition, and an approximate analytic theory for the growth and shrink dynamics of pinned interfaces. Our results show that pinned interfaces grow to a macroscopic size preventing the polar order in the regime where the particle diffusion rate is sufficiently smaller than the self-propulsion rate. The growth behavior in the opposite regime and its implications on the polar order remain unresolved and require further investigation.

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

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

Modelling of planetary accretion and core-mantle structure formation

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

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

Systems of fixpoint equations over complete lattices, which combine least and greatest fixpoints, often arise from verification tasks such as model checking and behavioural equivalence checking. In this paper we develop a theory of approximation in the style of abstract interpretation, where a system over some concrete domain is abstracted into a system on a suitable abstract domain, ensuring sound and possibly complete over-approximations of the solutions. We also show how up-to techniques, commonly used to simplify coinductive proofs, fit into this framework, interpreted as abstractions. Additionally, we characterise the solution of fixpoint equation systems through parity games, extending prior work limited to continuous lattices. This game-based approach allows for local algorithms that verify system properties, such as determining whether a state satisfies a formula or two states are behaviourally equivalent. We describe a local algorithm, that can be combined with abstraction and up-to techniques to speed up the computation.

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

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

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

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

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

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