Approximation Theory Research Articles

Approximate Set Theory

Abstract This article describes an approximate set theory modeling intuitions shared by musicians such as Cowell, Schoenberg, Messiaen, and Persichetti. The author considers five approximation strategies, showing that in each case the result resembles an exact seven-tone set theory. Since most seven-tone sets are interval cycles, approximate twelve-tone sets are approximately cyclic as well. The theory explains how to highlight this cyclic structure using voicings, modeled by intervals in the intrinsic scale formed from a chord's own notes. This connection to voicing is what gives approximate chord categories much of their significance. The approach is most useful for chords with five or fewer notes and works tolerably for hexachords, but it breaks down with larger collections. This is not a failure of the model but a reflection of the fact that quality space contracts as cardinality increases.

AAA interpolation of equispaced data

We propose AAA rational approximation as a method for interpolating or approximating smooth functions from equispaced samples. Although it is always better to approximate from large numbers of samples if they are available, whether equispaced or not, this method often performs impressively even when the sampling grid is coarse. In most cases it gives more accurate approximations than other methods. We support this claim with a review and discussion of nine classes of existing methods in the light of general properties of approximation theory as well as the “impossibility theorem” for equispaced approximation. We make careful use of numerical experiments, which are summarized in a sequence of nine figures. Among our new contributions is the observation, summarized in Fig. 7, that methods such as polynomial least-squares and Fourier extension may be either exponentially accurate and exponentially unstable, or less accurate and stable, depending on implementation.

Effect of thermal radiation on convective heat transfer in MHD boundary layer Carreau fluid with chemical reaction

The temperature dependent thermophysical fluid properties have numerous aspects in different industries and engineering processes in which heat transmission is based on fluid flow. For such heat transmission processes, heat transmission system is highly fluctuated with variation of viscosity. Thus, the aim of this study is to investigate the transfer of heat in magnetized Carreau fluid with chemical reaction and under influence of thermal radiation over nonlinear stretching/shrinking surface. Additionally, we have incorporated variable heat dependent thermophysical properties to analyze the heat transfer in magnetized Carreau fluid. Set of flow governing non linear PDE’s are obtained using Carreau fluid tensor and boundary layer approximation (BLA) theory. Dimensionless set of ODE’s are obtained using suitable similarity transforms. Shooting method in conjunction with Newton's method have been utilized to solve the problem. It is noted that when stretching Bge 0 is significant with strictly increasing mass suction S shear stress rate increase with minor levels and sharp increase has been observed in Nusselt number, whereas in shrinking case B<0 shear stress and heat transfer coefficient values are improved raising the value of S mass suction. Further, raising the values of power law index n produce reduced skin friction over stretching surface Bge 0 while skin friction dramatically enhance in shrinking case B<0. It is observed that raising the non-linearity m values for stretching or shrinking, skin friction and Nusselt number considerably improved. Moreover, computational outcomes of the study are validated with already published previous results and the results obtained in this study are found in good agreement.

Mini-Workshop: Interpolation, Approximation, and Algebra

This report involves two concepts of geometric modeling: multivariate data interpolation by polynomials, and the study of generalized barycentric coordinates. These topics are connected to a wide range of of applications, from computer aided design (CAD) systems for designing airplanes and automobiles to animation in movies to problems in numerical analysis and partial differential equations. Traditionally these topics were studied mostly from an analytic standpoint, but recently advanced algebraic tools have come into the picture. The purpose of the mini-workshop was to bring together a diverse group of researchers with different areas of expertise, drawing from both the approximation theory and algebraic geometry communities, and to explore the connections between the two areas in greater detail.

Research and Application of Wavelet Transform-Based Two-Dimensional Pinning Potential Stochastic Resonant System

In stochastic resonance (SR) weak signal detection, there is no literature currently report on the study and comparison of one-dimensional (1D) and 2D pinning potential worldwide which has potential research necessity. A one (ODPPBSR) and a 2D pinning potential bistable SR (TDPPBSR) are proposed. The expressions for MFPT, SPD and SNR are derived based on adiabatic approximation theory. To investigate the correctness of the theoretical results, numerical simulations are carried out with the Runge–Kutta algorithm and the genetic algorithm (GA) is used to optimize the system. The system has exceptional ability to restore signal periodicity and amplitude amplification at low frequency, high frequency and multi-frequency. The two systems are compared of the detection capabilities on weak signals through wavelet transform denoising and applied to the 6205-2RS JEM SKF and HRB 6205-2Z for bearing fault detection. The experimental results show that the 2D system is superior to the 1D system.

Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions

We study the approximation by tensor networks (TNs) of functions from classical smoothness classes. The considered approximation tool combines a tensorization of functions in $L^p([0,1))$, which allows to identify a univariate function with a multivariate function (or tensor), and the use of tree tensor networks (the tensor train format) for exploiting low-rank structures of multivariate functions. The resulting tool can be interpreted as a feed-forward neural network, with first layers implementing the tensorization, interpreted as a particular featuring step, followed by a sum-product network with sparse architecture. In part I of this work, we presented several approximation classes associated with different measures of complexity of tensor networks and studied their properties. In this work (part II), we show how classical approximation tools, such as polynomials or splines (with fixed or free knots), can be encoded as a tensor network with controlled complexity. We use this to derive direct (Jackson) inequalities for the approximation spaces of tensor networks. This is then utilized to show that Besov spaces are continuously embedded into these approximation spaces. In other words, we show that arbitrary Besov functions can be approximated with optimal or near to optimal rate. We also show that an arbitrary function in the approximation class possesses no Besov smoothness, unless one limits the depth of the tensor network.

Efficient estimation of average derivatives in NPIV models: Simulation comparisons of neural network estimators

Artificial Neural Networks (ANNs) can be viewed as nonlinear sieves that can approximate complex functions of high dimensional variables more effectively than linear sieves. We investigate the performance of various ANNs in nonparametric instrumental variables (NPIV) models of moderately high dimensional covariates that are relevant to empirical economics. We present two efficient procedures for estimation and inference on a weighted average derivative (WAD): an orthogonalized plug-in with optimally-weighted sieve minimum distance (OP-OSMD) procedure and a sieve efficient score (ES) procedure. Both estimators for WAD use ANN sieves to approximate the unknown NPIV function and are n-asymptotically normal and first-order equivalent. We provide a detailed practitioner’s recipe for implementing both efficient procedures. We compare their finite-sample performances in various simulation designs that involve smooth NPIV function of up to 13 continuous covariates, different nonlinearities and covariate correlations. Some Monte Carlo findings include: (1) tuning and optimization are more delicate in ANN estimation; (2) given proper tuning, both ANN estimators with various architectures can perform well; (3) easier to tune ANN OP-OSMD estimators than ANN ES estimators; (4) stable inferences are more difficult to achieve with ANN (than spline) estimators; (5) there are gaps between current implementations and approximation theories. Finally, we apply ANN NPIV to estimate average partial derivatives in two empirical demand examples with multivariate covariates.

Use of DFT+U+J with linear response parameters to predict non-magnetic oxide band gaps with hybrid-functional accuracy

First-principles Hubbard-corrected approximate density-functional theory (DFT+U) is a low-cost, potentially high throughput method of simulating materials, but it has been hampered by empiricism and inconsistent band-gap correction in transition-metal oxides. DFT+U property prediction of non-magnetic systems such as d0 and d10 transition-metal oxides is typically faced with excessively large calculated Hubbard U values, and with difficulty in obtaining acceptable band-gaps and lattice volumes. Meanwhile, Hund's exchange coupling J is an important but often neglected component of DFT+U, and the J parameter has proven challenging to directly calculate by means of linear response. In this work, we provide a revised formula for computing Hund's J using established self-consistent field DFT+U codes. For non-magnetic systems, we introduce a non-approximate technique for calculating U and J simultaneously in such codes, at no additional cost. Using unmodified Quantum ESPRESSO, we assess the resulting values using two different DFT+U functionals incorporating J, namely the widely used DFT+(U-J) and the readily available DFT+U+J. We assess a test set comprising TiO2, ZrO2, HfO2, Cu2O and ZnO, and apply the corrections both to metal and oxygen centered pseudoatomic subspaces. Starting from the PBE functional, we find that DFT+(U-J) is significantly out-performed in band-gap accuracy by DFT+U+J, the RMS band-gap error of which matches that of the hybrid functional HSE06. ZnO, a long-standing challenge case for DFT+U, is addressed by means of Zn 4s instead of Zn 3d correction, in which case the first-principles DFT+U+J band-gap error is half of that reported for HSE06.

Theoretical Simulation of the High–Order Harmonic Generated from Neon Atom Irradiated by the Intense Laser Pulse

Based on the strong field approximation theory and numerical solution of Maxwell’s propagation equations, the high–order harmonic is generated from a neon (Ne) atom irradiated by a high–intensity laser pulse whose central wavelength is 800 nm. In the harmonic spectrum, it is found that in addition to the odd harmonics of the driving laser, a new frequency peak appeared. By examining the time–dependent behavior of the driving laser, it is found that the symmetry of the laser field is broken. We demonstrated that these new spectrum peaks are caused by the intensity reduction and frequency blue shift of the high–intensity laser during propagation. Our results reveal that it is feasible to modulate the harmonics of the specific energy to produce high–intensity harmonic emission by changing the gas density and the position of the gas medium interacting with the laser pulse.

Spatial Differentiation Characteristics of Rural Areas Based on Machine Learning and GIS Statistical Analysis—A Case Study of Yongtai County, Fuzhou City

With the development of machine learning and GIS (geographic information systems) technology, it is possible to combine them to mine the knowledge rules behind massive spatial data. GIS, also known as geographic information systems, is a comprehensive discipline, which combines geography and cartography and has been widely used in different fields. It is a computer system for inputting, storing, querying, analyzing, and displaying geographic data. This paper mainly studies the spatial differentiation characteristics of rural areas based on machine learning (ML) and GIS statistical analysis. This paper studies 21 township units in Yongtai County. In this paper, ENVI remote sensing image processing software is used to carry out the geometric correction of Landsat-8 remote sensing data. ML is multidisciplinary and interdisciplinary, involving probability theory, statistics, approximation theory, convex analysis, algorithm complexity theory, and other disciplines. It is specialized in studying how computers simulate or realize human learning behavior to obtain new knowledge or skills, and reorganize existing knowledge structures to continuously improve its own performance. The purpose of using band fusion is to provide more data information for the study and improve the accuracy of land classification results. Through the extraction of evaluation elements, this paper preliminarily confirms the evaluation index object of a rural human settlement environment evaluation system from the perspective of spatial layout rationality. This paper uses a VMD-GWO-ELM-based three-stage evolutionary extreme learning machine evaluation method to simulate the model. In the same way, when the model is trained again, extra weight is given to extract the feature points to reduce the similarity. Experimental data show that GWO-SVM has good classification performance, with the cross-validation rate reaching 91.66% and the recognition rate of test samples reaching 82.41%. The results show that GIS statistics can provide a reference for environmental protection, which is conducive to land-use planning, implementation of environmental impact assessment of land-use planning, and ultimately achieving sustainable development.

A Study of Tate Homology via the Approximation Theory with Applications to the Depth Formula

A Study of Tate Homology via the Approximation Theory with Applications to the Depth Formula

Comparison of semi-analytical and numerical methods for the scattering of acoustic waves by rough, sloped seabed geometries

The seabed possesses variations in its material properties, layering structure, slope and roughness making prediction, and analysis of sound propagation in shallow water highly complex. For certain frequency ranges, the effect of roughness becomes significant. Numerous models have been adopted to account for the scattering of acoustic waves by rough surfaces, including Kirchhoff approximation (for relatively large scales of roughness), small perturbation theory (for relatively small scales of roughness), and small slope approximation, which bridges across the two regimes under certain conditions. Although grounded in analytical theory, these approaches invariably involve the computation of numerical integrals so additional approximation theories have been explored including stationary phase methods and Taylor series approximations. In this study, these semi-analytical methods are incorporated within a new sound propagation model and compared with a numerical graphics processing unit (GPU)-accelerated finite element model incorporating roughness parameters (root mean square height and correlation length) for rough, sloped seabed geometries (with gradients ranging from 0° to 45°) in shallow water environments. The effect of large- and small-scale roughness on sound propagation is investigated in two- and three-dimensional space. The finite element modelling is used to validate the optimal choice of semi-analytical method for the different environments considered.

Modification of Bekenstein-Hawking entropy of cylindrical symmetric black hole by Lorentz breaking theory

According to Lorentz breaking theory, WKB approximation theory, quantum tunneling radiation theory and the beyond semi-classical theory, the fermions dynamic equation in the space-time of a cylindrical symmetric black hole is modified more accurately. On this basis, the characteristics of the quantum tunneling radiation of such black holes and the new expressions of the modified temperature and Bekenstein-Hawking entropy of the black hole are studied. A series of results obtained in this paper are discussed in depth.

Surface crossing and energy flow in many-dimensional quantum systems

Energy flow in molecules, like the dynamics of other many-dimensional finite systems, involves quantum transport across a dense network of near-resonant states. For molecules in their electronic ground state, the network is ordinarily provided by anharmonic vibrational Fermi resonances. Surface crossing between different electronic states provides another route to chaotic motion and energy redistribution. We show that nonadiabatic coupling between electronic energy surfaces facilitates vibrational energy flow and, conversely, anharmonic vibrational couplings facilitate nonadiabatic electronic state mixing. A generalization of the Logan-Wolynes theory of quantum energy flow in many-dimensional Fermi resonance systems to the two-surface case gives a phase diagram describing the boundary between localized quantum dynamics and global energy flow. We explore these predictions and test them using a model inspired by the problem of electronic excitation energy transfer in the photosynthetic reaction center. Using an explicit numerical solution of the time-dependent Schrödinger equation for this ten-dimensional model, we find quite good agreement with the expectations from the approximate analytical theory.

A Regularity Theory for Static Schrödinger Equations on \(\boldsymbol{\mathbb{R}}\)d in Spectral Barron Spaces

Spectral Barron spaces have received considerable interest recently, as it is the natural function space for approximation theory of two-layer neural networks with a dimension-free convergence rate. In this paper, we study the regularity of solutions to the whole-space static Schrödinger equation in spectral Barron spaces. We prove that if the source of the equation lies in the spectral Barron space and the potential function admitting a nonnegative lower bound decomposes as a positive constant plus a function in , then the solution lies in the spectral Barron space .

Characterization of the Variation Spaces Corresponding to Shallow Neural Networks

We study the variation space corresponding to a dictionary of functions in $$L^2(\Omega )$$ for a bounded domain $$\Omega \subset {\mathbb {R}}^d$$ . Specifically, we compare the variation space, which is defined in terms of a convex hull with related notions based on integral representations. This allows us to show that three important notions relating to the approximation theory of shallow neural networks, the Barron space, the spectral Barron space, and the Radon BV space, are actually variation spaces with respect to certain natural dictionaries.

Evaluation of irreducible water saturation by electrical imaging logging based on capillary pressure approximation theory

One of the most important metrics for evaluating oil and gas reservoirs is irreducible water saturation (Swir). The completion of oil and gas reservoir development tasks, such as fluid identification, productivity prediction, and water-flooded reservoir discrimination, requires the accurate evaluation of Swir. Nuclear magnetic resonance (NMR) logging can reflect the intricate pore structure of reservoirs, so it offers a natural advantage in the evaluation of Swir, but its high measurement cost leads to infrequent use. In geophysical logging, electrical imaging logging can obtain pore size distribution information at a lower cost. Hence, a method based on electrical imaging logging data is proposed to predict Swir. This method not only has a high degree of accuracy but is also cost-effective. First, the pore characteristics of the bound fluid were analyzed with the mercury injection capillary pressure curve. Based on the capillary pressure approximation theory, the reverse cumulative curve of the porosity spectrum was derived. And the physical meaning of its “inflection point” was clarified. Then, the resistivity of the formation containing just irreducible water was computed using the conversion between the core displacement pressure and resistivity. Finally, based on Archie's formula, the Swir at each depth of the reservoir was predicted. By using ultradeep carbonate reservoir logging data from the Yuanba gas field, the model was validated. The results demonstrated that the calculated values using this method agreed with the measured values from the cores. Compared with the core fitting method, the average absolute error is reduced by 2.61%, and the average relative error decreases by 12.0%. If NMR logging information is not present, this method serves as an effective supplement to accurately predict Swir.

Crude oil price prediction using deep reinforcement learning

Crude oil price forecasting has received considerable attention owing to its significance in the commodity market and non-linear complexity in forecasting tasks. This study aims to develop a novel deep reinforcement learning algorithm for multi-step ahead crude oil price forecasting in three major commodity exchanges. The proposed algorithm includes two main improvements: (a) A dynamic action exploration mechanism based on the stochastic processes conforming to commodity price fluctuations is designed for accuracy and generalization. (b) A dynamic update policy of network parameters based on approximate optimization theory is developed to improve the network's learning efficiency. The algorithm's effectiveness is experimentally verified and compared with five state-of-the-art algorithms. The main findings are as follows. (a) DRL's forecasting ability is developed in crude oil price forecasting, which may be extended to the forecasting of other natural resource prices. (b) The proposed algorithm can be applied to the data of the world's three major crude oil price benchmarks with considerable universality. (c) The accuracy of the proposed algorithm declines indistinctively with the expansion of the forecasting step; however, it reflects the actual price and fluctuation. These findings have implications in accelerating the global economic recovery and exploring AI in the energy market.

On the Exact Evaluation of Integrals of Wavelets

Wavelet expansions are a powerful tool for constructing adaptive approximations. For this reason, they find applications in a variety of fields, from signal processing to approximation theory. Wavelets are usually derived from refinable functions, which are the solution of a recursive functional equation called the refinement equation. The analytical expression of refinable functions is known in only a few cases, so if we need to evaluate refinable functions we can make use only of the refinement equation. This is also true for the evaluation of their derivatives and integrals. In this paper, we detail a procedure for computing integrals of wavelet products exactly, up to machine precision. The efficient and accurate evaluation of these integrals is particularly required for the computation of the connection coefficients in the wavelet Galerkin method. We show the effectiveness of the procedure by evaluating the integrals of pseudo-splines.

Chiral perturbation theory: reflections on effective theories of the standard model

The pseudoscalar particles pions, kaons and the $\eta$-particle are considerably lighter than the other hadrons such as protons or neutrons. Their lightness was understood as a consequence of approximate chiral symmetry breaking. This led to current algebra, a way to express the relations imposed by the symmetry breaking. It was realized by Weinberg that because of their low mass, it is possible to formulate a purely pionic (effective) field theory at experimental energies, which carries all information on the (non-perturbative) dynamics, symmetries, and their spontaneous breaking of quantum chromodynamics (QCD) and allows for systematic calculations of observables. In this review, we trace these developments and present recent activities in this field. We make the connection to other effective theories, more generally introduced by Wilson, as approximate field theories at low energies. Indeed, principles and paradigms introduced first for pions have become ubiquitous in particle physics and the standard model. Lastly, we turn to the latest development where the present (fundamental) standard model itself is considered as an effective field theory of a - yet to be formulated - even more fundamental theory. We also discuss important techniques that were developed in order to turn chiral perturbation theory into a predictive framework and briefly review some connections between lattice QCD and chiral perturbation theory (ChPT).