Approximation Theory Research Articles

DETERMINACY OF SCHMIDT’S GAME AND OTHER INTERSECTION GAMES

AbstractSchmidt’s game and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games, $\mathsf {AD}_{\mathbb R}$ , which is a much stronger axiom than that asserting all integer games are determined, $\mathsf {AD}$ . One of our main results is a general theorem which under the hypothesis $\mathsf {AD}$ implies the determinacy of intersection games which have a property allowing strategies to be simplified. In particular, we show that Schmidt’s $(\alpha ,\beta ,\rho )$ game on $\mathbb R$ is determined from $\mathsf {AD}$ alone, but on $\mathbb R^n$ for $n \geq 3$ we show that $\mathsf {AD}$ does not imply the determinacy of this game. We then give an application of simple strategies and prove that the winning player in Schmidt’s $(\alpha , \beta , \rho )$ game on $\mathbb {R}$ has a winning positional strategy, without appealing to the axiom of choice. We also prove several other results specifically related to the determinacy of Schmidt’s game. These results highlight the obstacles in obtaining the determinacy of Schmidt’s game from $\mathsf {AD}$ .

Numerical analysis of two-layered isothermal calendering of viscoplastic and Newtonian fluids with different viscosity ratios

Co-extruded multi-layer plastic sheets and polymer structures formed by calendering process or by cold rolling are widely used in the packaging industry and thin-film transistor manufacturing. The different materials are extruded from separate extruders into the single sheet die which delivers a multi-layer sheet with uniform layer thickness at die exit. This multi-layer sheet is then stretched between counter-rotating rolls to obtain final uniform multi-layer sheet. There are many factors which can influence this process. In this article, calendering a single layer Newtonian or Non-Newtonian material has been extended to analyze a two-layer calendering process for an incompressible Viscoplastic and Newtonian fluids as upper and lower layers with different viscosity ratios. To simplify the equations of motion, the lubrication approximation theory is used. The expressions of non-dimensional pressure gradient, pressure and velocity distribution of both layers are obtained analytically by using proper no slip boundary conditions and dimensionless variables. The dimensionless detachment point is approximated by Regula-Falsi (false position) method. The important engineering factors including detachment point, calendered sheet thickness, roll separation force, power input by rolls, torque on each roll, and adiabatic temperature are all computed. In addition, how the viscosity ratios and viscoplastic casson parameter affect these factors have been investigated. Moreover all established results in literature for single layer calendering Newtonian fluids are also validated at casson parameter β tending towards infinity.

Copositive approximation by Hermits extension

In this paper, we have found a new expansion of the Hermits polynomial of degree ≤ 3n + 2, n ≥ 1, and by this expansion we have studied the copositive approximation of the functions f in space Lψ ,p (I), 0 < p < 1, and by using the quasi normed space, we presented a development of Whitney theory for the copositive approximation using the modulus of smoothness and average modulus of smoothness τ 3n+3(f,|I|,I) ψ ,p ,n ≥ 1 by using the Hermits polynomial expansion.

Control Analysis of Stochastic Lagging Discrete Ecosystems

In this paper, control analysis of a stochastic lagging discrete ecosystem is investigated. Two-dimensional stochastic hysteresis discrete ecosystem equilibrium points with symmetry are discussed, and the dynamical behavior of equilibrium points with symmetry and their control analysis is discussed. Using the orthogonal polynomial approximation theory, the stochastic lagged discrete ecosystems are approximately transformed as its equivalent deterministic ecosystem. Based on the stability and bifurcation theory of deterministic discrete systems, through mathematical analysis, asymptotic stability and Hopf bifurcation are existent in the ecosystem, constructing control functions, controlling the behavior of the system dynamics. Finally, the effects of different random strengths on the bifurcation control and asymptotic stability control are verified by numerical simulations, which validate the correctness and effectiveness of the main results of this paper.

A NEW FRAMEWORK FOR ACCURACY ASSESSMENT OF LIDAR-DERIVED DIGITAL ELEVATION MODELS

Abstract. Existing approaches to accuracy assessment of LiDAR-derived DEM are typically based on statistical methodology and focus on vertical accuracy. This paper presents a new framework which calls for assessment of not only a DEM’s vertical accuracy, but also its ability to preserve a point’s elevation rank in the bare earth topographic surface. New methods to assess each aspect are presented. For DEM’s vertical accuracy, approximation theory from numerical analysis is used to quantify the total error at each DEM point as well as its three error components – sensor error, ground error, and interpolation error. For DEM’s elevation order which is critical to model terrain structure, the concept of isomorphism in set theory is drawn as the mathematical rationale and Kendall’s rank correlation efficient is used to quantify the accuracy. The new framework is illustrated using a DEM derived from LiDAR for sea-level rise vulnerability assessment of a tidal salt marsh. Compared to conventional methods based on statistics, the new framework and methods produce detailed mapping of error distribution thus enable the identification of the main sources of error and where improvement is needed most. Results call for re-evaluation of the current practice of assessing filtering accuracy in LiDAR data processing and further research on relative elevation and isomorphism.

Numerical analysis of damage mechanisms for 3D-printed sandwich structures using a meshless method

Finite element analysis is a widely used simulation technique to analyze structural components and the mechanical behavior of materials under different loading conditions. The major steps involved in finite element simulations are the definition of the basic parameters and the discretization in elements of the component, also known as preprocessing, the analysis of the models and the post-processing by the representation and the interpretation of the obtained results. On the particular case of complex geometries obtained by fused deposition modeling, the most time consuming of these three steps is that of preprocessing, because on traditional finite element technology is require that the real geometry of the CAD model is simplified to be meshed and analyzed. Nowadays, different techniques have been proposed and developed with this objective: reduce costs in terms of time and specialized human resources because the workflow is easier and simpler. One of these methods is known as the meshless method, which do not require connection between nodes and are based on the interaction of the points of the geometry with the neighbors, so the processor analyzes the real CAD geometry and no simplifications are needed. The present work uses the meshless method based on the theory of external approximation as an alternative to the classic finite element method. In numerical simulations, the influence of core shape is evaluated on the failure mode of the 3D-printed lightweight structures. Three cell core patterns were numerically evaluated under tensile and three-point bending tests: out-of-plane hexagonal honeycomb, S-shape corrugated, and in-plane hexagonal honeycomb cores. The nucleation and propagation of cracks had more heterogeneous profiles for in-plane hexagonal honeycomb cores, showing greater unpredictability in the susceptible areas to failure. The numerical analysis with the meshless method showed a potential for a fast prediction and a low computational cost analysis.

Ligand Additivity and Divergent Trends in Two Types of Delocalization Errors from Approximate Density Functional Theory.

The predictive accuracy of density functional theory (DFT) is hampered by delocalization errors, especially for correlated systems such as transition-metal complexes. Two complementary strategies have been developed to reduce delocalization error: eliminating the global curvature with change in charge, and applying a linear response Hubbard U as a measure of local curvature at a metal center at fixed charge in a DFT+U framework. We investigate the relationship between the two delocalization error measures as the ligand field strength is varied with the number of strong-field ligands in a series of heteroleptic complexes or by geometrically constraining the metal-ligand bond length in homoleptic octahedral complexes. We show that across these sets of complexes an inverse relationship generally exists between global and local curvatures. We find that effects of ligand substitution on both measures of delocalization are typically additive, but the quantities seldom coincide.

Establishment of a corrective geoid surface by spline approximation of Iranian GNSS/levelling network

The performance of the gravimetric geoid models is usually evaluated by comparison of geoid heights with the GNSS/levelling derived geoid. But the GNSS/levelling network can be infected by significant systematic biases and random errors, especially in large and uneven areas. This contribution addresses the challenging problem of the corrector surface development along with the elimination of biases. To this end, fitting an appropriate geometric surface to the GNSS/levelling geoid heights is required, which is accomplished by applying the least squares B-spline approximation theory to the GNSS/levelling data. In addition, the 3D affine transformation is used to detect systematic effects of the GNSS/levelling network compared to a global geoid model. This strategy is applied to the adjustment of the Iranian GNSS/levelling network. A significant tilt is observed across the country ranging from −0.35 m to 0.04 m. The entire study area is divided into four zones and the corrector surfaces are obtained in each zone. The accuracy of three global geoid models, EGM2008, EIGEN-6C4 and SGG-UGM-1, along with that of the Iranian regional geoid model IRG2016, are then investigated based on the raw GNSS/levelling heights and the corrector surfaces. The evaluations show the reliable results regarding the corrector surfaces against the raw data set. All geoid models show their maximum RMSE values of discrepancies in the mountainous zones and their minimum RMSEs in the zones having lower variety in topography. This indicates that the irregular topographies are not well detected by the global models over the study area.

On some properties of linear functionals

In this paper, some properties of linear functionals are studied, which are used in functional analysis, optimization theory, numerical analysis, approximation theory, various applications, etc.

Time-Varying Pilot’s Intention Identification Based on ESAX-CSA-ELM Classification Method in Complex Environment

Dynamic and accurate identification of pilot intention is an important prerequisite for more accurate identification of control behavior, automatic flight early warning, and human–aircraft shared autonomy. Meanwhile, it is also the basic requirement of microscopic research on flight safety. In response to these demands, the airfield traffic pattern flight simulation experiment was carried out to obtain the pilot’s physiological data, such as electrocardiogram, respiration, and skin electricity, under different intentions. The extended symbol aggregation approximation theory (ESAX) and the intelligent icon method were utilized to analyze and extract the characteristics of the pilot’s intention. Furthermore, combined with the crow search algorithm (CSA) and extreme learning machine (ELM), a CSA-ELM pilot intention identification model was constructed and it is applied to climb, descend, level flight, and other situations in airfield traffic pattern missions to effectively identify whether the pilot has an intention. The rationality and validity of the identification model were verified through experiments with interactive computer simulations. In addition, compared with the traditional machine learning method, the accuracy of the identification method proposed in this paper is improved by about 10%. The above shows that the research results in this paper can provide support for improving the flight safety early-warning system and the pilot’s micro-behavior evaluation system.

Fuzzy-weighted differential evolution computing paradigm for fractional order nonlinear wiener systems

The parameter estimation of fractional order nonlinear Wiener system is complex and challenging task due to the presence of significant nonlinearity at the output block, unknown fractional order, unknown parameters of the linear/nonlinear blocks, and the unmeasurable intermediate variables as well as the states. In this study, a novel design of fuzzy-weighted differential evolution is presented for parameter estimation of fractional order nonlinear Wiener (FO-NW) systems which are designed as an extension of conventional Wiener type models subjected to be fractional order system by exploiting Grunwald-Letnikov fractional derivative. The parameter estimation problem of FO-NW systems is constructed by defining a merit/error function between the true and estimated response via knacks of approximation theory in mean square error sense. Fuzzy Weighted Differential evolution algorithm is employed as an optimization mechanism to estimate the parameters of FO-NW systems with various output nonlinearities of polynomial, sinusoidal and sigmoidal kernels for low/high noisy environments in the system dynamics. Comparative studies based on rigorous statistics endorse the accurate, effective, stable and robust performance of fuzzy-weighted differential evolution algorithm.

Analysis of Gaussian beam broadening and scintillation index in anisotropic plasma turbulence

In this paper, new expressions for the long-term beam spreading and scintillation index of Gaussian beams in anisotropic plasma turbulent media are derived on the basis of the anisotropic hypersonic plasma turbulent refractive index undulation power spectrum combined with the Rytov approximation theory. Considering the asymmetry of turbulent eddies in the orthogonal xy-plane throughout the path, two anisotropy factors are introduced to parameterize the anisotropy properties of turbulent cells in the horizontal and vertical directions. The spreading and scintillation index of Gaussian beams in plasma turbulence are calculated, and the propagation characteristics of Gaussian beams in anisotropic plasma turbulence are analyzed. Results show that enhancing the anisotropic properties of hypersonic plasma turbulence can significantly improve the propagation performance of Gaussian beams in hypersonic plasma turbulence. The effect of hypersonic plasma turbulence on Gaussian beam increases with the increase of the refractive index undulation variance and the decrease of the turbulent outer scale and anisotropy parameters. The results provide a theoretical basis for the further study of the propagation properties of electromagnetic waves in hypersonic plasma turbulence.

Correction Factors of the Approximate Theories for Axisymmetric Modes of Longitudinal Waves in Circular Rods

Correction Factors of the Approximate Theories for Axisymmetric Modes of Longitudinal Waves in Circular Rods

The Maximum of the Periodogram of a Sequence of Functional Data

We study the periodogram operator of a sequence of functional data. Using recent advances in Gaussian approximation theory, we derive the asymptotic distribution of the maximum norm over all fundamental frequencies. We consider the case where the noise variables are independent and then generalize our results to functional linear processes. Our theory can be used for detecting periodic signals in functional time series when the length of the period is unknown. We demonstrate the proposed methodology in a simulation study as well as on real data. Supplementary materials for this article are available online.

Correction of a relativistic impulse approximation expression used to obtain Compton profiles from photon scattering doubly differential cross sections

A Compton profile (CP) can provide useful information about the electron populations and distributions in atoms, molecules, and ions to assess many physical properties of matter. However, a CP cannot be measured directly, but must be obtained from scattering data. The CPs discussed in this study are derived from photon-atom doubly differential cross sections (DDCS) via the following expression which is derived from an impulse approximation (IA) theory given by DDCS = KJ, where K represents a kinematic factor and J represents the CP. A relativistic version of this expression (i.e., RKJ)—an approximation of the full relativistic IA expression—is used for relativistic regimes; however, it does not yield accurate results for the inner and middle shells of moderate to heavy atoms. In this study, expressions from nonrelativistic (NR) hydrogen-like wavefunctions with a relativistic QED kinematic factor K rel and relativistic electron energy were derived to correct the RKJ expression for the K, L, M, and N atomic subshells. This derivation made it possible for relativistic contributions and screening effects to largely cancel, for any regime of energy angle and Z. Thus, the RKJ error which can be greater than 30% is reduced to within few percent over 99% of the electron momentum distribution range of any subshell CP when compared to published tabulated theoretical values. Two simple versions of the relativistic QED kinematic component of the corrected RKJ expressions were obtained and tested: one valid at high photon energies, and the other at small scattering angles. RKJ corrections were applied to the extraction of CP from the full spectrum K-N shell DDCS, which resulted in much improved accuracy for the K-shell. Good agreement was observed with tabulated beyond K-shell CPs around the tail regions, but systematic differences were found to occur at the maxima. The details of this phenomenon are illustrated and discussed in this article.

Acoustic performance of cavities with gradient changes of cavity numbers under low lattice constant in a soft elastic medium

Air cavities with particular arrangements embedded in elastic layers are generally used as anechoic coatings. In this paper, an idea of cavity numbers with gradient changes is introduced in a soft elastic medium that comprises periodical cylindrical cavities to enhance the low-frequency absorption capacity effectively. The presented model provides a sound absorption coefficient that is higher than 0.5 in the frequency ranges of 75 to 95 Hz and 380 to 720 Hz. Thus, validating the idea can effectively enhance the low-frequency absorption capacity compared with those in previous literature. Analytical results based on effective medium approximation theory are compared with those obtained numerically using the finite element method. These results indicate that the invalidity of the analytical model is due to the coupling between cavities under low lattice constant. Power density dissipation and displacement vectors are plotted to study sound absorption mechanism, thereby showing that the dissipated energy is mainly resulted from the multiple-scattering effect. The scattering between cavities induces the transverse wave caused by the wave mode conversion dissipates in the elastic medium and results in sound absorption. Simulation results show that the absorption peaks can be effectively manipulated by tuning the gradient profile of the cavity numbers. The proposed model is then verified by the corresponding experiment. Results obtained numerically and experimentally are agreed well, and its feasibility is verified. The presented model would offer a new approach for the metamaterial design on low-frequency broadband sound absorption and could have potential applications on the design of anechoic coatings.

A Long Short-Term Memory for AI Applications in Spike-based Neuromorphic Hardware

Spike-based neuromorphic hardware holds promise for more energy-efficient implementations of deep neural networks (DNNs) than standard hardware such as GPUs. But this requires us to understand how DNNs can be emulated in an event-based sparse firing regime, as otherwise the energy advantage is lost. In particular, DNNs that solve sequence processing tasks typically employ long short-term memory units that are hard to emulate with few spikes. We show that a facet of many biological neurons, slow after-hyperpolarizing currents after each spike, provides an efficient solution. After-hyperpolarizing currents can easily be implemented in neuromorphic hardware that supports multi-compartment neuron models, such as Intel’s Loihi chip. Filter approximation theory explains why after-hyperpolarizing neurons can emulate the function of long short-term memory units. This yields a highly energy-efficient approach to time-series classification. Furthermore, it provides the basis for an energy-efficient implementation of an important class of large DNNs that extract relations between words and sentences in order to answer questions about the text. Deep learning could be less energy intensive when implemented on spike-based neuromorphic chips. An approach inspired by a characteristic feature of biological neurons, the presence of slowly changing internal currents, is developed to emulate long short-term memory units in a sparse spiking regime for neuromorphic implementation.

Investigation on molecular alignment induced by the few-cycle laser pulse and its phase evolution

BackgroundMolecular alignment and orientation are the hot topics in the ultrafast optics and the atomic, molecular and optical physics, since they are involved the laser field interaction with the atoms and the molecules. Objective and methodHere, we investigate the alignment of gaseous carbon monoxide induced by the strong few-cycle laser pulse based on the rigid rotor approximation theory. ResultsThe simulations show that there is a periodic recurrence in the alignment degree depending on the laser intensity, which is reduced by the dephasing-rephasing processes of molecules’ rotation. The carrier-envelope phase of the few-cycle laser also affects the alignment of the molecules, and produces a phase evolution alignment. The underlying physical mechanisms are discussed. SignificanceOur results might give a clue to understand the controlling the rotations of the gaseous molecules by the few-cycle laser pulse, and other revolved high-density physics processes.

A UNIFIED THREE-DIMENSIONAL EXTENDED FRACTIONAL ANALYTICAL SOLUTION FOR AIR POLLUTANTS DISPERSION

A novel technique is presented to analytically solve the fractional diffusion equation for non-reactive air pollutants emitted from an elevated continuous source into the air. A generalized methodical solution combining first-order Wentzel–Kramers–Brillouin (WKB) approximation theory and the Sturm–Liouville problem is used to solve the air pollutants’ fractional dispersion equation. Drawing insight from previous analysis, we expanded the initial issue assuming the turbulent flow characteristics appearing in the diffusion process in a non-integer dimensional space. We solved the transformed problem and compared the solutions against data from real experiment. Physical consequences connecting to the conventional generalized diffusion equations are presented. The results indicate that the present solutions are in accordance with those obtained in literature. This report demonstrates that fractional equations can be applied in a practical prediction of pollutant dissemination in a turbulent atmosphere.

Validated analysis of modulated signals: From de Prony to Padé and beyond

The spectral analysis of modulated signals has attracted quite some research, mainly because of the fact that Fourier methods are not particularly suitable. Among the challenges, we mention the separation of close components that differ significantly in magnitude, the limitation of the sampling duration, the probable ill-conditioning of certain structured matrices.We show how a validated exponential analysis add-on, for use with any standard exponential analysis method, offers a lot of advantages in the context of these challenges. The add-on uses an alias-free decimation technique and essentially combines the basics of de Prony’s method for exponential fitting with the theory of Padé approximation theory.