Orthogonal Expansion Research Articles

Multimodal discourse analysis of English reading instruction in colleges and universities based on weighted function algorithm

Abstract This paper proposes a kernel clustering method that uses the weighted function method to address the problem of excessive similarity calculation in the kernel function clustering process. Based on function-based clustering with orthogonal basis expansion, function-based principal component analysis is used to reduce the dimensionality of function-based data and extract the first few principal components that can contain the most information of the original data. The factors of each component are assigned with characteristic weights so that the scores of the principal components replace the original functional data for clustering analysis of English reading teaching in colleges and universities. A systematic functional language framework was used to conduct multimodal discourse analysis based on clustering results. The symbolic annotation of pure language reached 14.5, and the symbolic resources of visual imagery reached 85.7 during the CB stage of English reading instruction. The number of semiotic resources was relatively small, so the complexity of the modality was lower than in other stages.

Dynamic analysis for V-notch in a piezomagnetic bi-material strip under SH guided wave

In this paper, the problem of a V-notch with complex boundary conditions in a piezomagnetic bi-material strip is studied. Firstly, SH guided wave is considered as an external load acting on piezomagnetic bi-material strip, on the basis of repeated image superposition, the analytical expression of scattering wave is conducted, which satisfies the stress free and magnetic insulation conditions on the upper and lower horizontal boundaries of the strip. Then, the analytical expression of standing wave is established, which satisfies the stress free and magnetic insulation conditions on the boundaries of V-notch by the fractional Bessel function expansion method and Graf addition theorem. Finally, Green’s function method is applied, the bi-material strip is divided into two parts along the vertical interface, a pair of in-plane magnetic field and out-plane forces are applied on the vertical interface, and the first kind of Fredholm integral equations are set up and solved by applying orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor and magnetic field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Універсальний сплайн-пертурбаційний розподіл

This paper considers the problem of probability distribution construction for a random variable from known numerical characteristics. The problem is of importance in determining the parametric reliability of engineering systems when the numerical characteristics (in particular, the bias and the kurtosis) of an output parameter (state variable) are determined by analytical methods and its distribution must be recovered. This may be done using a four-parameter universal distribution, which allows one to cover certain ranges (preferably, as wide as possible) of the bias and kurtosis coefficients using a single analytical form. The most familiar universal distribution is Gram-Charlier’s, which is a deformation of the normal distribution obtained using a Chebyshev-Hermite orthogonal polynomial expansion. However, in the general case, Gram-Charlier’s distribution function is not a steadily increasing one. For some combinations of the bias and kurtosis coefficients, the density curve may exhibit negative values and multiple modes. Because of this, a search for other universal distributions to cover wider ranges of the bias and kurtosis coefficients is of current importance. The paper analyzes a method of universal probability distribution construction by multiplying the normal density by a perturbing polynomial in the form of a spline (referred to as the spline-perturbed distribution). The idea of a distribution of this type was proposed earlier to account for a nonzero bias coefficient. The spline is constructed based on Hermite’s interpolating polynomials of the third degree with two knots, which have a minimum of parameters and possess a locality property The basic distribution is constructed for a four-knot spline. The paper further develops and generalizes the spline-perturbed distribution to nonzero bias and kurtosis coefficients. Two cases are considered. The first case is a composition of two splines that have four and five knots, respectively. The former and the latter allow one to account for the bias and the kurtosis, respectively. Integral equations are obtained to find the values at the knots of both splines and construct the distribution. The second case is more general and uses one five-knot Hermite spline. The paper shows a way to construct a generalized spline-perturbed distribution without any negative density values or any multiple modes. The knot points are chosen using an enumerative technique. Conditions for the absence of negative density values and multiple nodes are identified.

Synthetic Aperture Passive Localization Method Based on Slant Range Orthogonal Expansion

The synthetic aperture passive localization system generally compensates for the second-order phase term of the received signal with the Taylor series of the range history and then uses the focusing result of the compensated signal to obtain the position of the emitter. However, the existence of a higher-order residual phase causes the mismatch of reference function, leading to the bias of localization results. To solve the problem, this paper proposes a slant range expansion method based on an orthogonal basis. The optimal expansion of the range history is obtained by constructing a set of orthogonal bases in the space composed of quadratic polynomials so that the residual phase after integration is minimized. The proposed method can effectively mitigate the localization bias caused by the model approximation of a synthetic aperture localization system. Simulations and Monte Carlo tests show that the proposed method outperforms the traditional synthetic aperture localization method.

On the use of L-functionals in regression models

Abstract In this article, we survey and unify a large class or L L -functionals of the conditional distribution of the response variable in regression models. This includes robust measures of location, scale, skewness, and heavytailedness of the response, conditionally on covariates. We generalize the concepts of L L -moments (G. Sillito, Derivation of approximants to the inverse distribution function of a continuous univariate population from the order statistics of a sample, Biometrika 56 (1969), no. 3, 641–650.), L L -skewness, and L L -kurtosis (J. R. M. Hosking, L-moments: analysis and estimation of distributions using linear combinations or order statistics, J. R. Stat. Soc. Ser. B Stat. Methodol. 52 (1990), no. 1, 105–124.) and introduce order numbers for a large class of L L -functionals through orthogonal series expansions of quantile functions. In particular, we motivate why location, scale, skewness, and heavytailedness have order numbers 1, 2, (3,2), and (4,2), respectively, and describe how a family of L L -functionals, with different order numbers, is constructed from Legendre, Hermite, Laguerre, or other types of polynomials. Our framework is applied to models where the relationship between quantiles of the response and the covariates follows a transformed linear model, with a link function that determines the appropriate class of L L -functionals. In this setting, the distribution of the response is treated parametrically or nonparametrically, and the response variable is either censored/truncated or not. We also provide a framework for asymptotic theory of estimates of L L -functionals and illustrate our approach by analyzing the arrival time distribution of migrating birds. In this context, a novel version of the coefficient of determination is introduced, which makes use of the abovementioned orthogonal series expansion.

Tensions with the flat ΛCDM model from high-redshift cosmography

ABSTRACT The longstanding search for the cosmological model that best describes the Universe has been made more intriguing since the recent discovery of the Hubble constant, H0, tension observed between the value of H0 from the Cosmic Microwave Background and from type Ia supernovae (SNe Ia). Hence, the commonly trusted flat Λ cold dark matter (ΛCDM) model is under investigation. In this scenario, cosmography is a very powerful technique to investigate the evolution of the Universe without any cosmological assumption, thus revealing tensions between observational data and predictions from cosmological models in a completely model-independent way. We here employ a robust cosmographic technique based on an orthogonal logarithmic polynomial expansion of the luminosity distance to fit quasars (QSOs) alone and QSOs combined with Gamma-Ray Bursts (GRBs), SNe Ia, and Baryon Acoustic Oscillations. To apply QSOs and GRBs as probes we use, respectively, the Risaliti–Lusso relation between ultraviolet and X-ray luminosities and the ‘Dainotti GRB 3D relation’ among the rest-frame end time of the X-ray plateau emission, its corresponding luminosity, and the peak prompt luminosity. We also correct QSOs and GRBs for selection biases and redshift evolution and we employ both the traditional Gaussian likelihood and the newly discovered best-fitting likelihoods for each probe investigated. This comprehensive analysis reveals a strong tension ($\gt 4 \, \sigma$) between our data sets and the flat ΛCDM model proving the power of both the cosmographic approach and high-redshift sources, such as QSOs and GRBs, which can probe the Universe at early epochs.

Analytical formulas for option prices under time-changed CARMA process

We consider the option pricing problem when the underlying asset price is driven by a continuous time autoregressive moving average (CARMA) process, time changed a Lévy subordinator or/and an absolutely continuous time change process. We derive the analytical formulas for the option prices by employing the orthogonal polynomial expansion method. Our method is based on the observation that the CARMA process belongs to the class of polynomial diffusion and the time variable and underlying state variables enter the polynomial expansion separately. We demonstrate the accuracy of the method through a number of numerical experiments. We also investigate the price sensitivities with respect to the key parameters that govern the dynamics of the underlying state and time change variables.

APPLICATION OF SPLINE FUNCTIONS AND WALSH FUNCTIONS IN PROBLEMS OF PARAMETRIC IDENTIFICATION OF LINEAR NONSTATIONARY SYSTEMS

Context. In this article, a generalized parametric identification procedure for linear nonstationary systems is proposed, which uses spline functions and orthogonal expansion in a series according to the Walsh function system, which makes it possible to find estimates of the desired parameters by minimizing the integral quadratic criterion of discrepancy based on solving a system of linear algebraic equations for a wide class of linear dynamical systems. The accuracy of parameter estimation is ensured by constructing a spline with a given accuracy and choosing the number of terms of the Walsh series expansion when solving systems of linear algebraic equations by the A. N. Tikhonov regularization method. To improve the accuracy of the assessment, an algorithm for adaptive partitioning of the observation interval is proposed. The partitioning criterion is the weighted square of the discrepancy between the state variables of the control object and the state variables of the model. The choice of the number of terms of the expansion into the Walsh series is carried out on the basis of adaptive approximation of non-stationary parameters in the observation interval, based on the specified accuracy of their estimates. The quality of the management of objects with variable parameters is largely determined by the accuracy of the evaluation of their parameters. Hence, obtaining reliable information about the actual nature of parameter changes is undoubtedly an urgent task.
 Objective. Improving the accuracy of parameter estimation of a wide class of linear dynamical systems through the joint use of spline functions and Walsh functions.
 Method. A generalized parametric identification procedure for a wide class of linear dynamical systems is proposed. The choice of the number of terms of the expansion into the Walsh series is made on the basis of the proposed algorithm for adaptive partitioning of the observation interval.
 Results. The results of modeling of specific linear non-stationary systems confirm the effectiveness of using the proposed approaches to estimating non-stationary parameters.
 Conclusions. The joint use of spline functions and Walsh functions makes it possible, based on the proposed generalized parametric identification procedure, to obtain analytically estimated parameters, which is very convenient for subsequent use in the synthesis of optimal controls of real technical objects. This procedure is applicable to a wide class of linear dynamical systems with concentrated and distributed parameters.

Neural-network-based wavefront solution algorithm for a wide field survey telescope.

The wide field survey telescope (WFST) is a 2.5m optical survey telescope currently under construction in China. The telescope employs a primary-focus optical design to achieve a wide field of view of 3deg, and its focal plane is equipped with four pairs of curvature sensors to perform wavefront sensing and active optics. Currently, there are several wavefront solution algorithms available for curvature sensors, including the iterative fast Fourier transform method, orthogonal series expansion method, Green's function method, and sensitivity matrix method. However, each of these methods has limitations in practical use. This study proposes a solution method based on a convolutional neural network model with a U-Net structure for the curvature wavefront sensing of the WFST. Numerical simulations show that the model, when properly trained, has a high accuracy and performs a curvature wavefront solution effectively. Upon a comparison with the sensitivity matrix method, this new method demonstrates its superiority. Finally, the study is summarized, and the drawbacks of the proposed method are discussed, which leads to direction for future optimizations.

Extracting a function encoded in amplitudes of a quantum state by tensor network and orthogonal function expansion

There are quantum algorithms for finding a function f satisfying a set of conditions, such as solving partial differential equations, and these achieve exponential quantum speedup compared to existing classical methods, especially when the number d of the variables of f is large. In general, however, these algorithms output the quantum state which encodes f in the amplitudes, and reading out the values of f as classical data from such a state can be so time-consuming that the quantum speedup is ruined. In this study, we propose a general method for this function readout task. Based on the function approximation by a combination of tensor network and orthogonal function expansion, we present a quantum circuit and its optimization procedure to obtain an approximating function of f that has a polynomial number of degrees of freedom with respect to d and is efficiently evaluable on a classical computer. We also conducted a numerical experiment to approximate a finance-motivated function to demonstrate that our method works.

Strain fields of Ms >6.0 earthquakes in Menyuan, Qinghai, China

In predicting earthquakes, it is a major challenge to capture the time factor and spatial isoline anomalies, and understand their physical processes, of the seismic strain field before a strong earthquake. In this study, the seismic strain field was used as representative of seismic activity. The natural orthogonal function expansion method was used to calculate the seismic strain field before the Menyuan Ms 6.4 earthquakes in 1986 and 2016, and the Ms 6.9 earthquake in 2022. Time factor and spatial isoline anomaly of the strain field before each earthquake was extracted. We also compared the evolution of the strain field with numerical simulation results under the tectonic stress system at the source. The results showed that the time factor before the earthquakes had high or low value anomalies, exceeding the mean square error of the stable background. The anomalies were concentrated in the first four typical fields of the strain field, which has multiple components. The abnormal contribution rate of the first typical field is the largest (accounting for 42%–49% of the total field). The long- and medium-term anomalies appear 3-4, and 1-2 years before the earthquake, respectively. There were no short or immediate-term anomalies within 3 months of the earthquake. In addition, during the evolution of the strain field, the abnormal area of the spatial isoline changed with the change in time. Usually, the intersection area of the two isoseismic lines of strain accumulation and strain release becomes a potential location for strong earthquakes. Finally, we found that the high strain field values of the 1986 and 2016 Ms 6.4 earthquakes were equivalent to the numerical simulation results, while the high strain field values of the 2022 Menyuan Ms 6.9 earthquakes were slightly different, but within the accepted error range. These results indicate that the two methods are consistent. We have shown that the natural orgthagonal method can be used to obtain the spatiotemporal anomaly information of strain field preceding strong earthquakes.

Back to Almost Ricci Solitons

In the paper, we study complete almost Ricci solitons using the concepts and methods of geometric dynamics and geometric analysis. In particular, we characterize Einstein manifolds in the class of complete almost Ricci solitons. Then, we examine compact almost Ricci solitons using the orthogonal expansion of the Ricci tensor, this allows us to substantiate the concept of almost Ricci solitons.

Deep Classification with Linearity-Enhanced Logits to Softmax Function.

Recently, there has been a rapid increase in deep classification tasks, such as image recognition and target detection. As one of the most crucial components in Convolutional Neural Network (CNN) architectures, softmax arguably encourages CNN to achieve better performance in image recognition. Under this scheme, we present a conceptually intuitive learning objection function: Orthogonal-Softmax. The primary property of the loss function is to use a linear approximation model that is designed by Gram-Schmidt orthogonalization. Firstly, compared with the traditional softmax and Taylor-Softmax, Orthogonal-Softmax has a stronger relationship through orthogonal polynomials expansion. Secondly, a new loss function is advanced to acquire highly discriminative features for classification tasks. At last, we present a linear softmax loss to further promote the intra-class compactness and inter-class discrepancy simultaneously. The results of the widespread experimental discussion on four benchmark datasets manifest the validity of the presented method. Besides, we want to explore the non-ground truth samples in the future.

Inverse of initial stress in carbon fiber reinforced polymer laminates using lamb waves and deep neural network

The prediction of the initial stress in composites is essential for the non-destructive testing (NDT) and structural health monitoring (SHM) of carbon fibre reinforced polymer (CFRP). This paper examines the potential of Lamb waves in the inverse of initial stress by calculating the influence of initial stress on the dispersion characteristics of Lamb waves propagating in multilayered CFRP laminates. By introducing the mechanics of incremental deformation into the linear three-dimensional elasticity theory, the Legendre orthogonal polynomial expansion (LOPE) method is used to mathematically model the Lamb wave propagating in multilayered CFRP laminates subjected to horizontal and vertical homogeneous initial stresses. Then, a three-hidden-layers Feed Forward Deep Neural Network (DNN) with Back Propagation (BP) algorithm is constructed to invert the magnitude and direction of the initial stresses. The input features are the phase velocities of fundamental Lamb wave A0 mode at five different frequencies. Both training and testing samples are obtained by LOPE forward calculation. An ablation experiment is presented to compare the two different activation functions. Finally, the accuracy of the inverse is verified by comparing with the available outcomes of LOPE forward calculation.

Generalized Bessel Gaussian vortex in the turbulent media of absorbing seawater

We investigate the effects of turbulent absorbing seawater and generalized Bessel Gaussian (GBG) vortex structural parameters on the transmission of orbital angular momentum (OAM). The receiving probability spectrum and bit error probability of OAM mode carried by GBG vortex are deduced with the orthogonal spiral phase expansion of random GBG vortex and signal-to-noise ratio of OAM mode. By numerical analysis, the underwater transmission invariance of GBG vortex is studied, and the optimal aperture angle of GBG vortex corresponding to the maximum received probability of OAM signal is obtained. The impact of seawater parameters and optical transceiver systems on bit error probability is analyzed by using optimal aperture angle. We found that the transmission invariance of OAM modes of GBG vortex is well maintained even in weak turbulent absorbing seawater. Moreover, received probability and bit error probability are significantly affected even if the deviation of the aperture angle from its optimal position is minimal, and Durnin-ring radius should be the minimum within the experimental allowable range. According to signal transmission distance, the selection of high SNR transceiver system, optimal aperture angle, low order OAM topological charge, and minimum Durnin-ring radius allowed by the experiment can optimize the OAM signal transmission efficiency.

Numerical solution of neutral delay differential equations using orthogonal neural network

In this paper, an efficient orthogonal neural network (ONN) approach is introduced to solve the higher-order neutral delay differential equations (NDDEs) with variable coefficients and multiple delays. The method is implemented by replacing the hidden layer of the feed-forward neural network with the orthogonal polynomial-based functional expansion block, and the corresponding weights of the network are obtained using an extreme learning machine(ELM) approach. Starting with simple delay differential equations (DDEs), an interest has been shown in solving NDDEs and system of NDDEs. Interest is given to consistency and convergence analysis, and it is seen that the method can produce a uniform closed-form solution with an error of order 2^{-n}, where n is the number of neurons. The developed neural network method is validated over various types of example problems(DDEs, NDDEs, and system of NDDEs) with four different types of special orthogonal polynomials.

Pre-quake frequency characteristics of Ms ≥7.0 earthquakes in mainland China

In this study, natural orthogonal expansion was performed on earthquake frequencies to compute the pre-quake frequency fields of 9 Ms ≥7.0 earthquakes in mainland China from 1980 to 2020. The temporal and spatial pre-quake anomalies of these earthquakes were extracted from their frequency fields. We found that the majority of pre-quake temporal anomalies (i.e., variations exceeding two-times the absolute mean square error) of a strong earthquake are condensed within the first four frequency fields, and typically comprise multiple components. The temporal factor of the first frequency field usually accounts for the largest proportion of these anomalies (40%–60% of the entire field). Most Ms ≥7.0 earthquakes exhibited long-term anomalies 5–8 years before their occurrence; some presented medium-term anomalies 1–2 years prior to the quake, and only a few presented short-term and imminent anomalies (≤3 months before the quake). Anomalous seismic hazard zones have high-gradient turning points in regional frequency-field contour maps, and the epicenters of strong earthquakes are often located in areas containing active faults that have contour values. Through the comparison of seismic frequency field and the traditional method of regional seismic activity frequency (3 months), it is shown that the frequency-field time factor has the advantages of diversified and rich abnormal information. The slope comparison between the frequency field and the cumulative frequency curve shows that the frequency anomaly time of the two is consistent, and the conclusion is reliable. Therefore, the seismic frequency method can predict the occurrence time and location of strong earthquakes, which is closer to the predictable seismic model.

Application of Nonlinear Differential Equation in Electric Automation Control System

Abstract This article uses fifth-order nonlinear differential equations to describe the dynamic process of electrical automation control systems. This method first derives the equivalent system of the nonlinear fuzzy global system and then uses the orthogonal polynomial series expansion technique and its integral operation matrix. The local manifold at the dominant unstable equilibrium point of a single-machine infinite-bus system after a failure described by a two-dimensional quadratic nonlinear differential equation is calculated, and the stability boundary of the power system is obtained. The research results show that the output frequency fluctuation of the electrical automation control system is small after the algorithm is adopted, and the intelligent control system can accurately diagnose and warn the electrical faults. The system can meet the requirements of online voltage coordinated control.

The dynamic behavior of a piezoelectric strip with a salient and an inclusion under guided SH waves

Dynamic stress around the inclusion and salient in a piezoelectric strip subjected to dynamic incident anti-plane shearing (guided SH wave) is calculated. Using the repeated image superposition method, the analytical expression for scattering wave, satisfying stress-free and electric insulation conditions at the upper and lower horizontal boundaries of the strip, is obtained. The integral equations are set up with the boundary conditions and solved by orthogonal function expansion and effective truncation techniques. The obtained results clarify the characteristic features of the influence on the dynamic stress concentration factor and electric field intensity concentration factor. The analytical solutions are compared to the finite element modeling results to verify the accuracy of the conclusions of this paper.

Nonorthogonal Active Space Decomposition of Wave Functions with Multiple Correlation Mechanisms

The nonorthogonal active space decomposition (NO-ASD) methodology is proposed for describing systems containing multiple correlation mechanisms. NO-ASD partitions the wave function by a correlation mechanism, such that the interactions between different correlation mechanisms are treated with an effective Hamiltonian approach, while interactions between correlated orbitals in the same correlation mechanism are treated explicitly. As a result, the determinant expansion scales polynomially with the number of correlation mechanisms rather than exponentially, which significantly reduces the factorial scaling associated with the size of the correlated orbital space. Despite the nonorthogonal framework of NO-ASD, the approach can take advantage of computational efficient matrix element evaluation when performing nonorthogonal coupling of orthogonal determinant expansions. In this work, we introduce and examine the NO-ASD approach in comparison to complete active space methods to establish how the NO-ASD approach reduces the problem dimensionality and the extent to which it affects the amount of correlation energy recovered. Calculations are performed on ozone, nickel-acetylene, and isomers of μ-oxo dicopper ammonia.