Search Results (16)

A common and simple estimate of variability is the sample range, which is the difference between the maximum and minimum values in the sample. While other measures of variability are preferred in most instances, process owners and operators regularly use range (R) control charts to monitor process variability. The center line and limits of the R charts use constants that are based on the first two moments (mean and variance) of the distribution of the range of normal random variables. Historically, the computation of moments requires the use of tabulated constants approximated using numerical integration. We provide exact results for the moments for sample sizes 2 through 5. For sample sizes from 6 to 1000, we used the differential correction method to find Chebyshev minimax rational-function approximations of the moments. The rational function we recommend for the mean (R-chart constant d₂) has a polynomial of order two in the numerator and six in the denominator and achieves a maximum error of 4.4 × 10⁻⁶. The function for the standard deviation (R-chart constant d₃) has a polynomial of order two in the numerator and seven in the denominator and achieves a maximum error of 1.5 × 10⁻⁵. The exact and approximate expressions eliminate the need for table lookup in the control chart design phase. Full article

This paper investigates a class of Jacobsthal-type polynomials (JTPs) that involves one parameter. We present several new formulas for these polynomials, including expressions for their derivatives, moments, and linearization formulas. The key idea behind the derivation of these formulas is based on developing a new connection formula that expresses the shifted Chebyshev polynomials of the third kind in terms of the JTPs. This connection formula is used to deduce a new inversion formula of the JTPs. Therefore, by utilizing the power form representation of these polynomials and their corresponding inversion formula, we can derive additional expressions for them. Additionally, we compute some definite integrals based on some formulas of these polynomials. Full article

In order to achieve efficient recognition of 3D images and reduce the complexity of network parameters, we proposed a novel 3D image recognition method combining deep neural networks with fractional-order Chebyshev moments. Firstly, the fractional-order Chebyshev moment (FrCM) unit, consisting of Chebyshev moments and the three-term recurrence relation method, is calculated separately using successive integrals. Next, moment invariants based on fractional order and Chebyshev moments are utilized to achieve invariants for image scaling, rotation, and translation. This design aims to enhance computational efficiency. Finally, the fused network embedding the FrCM unit (FrCMs-DNNs) extracts depth features to analyze the effectiveness from the aspects of parameter quantity, computing resources, and identification capability. Meanwhile, the Princeton Shape Benchmark dataset and medical images dataset are used for experimental validation. Compared with other deep neural networks, FrCMs-DNNs has the highest accuracy in image recognition and classification. We used two evaluation indices, mean square error (MSE) and peak signal-to-noise ratio (PSNR), to measure the reconstruction quality of FrCMs after 3D image reconstruction. The accuracy of the FrCMs-DNNs model in 3D object recognition was assessed through an ablation experiment, considering the four evaluation indices of accuracy, precision, recall rate, and F1-score. Full article

The dynamic behavior of liquid storage tanks is one of the research issues about fluid–structure interaction problems. The analysis errors of the dynamics of multiple adjacent tanks can exist if neglecting soil–tank interaction since tanks are typically supported on flexible soil. In the present paper, the dynamics of a group of baffled cylindrical storage tanks supported on a circular surface foundation and undergoing horizontal excitation are analytically examined. For upper multiple tank–liquid–baffle subsystems, accurate solutions to the velocity potential for liquid sloshing are acquired according to the subdomain partition technique. A theoretical model is utilized to portray the continuous sloshing of each tank. For the soil–foundation subsystem, a lumped-parameter model is used to characterize the impacts of soil on upper-tank structures using Chebyshev complex polynomials that present the fitting results of horizontal, rocking, and coupling impedance functions. Then, a model of the soil–foundation–tank–liquid–baffle system is constructed on the basis of the substructure approach. The present sloshing frequencies, sloshing height, and hydrodynamic shear as well as the moment under rigid/soft soil foundations are compared to the available exact results and the numerical results to prove the validity of the present model. The error of the maximum sloshing height between the present and the numerical solutions is within 5.27%; the solution efficiency of system dynamics from the present model is 40–50 times faster than that from the ADINA model. A detailed parameter analysis of the dynamic characteristics and earthquake responses of the coupling system is presented. The research novelty is that an equivalent analytical model is presented, and it allows for investigating the dynamics of soil-supported multiple cylindrical tanks with a baffle, providing acceptable accuracy and high calculation efficiency. Full article

Around the globe, agriculture has grown to a point where it is now a financially feasible way to produce more sophisticated cultivation methods. Throughout the long tradition of agriculture, this represents a pivotal moment. The widespread adoption of data and the latest technological advances in the contemporary period allowed this paradigm change. However, pests remain to blame for significant harm done to crops, which has a detrimental impact on finances, the natural world, and society. This highlights the necessity of using automated techniques to apprehend pests before they cause widespread harm. Agriculture-related issues are currently the predominant subject for research that utilizes ML. The overarching aim of this investigation is the development of an economically feasible method for pest detection in vast fields of crops that IoT enables through the use of pest audio sound analytics. The recommended approach incorporates numerous acoustic preparation methods from audio sound analytics. The Chebyshev filter; the Welch method; the non-overlap-add method; FFT, DFT, STFT, and LPC algorithms; acoustic sensors; and PID sensors are among them. Eight hundred pest sounds were examined for features and statistical measurements before being incorporated into Multilayer Perceptron (MLP) for training, testing, and validation. The experiment’s outcomes demonstrated that the proposed MLP model triumphed over the currently available DenseNet, VGG-16, YOLOv5, and ResNet-50 approaches alongside an accuracy of 99.78%, a 99.91% sensitivity, a 99.64% specificity, a 99.59% recall, a 99.82% F1 score, and a 99.85% precision. The significance of the findings rests in their potential to proactively identify pests in large agricultural fields. As a result, the cultivation of crops will improve, leading to increased economic prosperity for agricultural producers, the country, and the entire globe. Full article

A hybrid method combining the adaptive cross approximation method (ACA) and the Chebyshev approximation technique (CAT) is presented for fast wideband BCS prediction of arbitrary-shaped 3D targets based on non-cooperative radiation sources. The incident and scattering angles can be computed by using their longitudes, latitudes and altitudes according to the relative positions of the satellite, the target and the passive bistatic radar. The ACA technique can be employed to reduce the memory requirement and computation time by compressing the low-rank matrix blocks. By exploiting the CAT into ACA, it is only required to calculate the currents at several Chebyshev–Gauss frequency sampling points instead of direct point-by-point simulations. Moreover, a wider frequency band can be obtained by using the Maehly approximation. Three numerical examples are presented to validate the accuracy and efficiency of the hybrid ACA-CAT method. Full article

In this study, the liquid sloshing in a cylindrical tank considering soil–structure interaction and undergoing horizontal excitation is investigated analytically. Multiple rigid annular baffles are positioned on the rigid wall to mitigate the liquid sloshing. Firstly, combined with the subdomain partition method for sloshing, the complex liquid domain is partitioned into simple subdomains with the single condition for boundary. Based on continuity conditions of velocity and pressure as well as the linear sloshing equation for free surface, the exact solution for convective velocity potential is derived with high accuracy. By yielding the similar hydrodynamic shear and moment as those of the original system, a mechanical model is developed to describe continuous sloshing, and parameters of the model are given in detail. Then, by means of the least squares approach, the Chebyshev polynomials are utilized to fit impedances for the circular surface foundation. A lumped parameter model is employed to represent influences of soil on the superstructure. Finally, by using the substructure method, a coupling model of the soil–tank system is developed to simplify the dynamic analysis. Comparison investigations are carried out to verify the effectiveness of the model. Detailed sloshing characteristics and dynamic responses of sloshing are analyzed with regard to different baffle sizes and positions as well as soil parameters, respectively. The novelty of the present study is that an equivalent analytical model for the soil–foundation–tank–liquid system with multiple baffles is firstly obtained and it allows the dynamic behaviors of the coupling system to be investigated with high computation efficiency and acceptable accuracy. Full article

This paper investigates certain Jacobi polynomials that involve one parameter and generalize the well-known orthogonal polynomials called Chebyshev polynomials of the third-kind. Some new formulas are developed for these polynomials. We will show that some of the previous results in the literature can be considered special ones of our derived formulas. The derivatives of the moments of these polynomials are derived. Hence, two important formulas that explicitly give the derivatives and the moments of these polynomials in terms of their original ones can be deduced as special cases. Some new expressions for the derivatives of different symmetric and non-symmetric polynomials are expressed as combinations of the generalized third-kind Chebyshev polynomials. Some new linearization formulas are also given using different approaches. Some of the appearing coefficients in derivatives and linearization formulas are given in terms of different hypergeometric functions. Furthermore, in several cases, the existing hypergeometric functions can be summed using some standard formulas in the literature or through the employment of suitable symbolic algebra, in particular, Zeilberger’s algorithm. Full article

This paper investigates the polynomial stability of neutral stochastic pantograph differential equations with Markovian switching (NSPDEsMS). Firstly, under the local Lipschitz condition and a more general nonlinear growth condition, the existence and uniqueness of the global solution to the addressed NSPDEsMS is considered. Secondly, by adopting the Razumikhin approach, one new criterion on the qth moment polynomial stability of NSPDEsMS is established. Moreover, combining with the Chebyshev inequality and the Borel–Cantelli lemma, the almost sure polynomial stability of NSPDEsMS is examined. The results derived in this paper generalize the previous relevant ones. Finally, two examples are provided to illustrate the effectiveness of the theoretical work. Full article

Copyright protection of medical images is a vital goal in the era of smart healthcare systems. In recent telemedicine applications, medical images are sensed using medical imaging devices and transmitted to remote places for screening by physicians and specialists. During their transmission, the medical images could be tampered with by intruders. Traditional watermarking methods embed the information in the host images to protect the copyright of medical images. The embedding destroys the original image and cannot be applied efficiently to images used in medicine that require high integrity. Robust zero-watermarking methods are preferable over other watermarking algorithms in medical image security due to their outstanding performance. Most existing methods are presented based on moments and moment invariants, which have become a prominent method for zero-watermarking due to their favorable image description capabilities and geometric invariance. Although moment-based zero-watermarking can be an effective approach to image copyright protection, several present approaches cannot effectively resist geometric attacks, and others have a low resistance to large-scale attacks. Besides these issues, most of these algorithms rely on traditional moment computation, which suffers from numerical error accumulation, leading to numerical instabilities, and time consumption and affecting the performance of these moment-based zero-watermarking techniques. In this paper, we derived multi-channel Gaussian–Hermite moments of fractional-order (MFrGHMs) to solve the problems. Then we used a kernel-based method for the highly accurate computation of MFrGHMs to solve the computation issue. Then, we constructed image features that are accurate and robust. Finally, we presented a new zero-watermarking scheme for color medical images using accurate MFrGHMs and 1D Chebyshev chaotic features to achieve lossless copyright protection of the color medical images. We performed experiments where their outcomes ensure the robustness of the proposed zero-watermarking algorithms against various attacks. The proposed zero-watermarking algorithm achieves a good balance between robustness and imperceptibility. Compared with similar existing algorithms, the proposed algorithm has superior robustness, security, and time computation. Full article

The principal objective of this article is to develop new formulas of the so-called Chebyshev polynomials of the fifth-kind. Some fundamental properties and relations concerned with these polynomials are proposed. New moments formulas of these polynomials are obtained. Linearization formulas for these polynomials are derived using the moments formulas. Connection problems between the fifth-kind Chebyshev polynomials and some other orthogonal polynomials are explicitly solved. The linking coefficients are given in forms involving certain generalized hypergeometric functions. As special cases, the connection formulas between Chebyshev polynomials of the fifth-kind and the well-known four kinds of Chebyshev polynomials are shown. The linking coefficients are all free of hypergeometric functions. Full article

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type $4F_3\left(1\right)$, which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method. Full article

In this work, we have presented a general framework for reconstruction of intensity images based on new sets of Generalized Fractional order of Chebyshev orthogonal Moments (GFCMs), a novel set of Fractional order orthogonal Laguerre Moments (FLMs) and Generalized Fractional order orthogonal Laguerre Moments (GFLMs). The fractional and generalized recurrence relations of fractional order Chebyshev functions are defined. The fractional and generalized fractional order Laguerre recurrence formulas are given. The new presented generalized fractional order moments are tested with the existing orthogonal moments classical Chebyshev moments, Laguerre moments, and Fractional order Chebyshev Moments (FCMs). The numerical results show that the importance of our general framework which gives a very comprehensive study on intensity image representation based GFCMs, FLMs, and GFLMs. In addition, the fractional parameters give a flexibility of studying global features of images at different positions and scales of the given moments. Full article

Accurate combustion state recognition of flame images not only plays an important role in social security, but also contributes to increasing thermal efficiency and product quality. To improve the accuracy of feature extraction and achieve the combustion state recognition, a novel method based on radial Chebyshev moment invariants (RCMIs) and an improved firefly algorithm-wavelet support vector machine (IFA-WSVM) model is proposed. Firstly, the potential flame pixels and the potential flame contour are obtained in the pre-processing phase. Then, the rotation, translation and scaling (RTS) invariants of radial Chebyshev moments are derived. Combing the region and contour moments, the RCMIs of pre-processed and edge images are calculated to construct multi-feature vectors. To enhance the recognition performance, an IFA-WSVM model is built, where the IFA is applied to search the best parameters of WSVM. Then, the IFA-WSVM model is used to recognize the combustion state. Finally, the result for case studies show that the proposed method is superior to methods based on HMIs and ZMIs, achieving the highest rate of 99.07% in real time. The IFA algorithm also outperforms other benchmark algorithms. Even for the images transformed by RTS and small size of training sets, the proposed method continues to exhibit the best performance. Full article

Automatic fire detection, which can detect and raise the alarm for fire early, is expected to help reduce the loss of life and property as much as possible. Due to its advantages over traditional methods, image processing technology has been applied gradually in fire detection. In this paper, a novel algorithm is proposed to achieve fire image detection, combined with Tchebichef (sometimes referred to as Chebyshev) moment invariants (TMIs) and particle swarm optimization-support vector machine (PSO-SVM). According to the correlation between geometric moments and Tchebichef moments, the translation, rotation, and scaling (TRS) invariants of Tchebichef moments are obtained first. Then, the TMIs of candidate images are calculated to construct feature vectors. To gain the best detection performance, a PSO-SVM model is proposed, where the kernel parameter and penalty factor of support vector machine (SVM) are optimized by particle swarm optimization (PSO). Then, the PSO-SVM model is utilized to identify the fire images. Compared with algorithms based on Hu moment invariants (HMIs) and Zernike moment invariants (ZMIs), the experimental results show that the proposed algorithm can improve the detection accuracy, achieving the highest detection rate of 98.18%. Moreover, it still exhibits the best performance even if the size of the training sample set is small and the images are transformed by TRS. Full article