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The article considers numerical algorithms for determining the coefficients of polynomials with a fixed leading coefficient, the algorithms supplying a minimum deviation from zero in a minimax norm with a given weight function. The polynomials serve as a useful tool in many numerical methods, in particular, in the Lanczos’ tau method which provides an approximate numerical analytic solution of ordinary differential equations with coefficients as polynomials in the independent variable. The well-known Chebyshev polynomials determined analytically are the special case of such polynomials, however, in most cases of weight functions, such polynomials can only be determined and tabulated numerically.

The privacy and security of the information exchanged between automobiles in 5G-enabled vehicular networks is at risk. Several academics have offered a solution to these problems in the form of an authentication technique that uses an elliptic curve or bilinear pair to sign messages and verify the signature. The problem is that these tasks are lengthy and difficult to execute effectively. Further, the needs for revoking a pseudonym in a vehicular network are not met by these approaches. Thus, this research offers a fog computing strategy for 5G-enabled automotive networks that is based on the Chebyshev polynomial and allows for the revocation of pseudonyms. Our solution eliminates the threat of an insider attack by making use of fog computing. In particular, the fog server does not renew the signature key when the validity period of a pseudonym-ID is about to end. In addition to meeting privacy and security requirements, our proposal is also resistant to a wide range of potential security breaches. Finally, the Chebyshev polynomial is used in our work to sign the message and verify the signature, resulting in a greater performance cost efficiency than would otherwise be possible if an elliptic curve or bilinear pair operation had been employed. [ABSTRACT FROM AUTHOR]

A reliability‐based design optimization (RBDO) method of a car body is presented on basis of dimension‐reduced Chebyshev polynomial method (DCM). To improve calculation efficiency and save computational time, complex models are often approximated by metamodels in reliability analysis. Traditional metamodels require a large number of sample points, which is time‐consuming. To improve the efficiency, DCM is proposed to approximate the performance function of the car body. First, the performance function is decomposed by the dimension‐reduction method into a sum of univariate functions, which are then fitted through Chebyshev polynomials. The reliability of the car body is predicted by the Taylor expansion method and the fourth‐moment method. Finally, the result of RBDO is obtained using an improved adaptive genetic algorithm. The proposed method saves on the calculation time with high precision. Besides, the improved adaptive genetic algorithm reduces the number of iterations in the car body optimization and improves the efficiency. [ABSTRACT FROM AUTHOR]

This paper explores the effect of meteorological factors such as rainfall, temperature, sunshine, wind speed, and relative humidity on the yield of maize (Zea mays L.) and sorghum (Sorghum bicolor L.) at different growth stages in Togo's Plateau, Central, and Savannah regions. For this purpose, data from 1990 to 2019 on weather variables and maize and sorghum yields were used. The study applied Fisher's meteorological regression and Chebyshev polynomial function. Our findings revealed that rainfall had a more beneficial than detrimental effect on maize and sorghum yield across stages and regions. Contrariwise, temperature influence was as beneficial as detrimental and more significant across all growth stages of maize and sorghum in the Savannah and Plateau regions. Furthermore, the sunshine effect on maize yield was more significant in the Central and Savannah regions, while negative on sorghum yield in all the growth stages in the Central region. Similarly, the wind speed was also beneficial and detrimental to maize and sorghum yields, although it was more significant for sorghum in Plateau and Savannah regions. Lastly, relative air humidity positively and negatively influenced maize and sorghum yields in all the growth stages and regions for maize and the Plateau and Savannah regions for sorghum. Therefore, there is a need for real-time agricultural meteorological information to help farmers plan crop production more efficiently and increase crop yield. [ABSTRACT FROM AUTHOR]

When one solves differential equations by a spectral method, it is often convenient to shift from Chebyshev polynomials Tn(x) with coefficients an to modified basis functions that incorporate the boundary conditions. For homogeneous Dirichlet boundary conditions, u(±1) = 0, popular choices include the "Chebyshev difference basis" ςn(x) ≡ Tn+2(x)−Tn(x) with coefficients here denoted by bn and the "quadratic factor basis" ϱn(x) ≡ (1 − x2)Tn(x) with coefficients cn. If u(x) is weakly singular at the boundary, then the coefficients an decrease proportionally to O (A (n) / n κ) for some positive constant κ, where A(n) is a logarithm or a constant. We prove that the Chebyshev difference coefficients bn decrease more slowly by a factor of 1/n while the quadratic factor coefficients cn decrease more slowly still as O (A (n) / n κ − 2 ) . The error for the unconstrained Chebyshev series, truncated at degree n = N, is O (| A (N) | / N κ) in the interior, but is worse by one power of N in narrow boundary layers near each of the endpoints. Despite having nearly identical error norms in interpolation, the error in the Chebyshev basis is concentrated in boundary layers near both endpoints, whereas the error in the quadratic factor and difference basis sets is nearly uniformly oscillating over the entire interval in x. Meanwhile, for Chebyshev polynomials, the values of their derivatives at the endpoints are O (N 2) , but only O (N) for the difference basis. Furthermore, we give the asymptotic coefficients and rigorous error estimates of the approximations in these three bases, solved by the least squares method. We also find an interesting fact that on the face of it, the aliasing error is regarded as a bad thing; actually, the error norm associated with the downward curving spectral coefficients decreases even faster than the error norm of infinite truncation. But the premise is under the same basis, and when involving different bases, it may not be established yet. [ABSTRACT FROM AUTHOR]

We present methods to write Tn(x), Chebyshev polynomials of the first kind, as a product of minimal polynomials of cos(2π/m) over integers, Ψm(x), and to write Ψm(x) as ratios of expressions in Tn(x). We also prove the relation Ψm(Tn(x)) = Ψn(Tm(x)) for m, n having the same prime divisors and use it to express Ψm(x) as a linear combination of Tn(x) for certain values of m. [ABSTRACT FROM AUTHOR]

The mode shapes of many beam-type structures, such as aircraft wings and wind turbine blades, involve a high degree of coupling between flap-wise and edge-wise bending deformations. In the case of wind turbine blades, the principal bending deformations (flap-wise deformation and edge-wise deformation) are typically easily recognized by visual observation. However, this visual approach is sometimes challenging for high-order mode shapes that have complex mode deformation. More importantly, visual observation cannot quantify the contribution of different deformation components for each mode. This work proposes a novel mode shape recognition algorithm, called Mode Shape Recognition Matrix (MSRM), for application to complicated spatial beam-type structures not only to identify the deformation components of the complex beam mode shapes, but more importantly, to quantify their respective relative contribution. In the application case studied for the MSRM method, a three-dimensional wind turbine blade is mapped into three-dimensional Chebyshev polynomial space. The blade mode shape is correlated to each polynomial shape function to find the contribution of each polynomial shape function for the mode shape. To validate the mode shape recognition performance, MSRM is applied on both numerical mode shapes from a fidelity blade finite element model and experimental mode shapes from a high spatial resolution 3D SLDV modal test. Both numerical and experimental studies demonstrate that MSRM can successfully recognize the quantitative contribution of flap-wise deformation and edge-wise deformation for each wind turbine blade mode shape. [ABSTRACT FROM AUTHOR]

The moment method technique has been developed to examine the scattering properties of a perfectly conducting microstrip patch which is printed on isotropic or uniaxial anisotropic substrate. The choice of the asymptotic basis functions defined over the patch is illustrated to model the unknown currents. The resonant frequency of a rectangular microstrip patch antenna using these different asymptotic basis functions is investigated. Also the CPU-time for the resonant frequency of a uniaxial anisotropy substrate is presented for these asymptotic currents. Comparisons are made, and show that the utilization of the asymptotic Chebychev basis function provides a significant improvement in the computation time over the sinusoid entire domain forms in the evaluation of the resonant frequency of a microstrip patch antenna. Convergent solutions are in good agreement with the exact sinusoid basis function without edge condition and with those obtained from literature. [ABSTRACT FROM AUTHOR]

In this paper we propose a novel family of weighted orthonormal rational functions on a semi-infinite interval. We write a sequence of integer-coefficient polynomials in several forms and derive their corresponding differential equations. These equations do not form Sturm-Liouville problems. We overcome this disadvantage by multiplying some factors, resulting in a sequence of irrational functions. We deduce various generating functions of this sequence and find the associated Sturm-Liouville problems, which bring orthogonality. Then we establish a Hilbert space of functions defined on a semi-infinite interval with an inner product induced by a weight function determined by the Sturm-Liouville problems mentioned above. The even subsequence of the irrational function sequence above forms an orthonormal basis for this space. The Fourier expansions of some power functions are deduced. The sequence of non-positive integer power functions forms a non-orthogonal basis for this Hilbert space. We give two examples, one example of Fourier expansion and one example of interpolation as applications. • The properties of a family of integer-coefficient polynomials are studied. • A novel family of orthonormal rational functions on a semi-infinite interval is proposed by applying Sturm-Liouville theory. • A new Hilbert space is established with two bases and the Fourier expansions of some functions therein are discussed. [ABSTRACT FROM AUTHOR]

• An adaptive dimension-reduction Chebyshev metamodel (ADC) is proposed to balance the accuracy and efficiency of dimension-reduction Chebyshev metamodels. • The ADC can be used to approximated the complex model with high dimension accurately. • The sample points are selected by the adaptive method which can reduce the number of sample points and approve the efficiency. An adaptive dimension-reduction Chebyshev metamodel (ADC) is proposed to balance the accuracy and efficiency of dimension-reduction Chebyshev metamodels. A univariate dimension-reduction Chebyshev metamodel (UDC) is constructed by the dimension-reduction method and the Chebyshev metamodel. Based on the UDC, the bivariate terms largely impacting the metamodel are selected using an adaptive selection method, and are combined with the UDC to construct the ADC. The ADC has higher accuracy than the UDC because more calculated sample points are added. Compared with the bivariate dimension-reduction Chebyshev metamodel, the ADC needs fewer sample points and has higher efficiency. The result of numerical examples illustrate that ADC has higher accuracy compared with other commonly-used metamodels and is more suitable for approximating high-dimensional complex models. [ABSTRACT FROM AUTHOR]

The inherent non-smoothness of the vibroimpact system leads to complex behaviors and a strong sensitivity to parameter changes. Unfortunately, uncertainties and errors in system parameters are inevitable in mechanical engineering. Therefore, investigations of dynamical behaviors for vibroimpact systems with stochastic parameters are highly essential. The present study aims to analyze the dynamical characteristics of the three-degree-of-freedom vibroimpact system with an uncertain parameter by means of the Chebyshev polynomial approximation method. Specifically, the vibroimpact system model considered is one with unilateral constraint. Firstly, the three-degree-of-freedom vibroimpact system with an uncertain parameter is transformed into an equivalent deterministic form using the Chebyshev orthogonal approximation. Then, the ensemble means responses of the stochastic vibroimpact system are derived. Numerical simulations are performed to verify the effectiveness of the approximation method. Furthermore, the periodic and chaos motions under different system parameters are investigated, and the bifurcations of the vibroimpact system are analyzed with the Poincaré map. The results demonstrate that both the restitution coefficient and the random factor can induce the appearance of the periodic bifurcation. It is worth noting that the bifurcations fundamentally differ between the stochastic and deterministic systems. The former has a bifurcation interval, while the latter occurs at a critical point.

Trajectory prediction is gaining attention as a form of situational awareness because it is an essential component of the support system of autonomous driving, particularly in urban areas. A promising application is cooperative driving automation, where the traffic scene is monitored by roadside sensors with undisrupted views. A critical problem is that these sensors are adversely affected by inclement weather, including drenching rain or large amounts of snow, in which case the reliability of the prediction results can be significantly compromised. To address these problems, this study proposes a framework for robust vehicle-trajectory predictions based on the Chebyshev transform. In the proposed framework, the original trajectory snippets (partial trajectories) are Chebyshev-transformed, and the resulting coefficients form new snippets. The LSTM (long-short term memory) encoder-decoder structure was trained and tested using these new coefficient snippets, which were extracted from a public vehicle trajectory dataset. The performance and robustness of the proposed framework were verified by emulating sensor data that were incomplete as a result of environmental factors. The proposed framework provides stable and accurate long-term trajectory prediction because the Chebyshev transform is robust to incomplete sensor data by virtue of its uniform nature.

A formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily nonnegative integers. The notion of formal weight enumerator was first introduced by Ozeki in connection with modular forms, and a systematic investigation of formal weight enumerators has been conducted by Chinen in connection with zeta functions and Riemann hypothesis for linear codes. In this paper, we establish a relation between formal weight enumerators and Chebyshev polynomials. Specifically, the condition for the existence of formal weight enumerators with prescribed parameters (n , ε , q) is given in terms of Chebyshev polynomials. According to the parity of n and the sign ε , the four kinds of Chebyshev polynomials appear in the statement of the result. Further, we obtain explicit expressions of formal weight enumerators in the case where n is odd or ε = - 1 using Dickson polynomials, which generalize Chebyshev polynomials. We also state a conjecture from a viewpoint of binomial moments, and see that the results in this paper partially support the conjecture. [ABSTRACT FROM AUTHOR]

We study the embedding of the Hamiltonian Cycle problem, on a symmetric graph, in a discounted Markov decision process. The embedding allows us to explore the space of occupational measures corresponding to that decision process. In this paper we consider a convex combination of a Hamiltonian cycle and its reverse. We show that this convex combination traces out an interesting "H-curve" in the space of occupational measures. Since such an H-curve always exists in Hamiltonian graphs, its properties may help in differentiating between graphs possessing Hamiltonian cycles and those that do not. Our analysis relies on the fact that the resolvent-like matrix induced by our convex combination can be expanded in terms of finitely many powers of probability transition matrices corresponding to that Hamiltonian cycle. We derive closed form formulae for the coefficients of these powers which are reduced to expressions involving the classical Chebyshev polynomials of the second kind. For regular graphs, we also define a function that is the inner product of points on the H-curve with a suitably defined center of the space of occupational measures and show that, despite the nonlinearity of the inner-product, this function can be expressed as a linear function of auxiliary variables associated with our embedding. These results can be seen as stepping stones towards developing constraints on the space of occupational measures that may help characterize non-Hamiltonian graphs. [ABSTRACT FROM AUTHOR]

In this paper, a new apodized fiber Bragg grating (FBG) structure, the Chebyshev apodization, is proposed. The Chebyshev polynomial distribution has been widely used for the optimal design of antennas and filters, but it has not been used for designing FBGs. Unlike the function of traditional Gaussian-apodized FBGs, the Chebyshev polynomial is a discrete function. We demonstrate a new methodology for designing Chebyshev-apodized FBGs: the grating region is divided by discrete n sections with uniform gratings, while the index change is used to express the Chebyshev polynomial. We analyze the Chebyshev-apodized FBGs by using coupled mode theory and the piecewise-uniform approach. The reflection spectrum and the dispersion of Chebyshev-apodized FBGs are calculated and compared with those of Gaussian FBGs. Moreover, a sidelobe suppression level (SSL), a parameter of the Chebyshev polynomial, along with the maximum ac-index change of FBGs are analyzed. Assume that the grating length is 20mm, SSL is 100 dB, the section number is 40, and the maximum ac-index change is 2 × 10−4. The reflection spectrum of Chebyshev apodized FBGs shows flattened sidelobes with an absolute SSL of −95.9 dB (corresponding to SSL=100 dB). The simulation results reveal that at the same full width at half maximum, the Chebyshev FBGs have lower sidelobe suppression than the Gaussian FBGs, but their dispersion is similar. We demonstrate the potential of using Chebyshev-apodized FBGs in optical filters, dispersion compensators, and sensors; Chebyshev apodization can be applied in the design of periodic dielectric waveguides. [ABSTRACT FROM AUTHOR]

One of the most critical aspects of this technologically progressive era is the propagation of information through an unsecured communication channel. The information is electronically transported as binary bits. One of the most important issues in the existing world is the secrecy of these digital contents. In this paper, we used Chebyshev's polynomial-based, chaotic maps to propose a new technique for encrypting images. The proposed encryption brings confusion as well as diffusion to the system presented, which is one of the utmost essential aspects of the encryption method. The confusion process is performed in two stages, XORing the original image with the Chebyshev polynomial generated matrices and scrambling, the chaotic scrambling matrix is also generated by the Chebyshev polynomial generated matrix, which increases the sensitivity of the cryptosystem to the key which initialize the parameters for Chebyshev polynomial. We checked our planned scheme against various performance analyses and contrasted them with established outcomes. The scheme is accomplished by offering excellent privacy to digital images. [ABSTRACT FROM AUTHOR]

In this paper, the exact rate of decrease of best approximations of non-integer numbers by polynomials with integer coefficients is established. [ABSTRACT FROM AUTHOR]

In this article, we discuss the Chebyshev Polynomial and its characteristics. The second order difference equation and the process obtaining the explicit solution of the Chebyshev polynomial have been given for each real number. The symmetry and orthogonality of the Chebyshev polynomial has also been demonstrated using the explicit solutions obtained. Furthermore, we have also given how to approx the polynomial function using the Chebyshev polynomials.

The aim of the present study is to investigate the nonlinear free vibration of spinning cylindrical shells under spinning and arbitrary boundary conditions. Artificial springs are used to simulate arbitrary boundary conditions. Sanders' shell theory is employed, and von Kármán nonlinear terms are considered in the theoretical modeling. By using Chebyshev polynomials as admissible functions, motion equations are derived with the Ritz method. Then, a direct iteration method is used to obtain the nonlinear vibration frequencies. The effects of the circumferential wave number, the boundary spring stiffness, and the spinning speed on the nonlinear vibration characteristics of the shells are highlighted. It is found that there exist sensitive intervals for the boundary spring stiffness, which makes the variation of the nonlinear frequency ratio more evident. The decline of the frequency ratio caused by the spinning speed is more significant for the higher vibration amplitude and the smaller boundary spring stiffness. [ABSTRACT FROM AUTHOR]

We study perfect state transfer in Grover walks, which are typical discrete-time quantum walk models. In particular, we focus on states associated to vertices of a graph. We call such states vertex type states. Perfect state transfer between vertex type states can be studied via Chebyshev polynomials. We derive a necessary condition on eigenvalues of a graph for perfect state transfer between vertex type states to occur. In addition, we perfectly determine the complete multipartite graphs whose partite sets are the same size on which perfect state transfer occurs between vertex type states, together with the time. [ABSTRACT FROM AUTHOR]

The security and privacy concerns in vehicular communication are often faced with schemes depending on either elliptic curve (EC) or bilinear pair (BP) cryptographies. However, the operations used by BP and EC are time-consuming and more complicated. None of the previous studies fittingly tackled the efficient performance of signing messages and verifying signatures. Therefore, a chaotic map-based conditional privacy-preserving authentication (CM-CPPA) scheme is proposed to provide communication security in 5G-enabled vehicular networks in this paper. The proposed CM-CPPA scheme employs a Chebyshev polynomial mapping operation and a hash function based on a chaotic map to sign and verify messages. Furthermore, by using the AVISPA simulator for security analysis, the results of the proposed CM-CPPA scheme are good and safe against general attacks. Since EC and BP operations do not employ the proposed CM-CPPA scheme, their performance evaluation in terms of overhead such as computation and communication outperforms other most recent related schemes. Ultimately, the proposed CM-CPPA scheme decreases the overhead of computation of verifying the signatures and signing the messages by 24.2% and 62.52%, respectively. Whilst, the proposed CM-CPPA scheme decreases the overhead of communication of the format tuple by 57.69%. [ABSTRACT FROM AUTHOR]

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The Human-Centered Internet of Things (HC-IoT) is fast becoming a hotbed of security and privacy concerns. Two users can establish a common session key through a trusted server over an open communication channel using a three-party authenticated key agreement. Most of the early authenticated key agreement systems relied on pairing, hashing, or modular exponentiation processes that are computationally intensive and cost-prohibitive. In order to address this problem, this paper offers a new three-party authenticated key agreement technique based on fractional chaotic maps. The new scheme uses fractional chaotic maps and supports the dynamic sensing of HC-IoT devices in the network architecture without a password table. The projected security scheme utilized a hash function, which works well for the resource-limited HC-IoT architectures. Test results show that our new technique is resistant to password guessing attacks since it does not use a password. Furthermore, our approach provides users with comprehensive privacy protection, ensuring that a user forgery attack causes no harm. Finally, our new technique offers better security features than the techniques currently available in the literature. [ABSTRACT FROM AUTHOR]

The privacy and security vulnerabilities in fifth-generation (5G)-enabled vehicular networks are often required to cope with schemes based on either bilinear pair cryptography (BPC) or elliptic curve cryptography (ECC). Nevertheless, these schemes suffer from massively inefficient performance related to signing and verifying messages in areas of the high-density traffic stream. Meanwhile, adversaries could launch side-channel attacks to obtain sensitive data protected in a tamper-proof device (TPD) to destroy the system. This paper proposes a Chebyshev polynomial-based scheme for resisting side-channel attacks in 5G-enabled vehicular networks. Our work could achieve both important properties of the Chebyshev polynomial in terms of chaotic and semi-group. Our work consists of five phases: system initialization, enrollment, signing, verification, and pseudonym renew. Moreover, to resist side-channel attacks, our work renews periodically and frequently the vehicle's information in the TPD. Security analysis shows that our work archives the privacy (pseudonym identity and unlikability) and security (authentication, integrity, and traceability) in 5G-enabled vehicular networks. Finally, our work does not employ the BPC or the ECC; its efficiency performance outperforms other existing recent works, making it suitable for use in vehicular networks. [ABSTRACT FROM AUTHOR]

The deceptive jamming technique can threaten a synthetic aperture radar (SAR) imaging system by repeatedly transmitting copied radar signals back to the radar. A conventional phased array can generate only an angle-dependent beam and cannot distinguish the jammer from other targets. Instead, a frequency diverse array (FDA) can offer a range-angle-dependent beampattern by introducing small frequency offsets to array elements, which can be adopted to distinguish signals from different range cells. Thus, considering that the position of the jammer is unknown, it is necessary to suppress the range peak sidelobe as much as possible. To lower this value, this communication proposes a weighted FDA to achieve the optimal solution at the expense of raising other range sidelobes. [ABSTRACT FROM AUTHOR]

Abstract This paper proposes a probabilistic small-signal stability analysis method based on the polynomial approximation approach. Since the correct determination of unknown coefficients has a direct effect on the accuracy of the polynomial approximation method, this paper presents a method for determining these coefficients, with high coverage on the probabilistic input domain of the problem. In this method, by increasing the number of random input variables, the proposed method can continue to maintain its efficiency. After determining the unknown coefficients, the load flow results and the system state matrix are determined for random changes of all loads based on the Hermite polynomial approximation. Then, the small-signal stability of the system is probabilistically evaluated based on stochastic analysis of the system eigenvalues. The consistency and validity of the proposed method are demonstrated based on the simulation studies in the MATLAB® software environment. In the simulation studies, the performance of the proposed method is examined by comparison with Point Estimation, Cumulant, Monte Carlo, and Chebyshev polynomial-based methods, for IEEE 14-bus and IEEE 39-bus test systems. Article highlights Probabilistic small-signal stability analysis of power systems Modeling of governing equations of power system based on Hermite polynomial approximation Forming the Collocation matrix based on a robust method

In this article, we propose Chebyshev spectral projection methods to solve the hypersingular integral equation of first kind. The presence of strong singularity in Hadamard sense in the first part of the integral equation makes it challenging to get superconvergence results. To overcome this, we transform the first kind hypersingular integral equation into a second kind integral equation. This is achieved by defining a bounded inverse of the hypersingular integral operator in some suitable Hilbert space. Using iterated Chebyshev spectral Galerkin method on the equivalent second kind integral equation, we obtain improved convergence of O (N − 2 r) , where N is the highest degree of Chebyshev polynomials employed in the approximation space and r is the smoothness of the solution. Further, using commutativity of projection operator and inverse of the hypersingular integral operator, we are able to obtain superconvergence of O (N − 3 r) and O (N − 4 r) , by Chebyshev spectral multi-Galerkin method (CSMGM) and iterated CSMGM, respectively. Finally, numerical examples are presented to verify our theoretical results. • Developed Chebyshev spectral projection methods for hypersingular integral equations. • Established error bounds and convergence rates for all the proposed methods. • Obtained superconvergence of O (N − 4 r) using iterated spectral multi-Galerkin method. • Verified theoretical superconvergence results through numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider discretely monitored double barrier option based on the Black-Scholes partial differential equation. In this scenario, the option price can be computed recursively upon the heat equation solution. Thus we propose a numerical solution by projection method. We implement this method by considering the Chebyshev polynomials of the second kind. Finally, numerical examples are carried out to show accuracy of the presented method and demonstrate acceptable accordance of our method with other benchmark methods.

• Comprehensive evaluation of ten cubic equation-solving methods. • Analytical technique transforms equations into Chebyshev polynomials. • Proposed iterative refinement strategy mitigates numerical errors. • Novel criterion using Vieta's formulas isolates error-prone areas. • Proposed solution method reduces computational time by 15–20 %. The study presents a comprehensive evaluation of ten distinct methods for solving cubic equations, encompassing both analytical, numerical, and linear algebra-based approaches. Among these, an analytical technique involving the transformation of cubic equations into Chebyshev polynomials emerged as the most efficient. To mitigate issues arising from numerical round-off errors, the study proposes an iterative refinement strategy. Furthermore, the research introduces a novel criterion grounded in Vieta's formulas to identify and isolate areas where round-off errors may occur. This criterion aids in the application of the proposed iterative refinement technique. The novel solution method, which involves the conversion of cubic equations into Chebyshev polynomials and subsequent error-checking and iterative refinement, exhibits significant advantages. Firstly, it proves efficient, reducing computational time by 15–20 % in comparison to the well-established Cardano analytical method. It also demonstrates robustness in effortlessly identifying regions of potential error and offers a method for iterative improvement when needed. Additionally, its reliability is underscored by its independence from the specific cubic equation of state under consideration, making it well-suited for the intensive solution of a wide range of cubic equations. [ABSTRACT FROM AUTHOR]

• Building high precision test platform for testing laboratory-developed ZrOxNy thin films temperature sensors. • Providing a Chebyshev polynomial-based BiLSTM (Bi-directional Long Short-Term Memory) algorithm (C-BiLSTM) and using it to process obtained data of temperature sensors. • C-BiLSTM are proved to be accurate and fast in prediction of temperature value. • The novel approach has a significant promotion for improving the accuracy of temperature measurement. Nonlinear error compensation is a significant factor that affects the measurement accuracy of temperature sensors. Cryogenic temperature sensors require a precise calibration technique to achieve accurate temperature measurements. BP (Back Propagation) neural networks are not suitable for sensor temperature prediction due to slow convergence and poor learning ability. In this study, a Chebyshev polynomial-based BiLSTM (Bi-directional Long Short-Term Memory) algorithm (C-BiLSTM) is proposed to improve the accuracy of measurement. Firstly, we demonstrate vacuum packaged temperature sensors based on zirconium oxynitride (ZrO x N y) thin films with ultra-high sensitivity at cryogenic temperatures. Secondly, a dataset consisting of sensor resistance and temperature values was obtained from five above-mentioned sensors, which contains 29 calibration points in the temperature range of 16 K-300 K measured by the Calibration System. Then, the dataset was divided into two temperature ranges (16 K-54.358 K and 40 K-300 K). In the range 16–54.358 K, 14 set-points are selected as training set and 3 set-points as testing set. In the range 40–300 K, 12 set-points are selected as training set and 2 set-points as testing set. Thirdly, a neural network model was built and trained using the TensorFlow framework. By comparing C-BiLSTM we proposed with the BP neural network and BiLSTM, the results show that the C-BiLSTM model converges faster and greatly improves the prediction accuracy after adding Chebyshev polynomial features. The fitting error and prediction error are less than 1 mK in the temperature range of 16 K-40 K. They can also keep less than 10 mK even at the wide temperature range of 40 K-300 K, which is a significant improvement respect to improving the accuracy of temperature measurement. [ABSTRACT FROM AUTHOR]

For large data fitting, the least squares progressive iterative approximation (LSPIA) methods have been proposed by Lin and Zhang (2013) and Deng and Lin (2014) , in which a constant step size is used. In this paper, we further accelerate the LSPIA method in terms of a Chebyshev semi-iterative scheme and present an asynchronous LSPIA (denoted by ALSPIA) method. The control points in ALSPIA are updated by using an extrapolated variant in which an adaptive step size is chosen according to the roots of Chebyshev polynomial. Our convergence analysis shows that ALSPIA is faster than the original LSPIA method in both singular and non-singular least squares fitting cases. Numerical examples show that the proposed algorithm is feasible and effective. • We propose a new variant of LSPIA (denoted by ALSPIA) by updating the control points with an adaptive step size. When the step size is the same constant, our method automatically reduces to the original LSPIA method. • We choose the step size according to the roots of Chebyshev polynomials. We prove that the ALSPIA method is convergent when the collocation matrix is rank deficient and of full-column rank. Convergence analysis reveals that ALSPIA has a faster convergence rate than LSPIA. • Numerical experiments show that the ALSPIA method is very robust and exhibits similar convergence behaviors with various selection orders for the step size, and outperforms classical LSPIA-type methods when fitting missing data, noisy data, and datasets of very large size. [ABSTRACT FROM AUTHOR]

Complementarity relations between various characterizations of a probability distribution are at the core of information theory. In particular, lower and upper bounds for the entropic function are of great importance. In applied topics, we often deal with situations where the sums of certain powers of probabilities are known. The main question is how to convert the imposed restrictions into two-sided estimates on the Shannon entropy. It is addressed in two different ways. The more intuitive of them is based on truncated expansions of the Taylor type. Another method is based on the use of coefficients of the shifted Chebyshev polynomials. We propose here a family of polynomials for estimating the Shannon entropy from below. As a result, estimates are more uniform in the sense that errors do not become too large in particular points. The presented method is used for deriving uncertainty and certainty relations for positive operator-valued measures assigned to a quantum design. Quantum designs are currently the subject of active researches due to potential use in quantum information science. It is shown that the derived estimates are applicable in quantum tomography and detecting steerability of quantum states. [ABSTRACT FROM AUTHOR]

Motivated by a famous question of Lehmer about the Mahler measure, we study and solve its analytic analogue. [ABSTRACT FROM AUTHOR]

The security of cryptosystems based on Chebyshev recursive relation, T n (x) = 2 x T n − 1 (x) − T n − 2 (x) , relies on the difficulty to find the large degree n of Chebyshev polynomial for given parameters. But this relation cannot be used to evaluate T n (x) , if n is very large. We investigate three other methods: matrix-multiplication-based evaluation, halve-and-square evaluation, and root-extraction-based evaluation. We find they have the same complexity O (log n log 2 p) over finite field p . The result shows that the hardness of some cryptosystems based on modular Chebyshev polynomials is almost equivalent to the general discrete logarithm. This partially answers the question why such chaos-based cryptosystems have rarely been put into practice. [ABSTRACT FROM AUTHOR]

The boundary value problem (BVP) for the varying coefficient linear Caputo-type fractional differential equation subject to the mixed boundary conditions on the interval 0 ≤ x ≤ 1 was considered. First, the BVP was converted into an equivalent differential–integral equation merging the boundary conditions. Then, the shifted Chebyshev polynomials and the collocation method were used to solve the differential–integral equation. Varying coefficients were also decomposed into the truncated shifted Chebyshev series such that calculations of integrals were only for polynomials and can be carried out exactly. Finally, numerical examples were examined and effectiveness of the proposed method was verified. [ABSTRACT FROM AUTHOR]

To explore the application of improved variational iterative method in solving fractional differential equations and the effect of DNA coding algorithm based on logistic chaos mapping in image encryption, combining Chebyshev polynomial with variational iterative method, a new algorithm for solving fractional differential equations is proposed. In order to solve the initial value problem, the simulation software is used to compare the approximate solution after the change of values of α and k in the solution process to investigate the influence of different parameters on the accuracy of fractional order system. Subsequently, the logistic chaos sequence is used to generate discrete binary sequences, and an image encryption algorithm based on DNA coding is proposed. In order to verify the encryption performance of the proposed algorithm, Matlab simulation software is used to simulate and verify the image encryption processing. The results show that when α approaches 1, the solution is closer to the exact solution than that when α approaches 0.5, 0.7, and 0.9; compared with 1, 3, 5, and 7, the constructed algorithm has the highest accuracy when k value is 9. The simulation results of the fuzzy partial fractional order system show that the chaotic sequence is sensitive to the initial value, and the logistic mapping sequence is suitable for the information encryption of the secure communication system. The simulation results of the image encryption algorithm based on DNA coding show that the algorithm can effectively encrypt the image, and the image histogram after encryption is approximately a horizontal line; slight changes in the initial value will affect the decryption effect of the image; the correlation analysis results show that there is no correlation between adjacent pixels in the image after encryption, which indicates that combining Chebyshev polynomials with variational iteration method can reduce the computational burden and improve the computational accuracy, and the image encryption algorithm proposed in this study can improve the reliability and security of image encryption. [ABSTRACT FROM AUTHOR]

Traffic flow forecasting is of great significance in the application of traffic management and urban planning. However, the existing forecasting methods cannot fully exploit the potential complex spatio-temporal correlations. In order to further explore the temporal and spatial characteristics of road network data to improve the prediction accuracy, this paper proposed an multi-spatial-temporal graph convolutional network (MST-GCN) model. Firstly, by using Chebyshev graph convolution (ChebNet) combined with gated recurrent unit (GRU) to construct spatio-temporal components to deeply mine the spatio-temporal correlation of nodes. Secondly, it extracted weekly, daily, and recent time sequence data separately, and entered three spatio-temporal components to deeply explore the time correlation between different time windows. Finally,it combined the spatio-temporal component and the encoder-decoder network structure to form the MST-GCN model. Experiments were conducted using the highway datasets PEMS04 and PEMS08 in the California Department of Transportation (Caltrans) performance evaluation system. The results show that the The new model has significantly better performance than the gated recurrent unit model and the recently proposed diffusion convolutional recurrent neural network (DCRNN), temporal graph convolutional network (T-GCN), attention based spatial-temporal graph convolutional networks (ASTGCN) and spatial-temporal synchronous graph convolutional network (STSGCN) models. [ABSTRACT FROM AUTHOR]

In order to increase the nonlinear fitting performance of functional link neural network (FLNN), a novel chebyshev functional link spline neural filter (CFLSNF) to apply in system identification is proposed. Compared with the weak nonlinearity and boundedness of the fixed activation function (e.g., $sigmoid$ and $tanh$), CFLSNF has stronger nonlinear approximation ability than FLNN due to the flexible interpolation ability of spline activation function. At the same time, the proposed CFLSNF eliminates the hidden layers by using Chebyshev polynomial to extend the input space into high dimensions, which shows certain computational advantages compared with the artificial neural network (ANN) structures. Moreover, to update the weights of the CFLSNF, the CFLSNF-LMS is also developed. The stability conditions and computational complexity are studied. Besides, in order to make CFLSNF structure suitable for impulsive noise interference environment, a robust algorithm based on maximum versoria criterion is also proposed. Finally, the validity of the proposed architecture and algorithm are verified by experiments. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and uniqueness of solution of the singular quadratic integral equation (SQIE). The Fredholm integral term is assumed in position with singular kernel. Under certain conditions and new discussions, the singular kernel will tend to a logarithmic kernel. Then, using Chebyshev polynomial, a main of spectral relationships are stated and used to obtain the solution of the singular quadratic integral equation with the logarithmic kernel and a smooth kernel. Finally, the Fredholm integral equation of the second kind is established and its solution is discussed, also numerical results are obtained and the error, in each case, is computed. [ABSTRACT FROM AUTHOR]

The Internet of Things (IoT) is the domain of interest for researchers with exponential growth in technology. This article develops an authentication protocol based on the Chebyshev polynomial, hashing function, session password, and encryption. The proposed authentication protocol is named as proposed Elliptic, Chebyshev, Session password, and Hash function (ECSH)‐based multilevel authentication. For authenticating the incoming user, there are two phases, registration and authentication. In the registration phase, the user is registered with the server and authentication center (AC), and the authentication follows, which is an eight‐step criterion. The authentication is duly based on the scale factor of the user and server, session password, and verification messages. The authentication at the eight levels assures the security against various types of attacks and renders secure communication in IoT with minimal communication overhead and packet loss. The performance is analyzed using black‐hole, and denial‐of‐service (DOS) attacks with 50 and 100 nodes in the simulation environment. The proposed ECSH‐based multilevel authentication acquired the maximal detection rate, PDR, and QoS of 15.2%, 35.7895%, and 26.4623%, respectively, for 50 nodes with DOS attacks. By contrast, the minimal delay of 135.922 ms is acquired in the presence of 100 nodes and DOS attacks. [ABSTRACT FROM AUTHOR]

We address aseismic fault slip and the onset of seismicity resulting from depletion-induced or injection-induced stresses in reservoirs with pre-existing vertical or inclined faults. Building on classic results, we discuss semi-analytical modelling techniques for fault slip including dislocation theory, Cauchy-type singular integral equations and the use of Chebyshev polynomials for their solution and an eigenvalue-based stability analysis. We consider slip patch development during depletion for faults with zero, constant static and slip-weakening friction, and our results confirm earlier findings based on numerical simulation, in particular the aseismic growth of two slip patches that may subsequently merge and/or become unstable resulting in nucleation of seismic slip. New findings include improved approximate expressions for the induced seismic moment per unit strike length and a description of the effect of coupling between the slip patches which affects both forward simulation and eigenvalue computation for high values of the ratio of fault throw to reservoir height. Our implementation based on analytical inversion and semi-analytical integration with Chebyshev polynomials is more efficient and more robust than our numerical integration approach. It is not yet well suited for Monte Carlo simulation, which typically requires sub-second simulation times, but with some further development that option seems to be within reach. Moreover, our results offer a possibility for embedded fault modelling in large-scale numerical simulation tools. [ABSTRACT FROM AUTHOR]

Excitation identification for nonlinear structures is still a challenging problem due to the convergence and accuracy in this process. In this study, a load estimation method is proposed with orthogonal decomposition, the order for which can be fairly accurately determined by a regression. In this process, the force time history is represented by the orthogonal basis and the coefficients of the orthogonal decomposition are taken as unknowns and augmented to the state variable, which can be identified recursively in state space. A general energy-conserving method is selected to a step-by-step integration to guarantee the convergence of this integration. The proposed method is first validated by numerical simulation studies of a truss structure considering its geometric property. The identification results of the numerical studies demonstrate that the proposed excitation identification method and the orthogonal decomposition order determination method work well for nonlinear structures. The laboratory work of a 7-story frame is investigated to consider the geometric nonlinearity in impact force identification. The results of experimental studies show that uncertainties such as measurement noise and model error are included in the investigation of the accuracy and robustness of the proposed force identification method, while the time history of external forces could be identified with promising results.