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Nowadays, wireless technology has been widely used in healthcare and communication systems. It makes our life easier in all respects. A radiofrequency identification device (RFID) has been deployed as a wireless and identity communication device. RFID is a low-resource device that requires cryptography with limited energy regarding the minimum key size. Security and authentication between the server and the tags are key challenges for an RFID system to maintain data privacy. This article presents the security vulnerabilities of recent existing RFID authentication schemes. Keeping the focus on stringent security, privacy, and low cost, we have designed a new lightweight group authentication protocol for the robust RFID system. A single server controls multiple tags by using the proposed lightweight protocol in the RFID system. Formal and informal security analysis is performed compared to other lightweight group authentication articles in which only informal security analysis is carried out. The formal security strength of our proposed protocol is analyzed using the AVISPA (Automated Verification of Internet Security Protocol Analysis) tool, confirming that it is safe from different security threats. The newly designed protocol's performance analysis results are measured in terms of computational cost, storage space, and communication cost. Finally, the combined consequence of security and lightweight of the proposed group authentication protocol is superior and outperforms compared to the existing scheme. [ABSTRACT FROM AUTHOR]

The dynamical system is one of the major research subjects, and many researchers and experts are attempting to evolve new models and approaches for its solution due to its vast applicability. Applied mathematics has been used to anticipate the chaotic behavior of some attractors using a novel operator termed fractal-fractional derivatives. They were made operating three distinct kernels: power low, exponential decay, and the generalized Mittag Leffler function. There are two parameters in the new operator. Fractional order is the first, while fractal dimension is the second. These derivatives will manage to detect self-similarities in chaotic attractors. We provided numerical approaches for solving such a nonlinear differential equation system. The solution’s existence and uniqueness are determined. Bifurcation analysis is also presented briefly. These new operators were tested in the chaotic attractor with numerical simulations for varied fractional order and fractal dimension, and the findings were quite interesting. We believe that this new notion is the way to go for modeling complexes with self-similarities in the future. [ABSTRACT FROM AUTHOR]

Copyright of Informatica (03505596) is the property of Slovene Society Informatika and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Basrah Journal of Science / Magallat Al-Barat Li-L-ulum is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order α (D α f) (y) = 1 Γ (m − α) ∫ 0 y (y − x) m − α − 1 f (m) (x) d x , y > 0 , with m − 1 < α ≤ m , m ∈ N. The numerical procedure is based on approximating f (m) by the m-th derivative of a Lagrange polynomial, interpolating f at Jacobi zeros and some additional nodes suitably chosen to have corresponding logarithmically diverging Lebsegue constants. Error estimates in a uniform norm are provided, showing that the rate of convergence is related to the smoothness of the function f according to the best polynomial approximation error and depending on order α. As an application, we approximate the solution of a Volterra integral equation, which is equivalent in some sense to the Bagley–Torvik initial value problem, using a Nyström-type method. Finally, some numerical tests are presented to assess the performance of the proposed procedure. [ABSTRACT FROM AUTHOR]

Penelitian ini bertujuan untuk mengeksplorasi penggunaan metode interpolasi polinom Lagrange dan interpolasi polinom Newton-Gregory forward dalam mengestimasi tren pendaftaran mahasiswa baru di beberapa program studi di Fakultas Tarbiyah IAIN Kediri. Data historis pendaftaran mahasiswa baru selama lima tahun terakhir digunakan sebagai dasar analisis. Metode penelitian ini menggunakan pendekatan kuantitatif dengan desain deskriptif dan komparatif. Pendekatan deskriptif digunakan untuk menggambarkan tren pendaftaran mahasiswa baru, sedangkan pendekatan komparatif digunakan untuk membandingkan akurasi prediksi antara kedua metode interpolasi. Hasil penelitian menunjukkan bahwa interpolasi Gregory Forward umumnya memberikan hasil yang lebih akurat dibandingkan interpolasi Lagrange, terutama untuk program studi seperti Manajemen Pendidikan Islam dan Pendidikan Bahasa Arab, dengan nilai MAE, MAPE, dan RMSE yang lebih rendah. Namun, interpolasi Lagrange menunjukkan performa yang lebih baik pada beberapa program studi, seperti Tadris Bahasa Indonesia dan Tadris IPA. Misalnya, untuk program studi Tadris Bahasa Indonesia, interpolasi Lagrange memiliki MAE sebesar 6.8, MAPE 1%, dan RMSE 13.05, yang secara signifikan lebih rendah dibandingkan dengan metode Gregory Forward. Kesimpulan dari penelitian ini menunjukkan bahwa tidak ada satu metode yang selalu unggul untuk semua program studi. Oleh karena itu, pemilihan metode interpolasi yang paling sesuai harus didasarkan pada karakteristik data spesifik dari setiap program studi. Penelitian ini memberikan kontribusi penting dalam bidang perencanaan akademik dengan menyediakan model prediksi yang lebih akurat, yang diharapkan dapat meningkatkan efisiensi dan efektivitas pengelolaan sumber daya di Fakultas Tarbiyah IAIN Kediri. This study explores the Lagrange polynomial and Newton Gregory Forward polynomial interpolation methods in estimating new student enrollment trends in several study programs at the Faculty of Tarbiyah, IAIN Kediri. Historical data on new student enrollments over the past five years was analyzed. This research employs a quantitative approach with descriptive and comparative designs. The descriptive approach is used to depict new student enrollment trends, while the comparative approach compares the prediction accuracy between the two interpolation methods. The results indicate that the Gregory Forward interpolation generally provides more accurate results than the Lagrange interpolation, particularly for study programs such as Islamic Education Management and Arabic Language Education, as evidenced by lower MAE, MAPE, and RMSE values. However, the Lagrange interpolation performs better in some study programs, such as Indonesian Language Teaching and Natural Sciences Teaching. For example, the Lagrange interpolation for the Indonesian Language Teaching program has an MAE of 6.8, MAPE of 1%, and RMSE of 13.05, significantly lower than the Gregory Forward method. The conclusion of this study suggests that no single method is consistently superior for all study programs. Therefore, selecting the most appropriate interpolation method should be based on the specific data characteristics of each study program. This research contributes significantly to academic planning by providing a more accurate predictive model, which is expected to enhance the efficiency and effectiveness of resource management at the Faculty of Tarbiyah, IAIN Kediri.

In this study, we constructed a numerical technique for the simulation of the convection–diffusion problem under convection dominancy. Lagrange interpolation technique is applied to obtain new expressions for the approximation of variable at the interfaces of control volume. Moreover, based on these interface approximations new numerical scheme is developed to approximate convection–diffusion phenomena. The Crank–Nicolson approach is applied for the temporal approximation. This newly constructed numerical scheme is unconditionally stable with second-order accuracy in time and space both. Numerical tests are carried out for the justification of the new algorithm. A comparison of numerical results produced by proposed technique and some other numerical approaches is presented. This comparison indicates that for convection dominant phenomena, the numerical solution of conventional finite volume method contains with non-physical oscillations which analyze that proposed numerical technique results in a high accurate and stable solution. [ABSTRACT FROM AUTHOR]

This paper proposes fractional delay filter-based repetitive controllers (FDF-RCs) for grid-tied converters. Finite impulse response filters and infinite impulse response filters are used to estimate the fractional-order delay filter in a conventional repetitive controller. To do this, both the Lagrange filter-based repetitive controller (LF-RC) and Thiran filter-based repetitive controller (TF-RC) are utilized as FDF-RCs to compensate a fractional delay. The operating principle, the ladder structure, and the performance analysis of FDF-RCs are addressed. The simulation and the experimental results on a single-phase grid-tied converter verify the harmonic suppression ability and reference-tracking performance of the proposed FDF-RC under the grid frequency variation. Moreover, the advantages of the TF-RC over the LF-RC are discussed by comparing and analyzing the simulation and experimental results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the approximation of functions using a higher-order Hermite-Fej´er interpolation process on the unit circle. The system of nodes is composed of vertically projected zeros of Jacobi polynomials onto the unit circle with boundary points at ±1. Values of the polynomial and its first four derivatives are fixed by the interpolation conditions at the nodes. Convergence of the process is obtained for analytic functions on a suitable domain, and the rate of convergence is estimated. [ABSTRACT FROM AUTHOR]

In this paper, we show that a real valued function f defined on any closed bounded interval [a, b] is totally positive if it is Taylor positive. Also, it is shown that the function f is totally bounded if it is Taylor bounded. [ABSTRACT FROM AUTHOR]

In all Global Navigation Satellite Systems (GNSS) applications, the determination of the satellite orbits is an important task. In this study, we present the equations given in the Interface Specification Document of GPS and the Runge-Kutta method in the computation of the position P, velocity V, and acceleration A of the GPS satellites using the broadcast ephemeris. The definition of the differential equation describing the GPS satellite's motion has enabled us to introduce the Runge-Kutta method in the GPS orbit computation; this method uses the initial conditions determined in this study from the Keplerian elements provided in the broadcast ephemeris files. The Lagrange interpolation method is used for comparison of the results, where the vectors P, V, and A are estimated using the precise ephemeris. The difference not exceeding 2.4 m was obtained in the X, Y, and Z axes during seven days on the position of the GPS satellite number 9 tested in this study. In velocity and acceleration, the difference is about a few mm/s and mm/s2, respectively. [ABSTRACT FROM AUTHOR]

In this article, an accurate computational approaches based on Lagrange basis and Jacobi-Gauss collocation method is suggested to solve a class of nonlinear stochastic Itô-Volterra integral equations (SIVIEs). Since the exact solutions of this kind of equations are not still available, so finding an accurate approximate solutions has attracted the interest of many scholars. In the proposed methods, using Lagrange polynomials and zeros of Jacobi polynomials, the considered system of linear and nonlinear stochastic Volterra integral equations is reduced to linear and nonlinear systems of algebraic equations. Solving the resulting algebraic systems by Newton's methods, approximate solutions of the stochastic Volterra integral equations are constructed. Theoretical study is given to validate the error and convergence analysis of these methods; the spectral rate of convergence for the proposed method is established in the L ∞ -norm. Several related numerical examples with different simulations of Brownian motion are given to prove the suitability and accuracy of our methods. The numerical experiments of the proposed methods are compared with the results of other numerical techniques. [ABSTRACT FROM AUTHOR]

Nonlinear Model Predictive Control (NMPC) is effective for local planning of automated vehicles, especially when there exist dynamical objects and multipe requirements. But it requires many computation resources for numerical optimization, which limits its practical application becase of the limited power of onboard unit. To extend the application range of the NMPC based local planner, the coupled nonlinear vehicle dynamics model is adopted based on the numerical analysis, which conversely requires much more discretization poits for acceptable accuracy. For better computation efficiency, Lagrange polynomials are used to discretize the vehicle dynamics model and objective function with less points and fine numerical accuracy. Furthermore, an adaptive strategy is designed to determine the order of Lagrange polynomials according to running state by numerical analysis of discretization error. Both acceleration effect and performance of the local planner designed by NMPC are validated by experimental tests under scenarios with multiple dynamical obstacles. The test results show that compared with the original one the accuracy and efficiency are improved by 74% and 60%, respectively. [ABSTRACT FROM AUTHOR]

The objective of reliability-based design optimization (RBDO) is to minimize the optimization objective while satisfying the corresponding reliability requirements. However, the nested loop characteristic reduces the efficiency of RBDO algorithm, which hinders their application to high-dimensional engineering problems. To address these issues, this paper proposes an efficient decoupled RBDO method combining high dimensional model representation (HDMR) and the weight-point estimation method (WPEM). First, we decouple the RBDO model using HDMR and WPEM. Second, Lagrange interpolation is used to approximate a univariate function. Finally, based on the results of the first two steps, the original nested loop reliability optimization model is completely transformed into a deterministic design optimization model that can be solved by a series of mature constrained optimization methods without any additional calculations. Two numerical examples of a planar 10-bar structure and an aviation hydraulic piping system with 28 design variables are analyzed to illustrate the performance and practicability of the proposed method. [ABSTRACT FROM AUTHOR]

Bivariate Mittag-Leffler (ML) functions are a substantial generalization of the univariate ML functions, which are widely recognized for their significance in fractional calculus. In the present paper, our initial focus is to investigate the fractional calculus properties of the integral and derivative operators with kernels including the Bivariate ML functions. Further, certain fractional Cauchy-type problems including these operators are considered. Also the numerical approximations of the Caputo type derivative operator are investigated. The theoretical results are justified by applications on examples. Furthermore, the theory of applying the same operators with respect to arbitrary monotonic functions is analyzed in this research. [ABSTRACT FROM AUTHOR]

In this paper, we study a method of polynomial interpolation that lies in-between Lagrange and Hermite methods. The novelty is that we use very general nodal systems on the unit circle as well as on the bounded interval only characterized by a separation property. The way in which we interpolate consists in considering all the nodes for the prescribed values and only half for the derivatives. Firstly, we develop the theory on the unit circle, obtaining the main properties of the nodal polynomials and studying the convergence of the interpolation polynomials corresponding to continuous functions with some kind of modulus of continuity and with general conditions on the prescribed values for half of the derivatives. We complete this first part of the paper with the study of the convergence for smooth functions obtaining the rate of convergence, which is slightly slower than that when equidistributed nodal points are considered. The second part of the paper is devoted to solving a similar problem on the bounded interval by using nodal systems having good properties of separation, generalizing the Chebyshev–Lobatto system, and well related to the nodal systems on the unit circle studied before. We obtain an expression of the interpolation polynomials as well as results about their convergence in the case of continuous functions with a convenient modulus of continuity and, particularly, for differentiable functions. Finally, we present some numerical experiments related to the application of the method with the nodal systems dealt with. [ABSTRACT FROM AUTHOR]

The application of nanofluids as an alternative to traditional insulating oils is one of the most promising new approaches to improving cooling and insulating efficiency and the design of electric transformers, which are bound by the state-of-the-art technology. Over the past two decades, and even now, research has been conducted on nanofluids with the goal of using them as a coolant and dielectric medium in transformers. This work is proposed to study the reliability of moringa oil (MAOL) as an alternate liquid insulation with a comprehensive study with different concentrations of alumina (Al2O3) nanoparticles and C12H26-C15H32 hydrocarbon solvents. The reliability of proposed nanofluid samples is analyzed with the measurement of critical properties such as breakdown voltage, kinematic viscosity, flash point and fire point, pour point, thermal conductivity, and acid numbers according to IEC and ASTM standards. Based on the properties of nanofluids, the best sample is statistically analyzed with survival rate and probability of failure along with raw moringa oil and mineral oil. Also further ageing analysis with best nanofluid, moringa oil, and mineral oil (MLOL) is performed in real exposure in 1 kVA, 440/220 V for 720 h (30 days). The ageing performance is examined with measurement of critical properties and subsequent prediction of properties after 60 days and 90 days operation using Lagrange interpolation (LI), neural network (NN), and principal component analysis (PCA) to compare the modification in the properties of each sample. With this proposed work, it is found that moringa oil–based nanofluids have the reliable nature as an alternate liquid insulation for transformers. [ABSTRACT FROM AUTHOR]

Ebola is a serious disease that affects people; in many cases, it results in death. Ebola outbreaks have also occurred in communities where residents keep pets, particularly dogs. Due to a lack of food, the dogs must hunt for food. Dogs eat the internal organs of wildlife that the locals have killed and eaten, as well as small dead animals that are found within the communities which may contain the Ebola virus. This study introduces a mathematical model based on the Caputo–Fabrizio derivative to describe the Ebola transmission dynamics between dogs and humans. The model’s existence and the uniqueness of its solution were investigated using fixed-point theory. Furthermore, through the Sumudu transform criterion, we established that the Caputo–Fabrizio Ebola model is Picard stable. Some qualitative analysis was also carried out to investigate the Ebola propagation trend in the dog-to-human model. The proposed model is fitted to the reported Ebola incidence in Uganda between October 15, 2022, and November 2, 2022. The Ebola reproduction number obtained using the cumulative data was 2.65. It is noticed that as the fractional order reduces, the Ebola reproduction number also reduces. We derived a numerical scheme for our model using the two-step Lagrange interpolation. It has been discovered that the fractional orders significantly influence the model, indicating that natural occurrences could affect the dynamics of Ebola. It is observed that when the recovery rate is enhanced, such as through the hospitalisation of Ebola-infected individuals, the disease will reduce. Finally, as we ensure a reduction in the contact rate among the dog’s compartments, the disease does not spread adiabatically. Therefore, we urge that quarantine measures be put in place to control interactions among the dogs during the outbreak.

Featured Application: This research contributes to the development of a technological method to obtain highly accurate vehicle trajectory data. The reconstructed trajectory data play a key role in traffic state prediction, traffic management and the decision making of autonomous vehicles and robots. Vehicle trajectory usually suffers from a large number of outliers and observation noises. This paper proposes a novel framework for reconstructing vehicle trajectories. The framework integrates the wavelet transform, Lagrange interpolation and Kalman filtering. The wavelet transform based on waveform decomposition in the time and frequency domain is used to identify the abnormal frequency of a trajectory. Lagrange interpolation is used to estimate the value of data points after outliers are removed. This framework improves computation efficiency in data segmentation. The Kalman filter uses normal and predicted data to obtain reasonable results, and the algorithm makes an optimal estimation that has a better denoising effect. The proposed framework is compared with a baseline framework on the trajectory data in the NGSIM dataset. The experimental results showed that the proposed framework can achieve a 45.76% lower root mean square error, 26.43% higher signal-to-noise ratio and 25.58% higher Pearson correlation coefficient. [ABSTRACT FROM AUTHOR]

In this research article, we brought into consideration the set of non-uniformly distributed nodes on the unit circle to investigate a Lagrange-Hermite interpolation problem. These nodes are obtained by projecting vertically the zeros of Jacobi polynomial onto the unit circle along with the boundary points of the unit circle on the real line. Explicitly representing the interpolatory polynomial as well as establishment of convergence theorem are the key highlights of this manuscript. The result proved are of interest to approximation theory. [ABSTRACT FROM AUTHOR]

We present a numerical method for solving Fredholm two-dimensional integral equations in this study. Our approach is based on two-dimensional Lagrange interpolation polynomials. The use of interpolation is that instead of the unknown function, we use the Lagrange interpolator polynomial, and then by solving these linear equation system, we obtain the Lagrange coefficients, which are the second components of the support points, approximately. By putting these co-efficients in the Lagrange finder function, we get an approximation to the exact answer. A numerical algorithm is described for this purpose, and two cases are solved using this algorithm. Furthermore, a theorem is proved to demonstrate the algorithm's convergence and to obtain an upper bound on the distance between the exact and numerical solutions. [ABSTRACT FROM AUTHOR]

In this paper, we focus on identifying the transmission rate associated with a COVID-19 mathematical model by using a predefined prevalence function. To do so, we use a Python code to extract the Lagrange interpolation polynomial from real daily data corresponding to an appropriate period in Morocco. The existence of a perfect control scheme is demonstrated. The Pontryagin maximum technique is used to explain these optimal controls. The optimality system is numerically solved using the 4th-order Runge-Kutta approximation.

Fuzzy logic is becoming one of the most-influential fields of modern mathematics with applications that impact not only other sciences, but society in general. This newly found interest in fuzzy logic is in part due to the crucial role it plays in the development of artificial intelligence. As a result, new tools and practices for the development of the above-mentioned field are in high demand. This is one of the issues this paper was composed to address. To be more specific, a sizable part of fuzzy logic is the study of fuzzy connectives. However, the current method used to generalize them is restricted to the use of basic automorphisms, which hinders the creation of new fuzzy connectives. For this reason, in this paper, a new method of generalization is conceived of that aims to generalize the fuzzy connectives using polynomial automorphism functions instead. The creation of these automorphisms is achieved through numerical analysis, an endeavor that is supported with programming applications that, using mathematical modeling, validate and visualize the research. Furthermore, the automorphisms satisfy all the necessary criteria that have been established for use in the generalization process and, consequently, are used to successfully generalize fuzzy connectives. The result of the new generalization method is the creation of new usable and flexible fuzzy connectives, which is very promising for the future development of the field. [ABSTRACT FROM AUTHOR]

Digital imaging is used in variety of applications. Together with the improvements in artificial intelligence and its sub-fields, improving computer vision methods to address inter- and multi-disciplinary problems is possible. Especially in medical applications, there are significant improvements related to imaging in the last decades. Digital image interpolation is a key operation in digital image processing where there are no sufficient samples during the acquisition process. Using the available samples in hand, digital interpolation techniques are predicting the missing samples. The paper addresses the problem of digital image interpolation and proposes a novel algorithm using multiplicative calculus. The main contribution of the paper is the application of multiplicative Lagrange interpolation to accomplish image interpolation task. The proposed method is tested on several datasets, and the results are comparable to the state-of-the-art methods. The paper presents encouraging results to the literature, and the proposed method is open for further improvements. [ABSTRACT FROM AUTHOR]

In this paper, we focus on identifying the transmission rate associated with a COVID-19 mathematical model by using a predefined prevalence function. To do so, we use a Python code to extract the Lagrange interpolation polynomial from real daily data corresponding to an appropriate period in Morocco. The existence of a perfect control scheme is demonstrated. The Pontryagin maximum technique is used to explain these optimal controls. The optimality system is numerically solved using the 4th-order Runge-Kutta approximation. [ABSTRACT FROM AUTHOR]

An environment-aware millimeter wave radar (EMWR) is one of the most important sensors in an autonomous driving system. With the increasing penetration of an EMWR on an autonomous vehicle, the possibility of radar interference increases accordingly. Interference may seriously affect the detection performance of an EMWR. When the transmitted signals from other EMWRs are received by the EMWR on an autonomous vehicle, the accuracy of target detection will be affected, which may affect the environment-aware performance. Therefore, more and more attention is paid to how to restrain the interference between EMWR signals. Radar interference can be divided into two types: same-frequency interference (SFI) and different-frequency interference (DFI). In this paper, an anti-DFI algorithm of an EMWR is proposed. Firstly, the causes and signal characteristics of DFI are analyzed. Secondly, the signal amplitude is obtained via Hilbert transform to locate the interference according to the signal characteristics. Then, the Lagrange interpolation based on empirical-mode decomposition (EMD) is used to reconstruct the interference signal to reduce the influence of DFI. Finally, the feasibility of the proposed algorithm is verified by the simulation results. The simulation results validate that the interference signal after EMD filtering and Lagrange interpolation can repair the interference region and achieve the purpose of anti-DFI. [ABSTRACT FROM AUTHOR]

It is well-known that the Lagrange interpolation based on the Chebyshev nodes may be divergent everywhere (for arbitrary nodes, almost everywhere), like the Fourier series of a summable function. On the other hand, any measurable almost everywhere finite function can be “adjusted” in a set of an arbitrarily small measure such that its Fourier series will be uniformly convergent. The question arises whether the class of continuous functions has a similar property with respect to any interpolation process. In the present paper, we prove that there exists the matrix of nodes $\mathfrak{M}_\gamma$ arbitrarily close to the Jacoby matrix $\mathfrak{M}^{(\alpha,\beta)}$, $\alpha,\beta>-1$ with the following property: any function $f\in{C[-1,1]}$ can be adjusted in a set of an arbitrarily small measure such that interpolation process of adjusted continuous function $g$ based on the nodes $\mathfrak{M}_\gamma$ will be uniformly convergent to $g$ on $[a,b]\subset(-1,1)$.

The reptile search algorithm (RSA) is a bionic algorithm proposed by Abualigah. et al. in 2020. RSA simulates the whole process of crocodiles encircling and catching prey. Specifically, the encircling stage includes high walking and belly walking, and the hunting stage includes hunting coordination and cooperation. However, in the middle and later stages of the iteration, most search agents will move towards the optimal solution. However, if the optimal solution falls into local optimum, the population will fall into stagnation. Therefore, RSA cannot converge when solving complex problems. To enable RSA to solve more problems, this paper proposes a multi-hunting coordination strategy by combining Lagrange interpolation and teaching-learning-based optimization (TLBO) algorithm's student stage. Multi-hunting cooperation strategy will make multiple search agents coordinate with each other. Compared with the hunting cooperation strategy in the original RSA, the multi-hunting cooperation strategy has been greatly improved RSA's global capability. Moreover, considering RSA's weak ability to jump out of the local optimum in the middle and later stages, this paper adds the Lens pposition-based learning (LOBL) and restart strategy. Based on the above strategy, a modified reptile search algorithm with a multi-hunting coordination strategy (MRSA) is proposed. To verify the above strategies' effectiveness for RSA, 23 benchmark and CEC2020 functions were used to test MRSA's performance. In addition, MRSA's solutions to six engineering problems reflected MRSA's engineering applicability. It can be seen from the experiment that MRSA has better performance in solving test functions and engineering problems.

The stability of the representation of finite rank operators in terms of a basis is analyzed. A conditioning is introduced as a measure of the stability properties. This conditioning improves some other conditionings because it is closer to the Lebesgue function. Improved bounds for the conditioning of the Fourier sums with respect to an orthogonal basis are obtained, in particular, for Legendre, Chebyshev, and disk polynomials. The Lagrange and Newton formulae for the interpolating polynomial are also considered. [ABSTRACT FROM AUTHOR]

A new definition of fractional differentiation of nonlocal and non-singular kernels has recently been developed to overcome the shortcomings of the traditional Riemann–Liouville and Caputo fractional derivatives. In this study, the dynamic behaviors of the fractional financial chaotic model have been investigated. Singular and non-singular kernel fractional derivatives are used to examine the proposed model. To solve the financial chaotic model with nonlocal operators, the fractional Adams–Bashforth method (ABM) is applied based on Lagrange polynomial interpolation (LPI). The existence and uniqueness of the solution of the model can be demonstrated using fixed point theory and nonlinear analysis. Further, the error analysis of the present method and Ulam–Hyers stability of the considered model have also been included. Obtained numerical simulations reveal that the model based on three different fractional derivatives shows various chaotic behaviors that may be useful in a practical sense which may not be observed in the integer case. [ABSTRACT FROM AUTHOR]

We give a geometric condition on principal lattices in that ensures that the corresponding Lagrange interpolation polynomials of any sufficient smooth function converges to a Taylor polynomial. [ABSTRACT FROM AUTHOR]

Inspired by the advancements in (fully) homomorphic encryption in recent decades and its practical applications, we conducted a preliminary study on the underlying mathematical structure of the corresponding schemes. Hence, this paper focuses on investigating the challenge of deducing bivariate polynomials constructed using homomorphic operations, namely repetitive additions and multiplications. To begin with, we introduce an approach for solving the previously mentioned problem using Lagrange interpolation for the evaluation of univariate polynomials. This method is well-established for determining univariate polynomials that satisfy a specific set of points. Moreover, we propose a second approach based on modular knapsack resolution algorithms. These algorithms are designed to address optimization problems in which a set of objects with specific weights and values is involved. Finally, we provide recommendations on how to run our algorithms in order to obtain better results in terms of precision. [ABSTRACT FROM AUTHOR]

In this article, we have studied the dynamics of the Ebola virus disease by means of fractional differentiation combined with a fractal dimension. It has been shown explicitly that the Ebola model is positively invariant. The fixed point theorem procedures check the solution’s existence and uniqueness to the model using the Mittag-Leffler kernel. The stability of the Ebola fractal–fractional model is studied using the Hyers–Ulam stable analysis. The numerical simulation for the trajectories of the proposed model is obtained using Lagrangian interpolation. Also, the numerical sensitivity analysis of the model is presented. For instance, a reduction in the rate of human-to-human contact results in a decline in the rate of infection, implying that the disease at a point will die out. Again, an increase in the loss of immunity rate to unity leads to a massive spread of the Ebola disease in the population. The findings in our work depict that we can achieve an Ebola-free state should the health sector give more education on maintaining a strong immune system and reducing human-to-human contact, especially during outbreaks of the Ebola disease, using measures like isolation and quarantine. Finally, using the fractal–fractional operator, we observed that any amount of memory variation and repetition in the outbreak of the disease influences the spread of the Ebola disease.

Fractional calculus is an essential tool in every area of science today. This work gives the quadratic interpolation-based L1-2 formula for the Caputo-Fabrizio derivative, a numerical technique for approximating the fractional derivative. To get quadratic and cubic convergence rates, respectively, we study the use of Lagrange interpolation in the L1 and L1-2 formulations. Our numerical analysis shows the accuracy of the theory’s predicted convergence rates. The L1-2 formula aims to enhance the accuracy and usability of a flexible tool for many applications in science and mathematics. We demonstrate the validity of the theory’s predicted convergence rates using numerical analysis. Several numerical examples are also given to show how the suggested approaches may be utilized to determine the Caputo-Fabrizio derivative of well-known functions. Lagrange interpolation is used in the L1 and L1-2 procedures to obtain quadratic and cubic convergence rates, respectively. The numerical study demonstrates that the L1-2 formula offers greater accuracy when compared to current approaches. In addition, it is a better apparatus for several applications in science and mathematics. Due to its higher convergence rate, the L1-2 formula outperforms other available numerical methods for scientific computations. The L1-2 formula, a novel numerical method for the Caputo-Fabrizio derivative that makes use of quadratic interpolation, is introduced in this study as a conclusion.

Wireless sensor systems often fail to provide measurements with uniform time spacing. Measurements can be delayed or even miss completely. Resampling to uniform intervals is necessary to satisfy the requirements of subsequent signal processing. Common resampling algorithms, based on symmetric finite impulse response (FIR) filters, entail a group delay of 10 s of samples, which is not acceptable regarding the typical interval of wireless sensors of seconds or minutes. The purpose of this paper is to verify the feasibility of single-delay resampling, i.e., the algorithm resamples the data without waiting for future samples. A new method to parametrize Kriging interpolation is presented and compared with two variants of Lagrange interpolation in detailed simulations for the resulting prediction error. Kriging provided the most accurate resampling in the group-delay scenario. The single-delay scenario required almost double the OSR to achieve the same signal-to-noise ratio (SNR). An OSR between 1.8 and 3.1 was necessary for single-delay resampling, depending on the required SNR and signal distortions in terms of jitter, missing samples, and noise. Kriging was the least noise-sensitive method. Especially for signals with missing samples, Kriging provided the best accuracy. The simulations showed that single-delay resampling is feasible, but at the expense of higher OSR and limited SNR. [ABSTRACT FROM AUTHOR]

In this paper, a rapid and high accurate numerical method for pricing discrete single and double barrier knock‐out call options is presented. With regard to the well‐known Black‐Scholes model, the price of an option in each monitoring date could be calculated by computing a recursive integral formula that is based on the heat equation solution. We have approximated these recursive solutions with the aid of Lagrange interpolation on Jacobi polynomial nodes. After that, an operational matrix, which makes our computation significantly fast, has been derived. In some theorems, the convergence of the presented method has been shown and the rate of convergence has been derived. The most important benefit of this method is that its complexity is very low and does not depend on the number of monitoring dates. The numerical results confirm the accuracy and efficiency of the presented numerical algorithm. [ABSTRACT FROM AUTHOR]

For the purpose of solving elliptic partial differential equations, we suggest a new approach using an h-adaptive local discontinuous Galerkin approximation based on Sinc points. The adaptive approach, which uses Poly-Sinc interpolation to achieve a predetermined level of approximation accuracy, is a local discontinuous Galerkin method. We developed an a priori error estimate and demonstrated the exponential convergence of the Poly-Sinc-based discontinuous Galerkin technique, as well as the adaptive piecewise Poly-Sinc method, for function approximation and ordinary differential equations. In this paper, we demonstrate the exponential convergence in the number of iterations of the a priori error estimate derived for the local discontinuous Galerkin technique under the condition that a reliable estimate of the precise solution of the partial differential equation at the Sinc points exists. For the purpose of refining the computational domain, we employ a statistical strategy. The numerical results for elliptic PDEs with Dirichlet and mixed Neumann-Dirichlet boundary conditions are demonstrated to validate the adaptive greedy Poly-Sinc approach. [ABSTRACT FROM AUTHOR]

We prove that the Newton product of efficient polynomial projectors is still efficient. Various polynomial approximation theorems are established involving Newton product projectors on spaces of holomorphic functions on a neighborhood of a regular compact set, on spaces of entire functions of given growth and on spaces of differentiable functions. Efficient explicit new projectors are presented. [ABSTRACT FROM AUTHOR]

Blockchain technology has aroused great interest from society and academia since the inception of Bitcoin. Its de-centralization and non-tampering can apply in broader scenarios, such as the Internet of Things, smart cities, and cloud computing. Among various core components, the consensus protocol is the core of maintaining blockchain networks' performance, stability, and security. However, with the increase of network nodes and the improvement of network complexity, these properties are difficult to meet simultaneously. In this paper, we propose an advancement of the practical Byzantine consensus algorithm (LRBFT). The algorithm uses Lagrange interpolation that all backups can participate in to generate random seeds, uses the seeds to optimize the election process of the primary set, improves consensus efficiency through delegated nodes, and prevents the primary from doing evil through the supervisory mechanism. The generation of random seeds has the characteristics of full participation, unpredictability, and verifiability. The election process of the primary set has randomness, uniform distribution, and supervision. Furthermore, we proved the feasibility of our proposed algorithm through theoretical analysis and experimental evaluations. Experimental analysis shows that when there are 70 nodes in the practical Byzantine fault tolerance (PBFT) consensus protocol. If LRBFT selects only 7 nodes as delegated nodes, the time it takes for LRBFT to reach 100 consensuses is only 0.83% of that of PBFT. [ABSTRACT FROM AUTHOR]

Abstract For the linear Lagrange interpolation over a triangular domain, we propose an efficient algorithm to rigorously evaluate the interpolation error constant under the maximum norm by using the finite-element method (FEM). In solving the optimization problem corresponding to the interpolation error constant, the maximum norm in the constraint condition is the most difficult part to process. To handle this difficulty, a novel method is proposed by combining the orthogonality of the space decomposition using the Fujino–Morley FEM space and the convex-hull property of the Bernstein representation of functions in the FEM space. Numerical results for the lower and upper bounds of the interpolation error constant for triangles of various types are presented to verify the efficiency of the proposed method.

Vehicle trajectory usually suffers from a large number of outliers and observation noises. This paper proposes a novel framework for reconstructing vehicle trajectories. The framework integrates the wavelet transform, Lagrange interpolation and Kalman filtering. The wavelet transform based on waveform decomposition in the time and frequency domain is used to identify the abnormal frequency of a trajectory. Lagrange interpolation is used to estimate the value of data points after outliers are removed. This framework improves computation efficiency in data segmentation. The Kalman filter uses normal and predicted data to obtain reasonable results, and the algorithm makes an optimal estimation that has a better denoising effect. The proposed framework is compared with a baseline framework on the trajectory data in the NGSIM dataset. The experimental results showed that the proposed framework can achieve a 45.76% lower root mean square error, 26.43% higher signal-to-noise ratio and 25.58% higher Pearson correlation coefficient.

In linguistics, phonemes are the atomic sound, called word segmentor play an important role to recognize the word properly. A novel approach of seven Bengali vowels and ten diphthongs (a syllable for the pronunciation of two consecutive vowels) phoneme recognition has been proposed in the paper. In the proposed method, before extracting the feature, a novel pre-processing technique using amplitude interpolation method has been developed to align the starting point of all the phonemes of the same class which in turn boosts the recognition rate. Here seven Bengali vowels and ten diphthongs audio clips uttered by twenty persons (ten times each) of different age group and sex have been recorded to create a data set of 3400 audio samples for the proposed experiment. For each class of phonemes and diphthongs one sample (selected by linguistic) have been considered as a benchmark. Then each of the recorded audio clips is interpolated to match with the benchmark clip of the corresponding phoneme by finding the valleys in the amplitude using Lagrange interpolation technique. After that, 19 MFCC (Mel Frequency Cepstral Co-Efficient) speech features have been extracted from each phoneme of the interpolated audio clips and feed to classify using Support Vector Machine (SVM), k- Nearest Neighbour (KNN) and Deep Neural Network (DNN) classifier and the average classification accuracy obtained for vowels and diphthongs are 94.93% and 94.56% respectively. To check the effectiveness of the proposed pre-processing technique same MFCC features have been extracted from the raw recorded phonemes and feed to same classifiers and average accuracy obtained for vowels and diphthongs are 89.21% and 88.56% respectively which shows the effectiveness of the proposed method. It is also to note that best accuracy obtained using the DNN classifier with the accuracy of 98.16% for vowels and 97% for diphthongs. [ABSTRACT FROM AUTHOR]

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