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This paper investigates the persistence of shocks on non-renewable and renewable energy consumption for 15 leading countries by renewable energy consumption, over the period 1980–2018. For this aim, we apply Christopoulos and Leon-Ledesma (J Int Money Finance 29(6):1076–1093, 2010) Fourier ADF unit root test which structural breaks are included by using a trigonometric function to allow for smooth temporary mean changes rather than jump functions. In application, we do not need to specify the numbers, locations and the forms of the structural breaks a priori. The results give that shocks on non-renewable energy consumption per capita are persistent for 13 countries except Sweden and the United States; shocks on renewable energy consumption per capita are persistent for 12 countries except Canada, Sweden and the United Kingdom. Overall, we conclude that persistent policy implications on non-renewable energy consumption are more effective tool than transitory policy stances for 13 countries except Sweden and the United States whereas persistent policy implications on renewable energy consumption are more effective for 12 countries except Canada, Sweden and the United Kingdom. In other words, since the energy consumption returns to its trend path quickly, any policy will not be effective on non-renewable energy consumption for Sweden and the United States, and also on renewable energy consumption for Canada, Sweden and the United Kingdom. [ABSTRACT FROM AUTHOR]

RNA molecules, essential for viruses and living organisms, derive their pivotal functions from intricate 3D structures. To understand these structures, one can analyze torsion and pseudo-torsion angles, which describe rotations around bonds, whether real or virtual, thus capturing the RNA conformational flexibility. Such an analysis has been made possible by RNAtango, a web server introduced in this paper, that provides a trigonometric perspective on RNA 3D structures, giving insights into the variability of examined models and their alignment with reference targets. RNAtango offers comprehensive tools for calculating torsion and pseudo-torsion angles, generating angle statistics, comparing RNA structures based on backbone torsions, and assessing local and global structural similarities using trigonometric functions and angle measures. The system operates in three scenarios: single model analysis, model-versus-target comparison, and model-versus-model comparison, with results output in text and graphical formats. Compatible with all modern web browsers, RNAtango is accessible freely along with the source code. It supports researchers in accurately assessing structural similarities, which contributes to the precision and efficiency of RNA modeling. [ABSTRACT FROM AUTHOR]

Differential subordination in the complex plane is the generalization of a differential inequality on the real line. In this paper, we consider two subclasses of univalent functions associated with the trigonometric function cos z. Using some properties of the hypergeometric functions, we determine the sharp estimate on the parameter β such that the analytic function p(z) satisfying p(0) = 1, is subordinate to cos z when the differential expression p(z) + βz(dp(z)/dz) is subordinate to the Janowski function. We compute sharp bounds on coefficient functional Hermitian-Toeplitz determinants of the third and the fourth order with an invariance property for such functions. In addition, we estimate bound on Hankel determinants of the second and the third order. [ABSTRACT FROM AUTHOR]

Trigonometric functions have gained considerable attention in recent studies for developing new lifetime distributions. This is due to their parsimonious framework, mathematical tractability, and flexibility of the newly developed distributions. In this paper, the Sine‐generated family is used to create a new bounded lifetime distribution, known as Sine‐Marshall–Olkin Topp–Leone distribution, for modeling data defined on the unit interval. Some statistical properties of the new bounded distribution, including quantile function, ordinary moments, incomplete moments, moment‐generating functions, inequality measures, Rényi entropy, and probability weighted moments are derived. Seven methods of parameter estimation, including maximum likelihood, ordinary least squares, weighted least squares, moment product spacing, Anderson–Darling, and Cramér–von Mises estimators are used to estimate the parameters of the new distribution. The behavior of the estimators obtained from the estimation methods are investigated using Monte Carlo simulation studies. The results show that the estimation methods are asymptotically efficient and consistent. The flexibility of the new bounded distribution is examined via data fitting using two proportional datasets. The results of the fittings show that the new bounded distribution performs significantly better than the competing bounded lifetime distributions. Finally, Sine‐Marshall–Olkin Topp–Leone regression model is presented as an alternative to the beta and Kumaraswamy regression models. [ABSTRACT FROM AUTHOR]

In this paper, for 0 < x < π/2 and r > 0, we consider the following Iyengar-Madhava Rao-Nanjundiah type inequality; cos (x√3 +αxr) < sinx/x < cos (x/√3 +βxr). Our main theorems shows that α and β depend on r > 0, and if 0 < r < 3 then β = (2/π)r (−2√3 +arccos 2/π) and if r > 4 then α = (2/π)r (− π/2√3 +arccos 2/π). [ABSTRACT FROM AUTHOR]

Digital technology is regarded as providing a promising means of moving production and consumption towards the circular economy. However, it is still unclear which functions of digital technologies are most useful to improving circularity, and how these functions could be used to enhance different circular economy strategies. This paper aims to address this knowledge gap by conducting a systematic literature review. After examining 174 papers, creating 782 original codes and 259 second‐round codes, the study identifies 13 critical functions of digital technologies which are most relevant to circular economy strategies. The paper then proposes a framework which reveals seven mechanisms of how these digital functions can enhance different circular economy strategies. The framework also reveals which combinations of the digital functions and circular economy strategies have already been studied extensively as well as where there may be gaps. This indicates which digital functions are more mature in terms of possible implementation for circular economy as well as what missing links there are in the empirical and theoretical research. The study advances the synergies between digital technologies and the circular economy paradigm through the lens of digital functions. The proposed framework and mechanisms build a theoretical foundation for future research, and we highlight five research areas for further studies. This study also provides a structured way for managers to explore the appropriate digital functions for their CE strategies, so as to identify required digital technologies and new value creation through digital functions. [ABSTRACT FROM AUTHOR]

This paper devotes to the study of the classical Abel equation d x d t = g (t) x 3 + f (t) x 2 , where g (t) and f (t) are trigonometric polynomials of degree m ≥ 1. We are interested in the problem that whether there is a uniform upper bound for the number of limit cycles of the equation with respect to m , which is known as the famous Smale-Pugh problem. In this work we generalize an idea from the recent paper (Yu, Chen and Liu, Disc. Cont. Dyn. Syst. Ser. B, 2023) and give a new criterion to estimate the maximum multiplicity of limit cycles of the above Abel equations. By virtue of this criterion and the previous results given by Álvarez et al. and Bravo et al., we completely solve the simplest case of the Smale-Pugh problem, i.e., the case when g (t) and f (t) are linear trigonometric, and obtain that the maximum number of limit cycles, is three, which gives a positive answer to the sixth of 33 open problem proposed by Gasull in paper (Gasull, SeMA J., 2021). [ABSTRACT FROM AUTHOR]

Let S be a semigroup. In the present paper, we determine the complex-valued solutions (f, g) of the functional equation g(xσ(y)) = g(x)g(y) − f(x)f(y) + αf(xσ(y)), x, y ∈ S, where σ : S → S is an automorphism that need not be involutive, and α ∈ C is a fixed constant. Our results generalize and extend the ones by Stetkær in The cosine addition law with an additional term. Aequat Math., no. 6, 90, 1147-1168 (2016), and also the ones by Aserrar and Elqorachi in A generalization of the cosine addition law on semigroups. Aequat Math. 97, 787–804 (2023). Some consequences of our results are presented. [ABSTRACT FROM AUTHOR]

This paper introduces a novel numerical scheme, the conformable finite difference method (CFDM), for solving time-fractional gas dynamics equations. The method was developed by integrating the finite difference method with conformable derivatives, offering a unique approach to tackle the challenges posed by time-fractional gas dynamics models. The study explores the significance of such equations in capturing physical phenomena like explosions, detonation, condensation in a moving flow, and combustion. The numerical stability of the proposed scheme is rigorously investigated, revealing its conditional stability under certain constraints. A comparative analysis is conducted by benchmarking the CFDM against existing methodologies, including the quadratic B-spline Galerkin and the trigonometric B-spline functions methods. The comparisons are performed using L2 and L∞ norms to assess the accuracy and efficiency of the proposed method. To demonstrate the effectiveness of the CFDM, several illustrative examples are solved, and the results are presented graphically. Through these examples, the paper showcases the capability of the proposed methodology to accurately capture the behavior of time-fractional gas dynamics equations. The findings underscore the versatility and computational efficiency of the CFDM in addressing complex phenomena. In conclusion, the study affirms that the conformable finite difference method is well-suited for solving differential equations with time-fractional derivatives arising in the physical model. [ABSTRACT FROM AUTHOR]

This paper deals with the well-known Becker-Stark inequality. By using variable replacement from the viewpoint of hypergeometric functions, we provide a new and general refinement of Becker-Stark inequality. As a particular case, the double inequality π² − (π² − 8) sin² x/π² − 4x² < tan x/x < π² − (4 − π²/3) sin² x/π² − 4x² for x ∈ (0, π/2) will be established. The importance of our result is not only to provide some refinements preserving the structure of Becker-Stark inequality but also that the method can be extended to the case of generalized trigonometric functions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce two new subclasses R sin m (ℑ) and R sin (ℑ) of analytic functions by making use of subordination involving the sine function and the modified sigmoid activation function ℑ (v) = 2 1 + e − v , v ≥ 0 in the open unit disc E. Our purpose is to obtain some initial coefficients, Fekete–Szego problems, and upper bounds for the third- and fourth-order Hankel determinants for the functions belonging to these two classes. All the bounds that we will find here are sharp. We also highlight some known consequences of our main results. [ABSTRACT FROM AUTHOR]

The main object of this paper is to study the traveling wave solutions of the fractional coupled Konopelchenko–Dubrovsky model by using the complete discriminant system method of polynomials. Firstly, the fractional coupled Konopelchenko–Dubrovsky model is simplified into nonlinear ordinary differential equations by using the traveling wave transformation. Secondly, the trigonometric function solutions, rational function solutions, solitary wave solutions and the elliptic function solutions of the fractional coupled Konopelchenko–Dubrovsky model are derived by means of the polynomial complete discriminant system method. Moreover, a two-dimensional phase portrait is drawn. Finally, a 3D-diagram and a 2D-diagram of the fractional coupled Konopelchenko–Dubrovsky model are plotted in Maple 2022 software. [ABSTRACT FROM AUTHOR]

The (2+1)-dimensional Chaffee-Infante equation (CIE) is a significant model of the ionacoustic waves in plasma. The primary objective of this paper was to establish and examine closedform soliton solutions to the CIE using the modified extended direct algebraic method (m-EDAM), a mathematical technique. By using a variable transformation to convert CIE into a nonlinear ordinary differential equation (NODE), which was then reduced to a system of nonlinear algebraic equations with the assumption of a closed-form solution, the strategic m-EDAM was implemented. When the resulting problem was solved using the Maple tool, many soliton solutions in the shapes of rational, exponential, trigonometric, and hyperbolic functions were produced. By using illustrated 3D and density plots to evaluate several soliton solutions for the provided definite values of the parameters, it was possible to determine if the soliton solutions produced for CIE are cuspon or kink solitons. Additionally, it has been shown that the m-EDAM is a robust, useful, and user-friendly instrument that provides extra generic wave solutions for nonlinear models in mathematical physics and engineering. [ABSTRACT FROM AUTHOR]

This work studied in detail for the first time the bending of laminated composite plates subjected t mechanical variations by new theory Trigonometric and Trigonometric-Hyperbolic functions of shear deformation. From the Euler-Lagrange hypothesis and the equations of the shear deformation theory, we will develop a present method. One of the most important problems of composite plates is the analysis of their bending behavior. The correct approach used to study their bending behavior includes two trigonometric and trigonometric-hyperbolic functions satisfying the null shear stress condition at the free edges. In this paper the bending problem is solved analytically by developing a computational code and numerically solved by Finite Element Method. In order to simplify the study of the bending behavior, an approach taking into consideration the effect of the transverse shear deformation without the shear coefficient of correction with only four unknowns has been developed while requiring five or more unknowns for other theories. Convergence analysis has been carried and the results are compared to open literature available for plate bending analysis. The approach proves to be simple and useful in analyzing the bending behavior of composite layered plates. [ABSTRACT FROM AUTHOR]

In this paper, the  (2 + 1) -dimensional seventh-order Caudrey–Dodd–Gibbon–KP equation is investigated through the Lie group method. The Lie algebra of infinitesimal symmetries, commutative and adjoint tables, and one-dimensional optimal systems is presented. Then, the seventh-order Caudrey–Dodd–Gibbon–KP equation is reduced to nine types of  (1 + 1) -dimensional equations with the help of symmetry subalgebras. Finally, the unified algebra method is used to obtain the soliton solutions, trigonometric function solutions, and Jacobi elliptic function solutions of the seventh-order Caudrey–Dodd–Gibbon–KP equation. [ABSTRACT FROM AUTHOR]

This paper proposes a procedure to square-free factorization of mixed trigonometricpolynomials and some examples are presented to show the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

Presented in this paper is a trigonometrically fitted scheme based on a class of improved hybrid method for the numerical integration of oscillatory problems. The trigonometric conditions are constructed through which a third algebraic order scheme is derived. Numerical properties of the scheme are analysed. A numerical experiment is conducted to validate the scheme. Results obtained reveal the superiority of the scheme over its equals in the literature. [ABSTRACT FROM AUTHOR]

Copulas are well-known tools for describing the relationship between two or more quantitative variables. They have recently received a lot of attention, owing to the variable dependence complexity that appears in heterogeneous modern problems. In this paper, we offer five new copulas based on a common original ratio form. All of them are defined with a single tuning parameter, and all reduce to the independence copula when this parameter is equal to zero. Wide admissible domains for this parameter are established, and the mathematical developments primarily rely on non-trivial limits, two-dimensional differentiations, suitable factorizations, and mathematical inequalities. The corresponding functions and characteristics of the proposed copulas are looked at in some important details. In particular, as common features, it is shown that they are diagonally symmetric, but not Archimedean, not radially symmetric, and without tail dependence. The theory is illustrated with numerical tables and graphics. A final part discusses the multi-dimensional variation of our original ratio form. The contributions are primarily theoretical, but they provide the framework for cutting-edge dependence models that have potential applications across a wide range of fields. Some established two-dimensional inequalities may be of interest beyond the purposes of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the (possible) solutions of the equation exp* (f) = g, where g is a slice regular never vanishing function on a circular domain of the quaternions H and exp* is the natural generalization of the usual exponential to the algebra of slice regular functions. Any function f which satisfies exp* (f) = g is called a *-logarithm of g. We provide necessary and sufficient conditions, expressed in terms of the zero set of the "vector" part gv of g, for the existence of a *-logarithm of g, under a natural topological condition on the domain Γ. By this way, we prove an existence result if gv has no non-real isolated zeroes; we are also able to give a comprehensive approach to deal with more general cases. We are thus able to obtain an existence result when the non-real isolated zeroes of gv are finite, the domain is either the unit ball, or H, or D (the solid torus obtained by circularization in H of the disc contained in C and centered in 2 √ -1 with radius 1), and a further condition on the "real part" g0 of g is satisfied (see Theorem 6.19 for a precise statement). We also find some unexpected uniqueness results, again related to the zero set of gv, in sharp contrast with the complex case. A number of examples are given throughout the paper in order to show the sharpness of the required conditions. [ABSTRACT FROM AUTHOR]

This paper's main goal is to introduce left and right exponential trigonometric convex interval-valued mappings and to go over some of their important characteristics. Additionally, we demonstrate the Hermite–Hadamard inequality for interval-valued functions by utilizing fractional integrals with exponential kernels. Moreover, we use the idea of left and right exponential trigonometric convex interval-valued mappings to show various findings for midpoint- and Pachpatte-type inequalities. Additionally, we show that the results provided in this paper are expansions of several of the results already demonstrated in prior publications The suggested research generates variants that are applicable for conducting in-depth analyses of fractal theory, optimization, and research challenges in several practical domains, such as computer science, quantum mechanics, and quantum physics. [ABSTRACT FROM AUTHOR]

Path planning, which is needed to obtain collision-free optimal paths in complex environments, is one key step within unmanned aerial vehicle (UAV) systems with various applications, such as agricultural production, target tracking, and environmental monitoring. A new hybrid gray wolf optimization algorithm—SSGWO—is proposed to plan paths for UAVs under three-dimensional agricultural environments in this paper. A nonlinear convergence factor based on trigonometric functions is used to balance local search and global search. A new relative-distance fitness adaptation strategy is created to increase the convergence speed of the SSGWO. Integrating the simulated annealing (SA) algorithm, an alternative position update strategy based on SA is proposed to improve the search process with diverse capabilities. Finally, a B-spline curve is introduced into a smooth path to ensure the path's feasibility. The simulation results show that the SSGWO algorithm has better convergence accuracy and stability, and can obtain higher-quality paths in a three-dimensional environment, compared with GWO, MGWO, IGWO, and SOGWO. [ABSTRACT FROM AUTHOR]

This paper applies two different types of Riemann–Liouville derivatives to solve fractional differential equations of second order. Basically, the properties of the Riemann–Liouville fractional derivative depend mainly on the lower bound of the integral involved in the Riemann–Liouville fractional definition. The Riemann–Liouville fractional derivative of first type considers the lower bound as a zero while the second type applies negative infinity as a lower bound. Due to the differences in properties of the two operators, two different solutions are obtained for the present two classes of fractional differential equations under appropriate initial conditions. It is shown that the zeroth lower bound implies implicit solutions in terms of the Mittag–Leffler functions while explicit solutions are derived when negative infinity is taken as a lower bound. Such explicit solutions are obtained for the current two classes in terms of trigonometric and hyperbolic functions. Some theoretical results are introduced to facilitate the solutions procedures. Moreover, the characteristics of the obtained solutions are discussed and interpreted. [ABSTRACT FROM AUTHOR]

In this paper, we establish two exponential double inequalities generalized Jordan's inequality. [ABSTRACT FROM AUTHOR]

This paper presents new analytical expressions for displacements Green's functions to a steady-state spatial BVP of thermoelasticity for an unbounded parallelepiped, subjected to a unit point heat source. These results are obtained on the base of special structural formulas for displacements Green's functions, which are expressed in terms of respective Green's functions for Poisson's equation. An example of the application of derived new analytical expressions is presented for a particular spatial BVP for a thermoelastic unbounded parallelepiped, subjected to a constant heat source, given inside of a rectangle. Both analytical expressions for displacements Green's functions and thermoelastic displacements in the case of a particular problem are obtained in the form of double infinite series, containing product between exponential and trigonometric functions, which satisfy basic equations, boundary conditions on the marginal strips and vanishes at infinity. The presented example of a new steady-state spatial BVP of thermoelasticity for an unbounded parallelepiped, subjected to a unit point heat source will permit readers to derive the other examples to new analytical expressions for Green's functions. These Green's functions can be applied as kernels in the method of the boundary integral equations to solution of many particular BVP for thermoelastic unbounded parallelepiped. All these analytical results can be used also as some test problems for different numerical methods. [ABSTRACT FROM AUTHOR]

In this paper, a summation formula for a general family of a finite secant sum has been extended by making use of a particularly convenient integration contour method. The main theorem derived from this approach is the finite sum involving the Hurwitz-Lerch zeta function. This theorem for particular values is used to derive the finite product of the fifth roots of the quotient product of the gamma function along with finite sums and functional equations involving trigonometric functions. [ABSTRACT FROM AUTHOR]

This paper investigates the soliton wave solution of the nonlinear biological population (NBP) model by employing a novel computational scheme. The selected model for this study describes the logistics of the population because of births and deaths. Some novel structures of the NBP model's solutions, are obtained such as exponential, trigonometric, and hyperbolic. These solutions are clarified through some distinct graphs in contour three plot, three-dimensional, and two-dimensional plots. The Hamiltonian system's characterizations are used to check the obtained solutions' stability. The solutions' accuracy is checked by handling the NBP model through the variational iteration (VI) method. The matching between analytical and semi-analytical solutions shows the accuracy of the obtained solutions. The method's performance shows its effectiveness, power, and ability to apply to many nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This paper proposed a united proportional navigation guidance (UPNG) method to alleviate the guidance command saltation with an impact angle constraint under the condition of no real-time distance between the vehicle and the target (line-of-sight (LOS) distance). Firstly, based on the biased proportional navigation guidance (BPNG), a smooth-biased proportional navigation guidance (SBPNG) method was proposed, whose bias term was designed as a trigonometric function. In SBPNG method, due to the continuous smooth change of the bias term, the guidance command would not saltus anymore, and the impact angle was controlled by the bias integral component. Secondly, biased on SBPNG method, the united proportional navigation guidance (UPNG) method combining SBPNG and variable coefficient proportional navigation guidance (VCPNG) was established. In UPNG method, because there was no LOS distance, the guidance coefficient was designed as a function of the difference between the expected impact angle and the estimated impact angle, so the closed-loop control of impact angle was realized. Finally, a lot of simulation experiments on different guidance laws were carried out without real-time LOS distance. The results verify that the UPNG method proposed in this paper solves the problem of guidance command saltation effectively and has better robustness in impact angle control. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate various formulas, identities and relations involving Apostol type numbers and parametric type polynomials. By using generating functions and their functional equations, we give many relations among the certain family of combinatorial numbers, the Vieta polynomials, the two-parametric types of the Apostol-Euler polynomials, the Apostol-Bernoulli polynomials, the Apostol-Genocchi polynomials, the Fibonacci and Lucas numbers, the Chebyshev polynomials, and other special numbers and polynomials. Moreover, we give some formulas related to trigonometric functions, special numbers and special polynomials. Finally, some remarks and observations on the results of this paper are given. [ABSTRACT FROM AUTHOR]

This paper discussed the indefinite plithogenic trigonometric integrals, where we presented the integrating products of plithogenic trigonometric function, also studying the plithogenic trigonometric identities, which facilitated finding the integral of the associated formulas. In addition to a set of exercises that clarify each idea. [ABSTRACT FROM AUTHOR]

In this paper, we propose a decision procedure of reachability for a linear system ξ ′ = A ξ + u , where the matrix A ′ s eigenvalues can be arbitrary algebraic number and the input u is a vector of trigonometric-exponential polynomials. If the initial set contains only one point, the reachability problem under consideration is reduced to the decidability of the sign of trigonometric-exponential polynomial and then achieved by being reduced to verification of a series of univariate polynomial inequalities through Taylor expansions of the related exponential functions and trigonometric functions. If the initial set is open semi-algebraic, we will propose a decision procedure based on OpenCAD and an algorithm of real root isolation derived from the sign-deciding procedure for the trigonometric-exponential polynomials. The experimental results indicate the efficiency of our approach. Under the assumption of Schanuel's Conjecture, the above procedures are complete for bounded time except for several cases. [ABSTRACT FROM AUTHOR]

In this paper, we establish two new inequalities of the Masjed Jamei type for inverse trigonometric and inverse hyperbolic functions and apply them to obtain some refinement and extension of Mitrinović–Adamović and Lazarević inequalities. The inequalities obtained in this paper go beyond the conclusions and conjectures in the previous literature. Finally, we apply the main results of this paper to the field of mean value inequality and obtain two new inequalities on Seiffert-like means and classical means. [ABSTRACT FROM AUTHOR]

This paper proposes a novel approach for power system dynamic simulation based on the differential transformation (DT). The DT is introduced to study power systems as high-dimensional nonlinear dynamical systems for the first time, and is able to avoid computations of high-order derivatives with nonlinear differential equations by its transform rules. This paper, first, proposes and proves several new transform rules for generic nonlinear functions that often appear in power system models, and then uses these rules to transform representative power system models such as the synchronous machine model with trigonometric functions and the exciter model with exponential and square root functions. This paper also designs a DT-based simulation scheme that allows significantly prolonged time steps to reduce simulation time compared to a traditional numerical approach. The numerical stability, accuracy, and time performance of the proposed new DT-based simulation approach are compared with widely used numerical methods on the IEEE 39-bus system and Polish 2383-bus system. [ABSTRACT FROM AUTHOR]

p , q -quasirung orthopair fuzzy set (p , q -QOFS) is an extension of the q -rung orthopair fuzzy set (q -ROFS) theory used to represent uncertainty and vagueness in decision-making processes. The paper aims to utilize robust sine-trigonometric operational laws to investigate the group decision-making process within the p , q -QOFSs framework. The p , q -QOFS possess a distinctive characteristic of handling uncertain information by utilizing a wider space for membership representation compared to q -ROFS. Consequently, the current paper has been categorized into three distinct phases. In the initial phase, novel operational laws will be presented for p , q -QOFSs. The fundamental concept behind these newly proposed operations is to integrate the properties of the sine function, which include being periodic and symmetric about the origin, into the decision-making process for objects. Then, based on these laws, several operators for aggregating information will be obtained, along with their necessary properties and relationships. Lastly, an algorithm will be presented for interpreting the problem of multi attribute group decision-making (MAGDM), utilizing the operators, and demonstrating it with an illustrative example. A comprehensive comparative analysis will be conducted with some of the existing methods to uncover their impacts. [ABSTRACT FROM AUTHOR]

Proposing new families of probability models for data modeling in applied sectors is a prominent research topic. This paper also proposes a new method based on the trigonometric function to derive the updated form of the existing probability models. The proposed family is called the cotangent trigonometric-G family of distributions. Based on the cotangent trigonometric-G method, a new version of the Weibull model, namely, the cotangent trigonometric Weibull distribution, is studied. Certain mathematical properties of the cotangent trigonometric-G family are derived. The estimators of the cotangent trigonometric-G distributions are obtained via the maximum likelihood method. The Monte Carlo simulation study is conducted to assess the performances of the estimators. Finally, two applications from the health sector are considered to illustrate the cotangent trigonometric-G method. Based on seven evaluating criteria, it is observed that the cotangent trigonometric-G significantly improves the fitting power of the existing models. [ABSTRACT FROM AUTHOR]

Sound rendering is the process of determining the sound propagation path from an audio source to a listener and generating 3D sound based on it. This task demands complex calculations, including trigonometric functions. This paper presents hardware-based inverse cosine function calculations using the table method and linear approximation. This approach maintains a high accuracy while limiting hardware size for suitability in sound rendering applications. Consequently, our proposed hardware-based inverse cosine calculation method is a valuable tool for achieving high efficiency and accuracy in 3D sound rendering. [ABSTRACT FROM AUTHOR]

The dynamic tracking error due to servo dynamic performance deficiency is a main cause of processing inaccuracy during precise high-speed five-axis machining. The rotation tool center point (RTCP) test is usually considered a valid test for dynamic control accuracy, and the key to determining the capability of the RTCP test is axis motion planning in the test process. In previous studies, axis motion planning, based on multiple trigonometric functions, has been proposed as a typical and valid method for axis motion planning for RTCP tests, which is included in the latest GB/T Chinese national recommended standard. In this paper, the multiple-trigonometric-function-based axis motion planning method is analyzed and extended. According to the calculation and analysis of the effects of trigonometric parameters on axis motions, the effects of several trigonometric parameter combinations on the feasibility of RTCP test planning are analyzed and summarized, and feasible trigonometric parameter combinations for RTCP test planning are selected. Using the simulation, the effects of different trigonometric parameter collections on the dynamic control accuracy test capability are summarized, and valid trigonometric parameter combinations are selected to guide the design of new tests. According to simulations and experiments, the tests created by the selected parameter combinations can be verified to have advantages in the dynamic control accuracy test capability. The analysis and summary in this paper can provide a guide for RTCP test axis motion planning aimed at dynamic control accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we presented the indefinite integrals of refined neutrosophic trigonometric functions, where we discussed the integrals by distinguishing several cases of these types of refined neutrosophic integrals. Also, we introduced the refined neutrosophic trigonometric identities, which makes it easier for us to find these types of integrals. In addition, to supporting this with appropriate examples. [ABSTRACT FROM AUTHOR]

Grid converters need a synchronization which is often a phase-locked loop (PLL) to get connected to the grid. In the literature, impedance-based models using small-signal technique have been widely utilized to describe the behaviors of PLL-synchronized grid converter. However, most of them target at specific problems and are able to represent the phenomenon under certain scenarios. For the analysis during phase transient, like phase jump due to grid fault, an accurate model is still missing. To address this issue, a general impedance-based model using Maclaurin expansions for its trigonometric function approximation has been proposed in this paper, so that it can well represent the system behavior during phase transient. Moreover, different deepness of the general model can be scaled, leading to different models including phasorial model, current source model, impedance-based model with/without PLL effects. A guideline on how to choose a proper model based on the problems to be studied and grid conditions has been provided in this paper. Simulation and experimental results are given to validate the effectiveness of the model and the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper presents fault detection, classification, and location for a PV-Wind-based DC ring microgrid in the MATLAB/SIMULINK platform. Initially, DC fault signals are collected from local measurements to examine the outcomes of the proposed system. Accurate detection is carried out for all faults, (i.e., cable and arc faults) under two cases of fault resistance and distance variation, with the assistance of primary and secondary detection techniques, i.e. second-order differential current derivative d 2 I 3 d t 2 and sliding mode window-based Pearson's correlation coefficient. For fault classification a novel approach using modified multifractal detrended fluctuation analysis (M-MFDFA) is presented. The advantage of this method is its ability to estimate the local trends of any order polynomial function with the help of polynomial and trigonometric functions. It also doesn't require any signal processing algorithm for decomposition resulting and this results in a reduction of computational burden. The detected fault signals are directly passed through the M-MFDFA classifier for fault type classification. To enhance the performance of the proposed classifier, statistical data is obtained from the M-MFDFA feature vectors, and the obtained data is plotted in 2-D and 3-D scatter plots for better visualization. Accurate fault distance estimation is carried out for all types of faults in the DC ring bus microgrid with the assistance of recursive least squares with a forgetting factor (FF-RLS). To verify the performance and superiority of the proposed classifier, it is compared with existing classifiers in terms of features, classification accuracy (CA), and relative computational time (RCT). [ABSTRACT FROM AUTHOR]

Memristor synapses have been widely introduced into neuronal models to investigate the effects of external magnetic fields. However, there is a relative lack of research on the external-induced electric fields in neurons. In this paper, a 4D-memristive Hindmarsh–Rose neuron model is constructed by introducing a memristor and an electric field variable, which can generate complex neural firing. Notably, numerical simulations reveal that the initial conditions of the memristor can induce different firing patterns, exhibiting a unique fractal structure in the basin of attractions. Remarkably, the offset parameters of the internal variables of the neuron can be canceled out so that the offset boosting of the variables can be achieved according to the initial values, giving rise to an uncountably many hidden attractors with homogeneous multistability. This model provides the first example of generating uncountably many attractors in a memristive neuron model without relying on trigonometric functions, significantly advancing our understanding of neuronal dynamics. Finally, a digital circuit is designed and implemented on the RISC-V platform to verify the numerical simulation and theoretical analysis. The findings of this study have a certain implication for the development of advanced neuromorphic computing systems and the understanding of complex neuronal behaviors in the presence of external electric fields. [ABSTRACT FROM AUTHOR]

Statistical methods have widespread applications in real-life scenarios. Researchers have expressed keen interest in implementing different functions to raise new probability distributions. However, very limited efforts have been made to raise new methodologies by applying trigonometric functions. In this paper, we propose a new method based on the trigonometric function. The suggested trigonometric-based method is called a new trigonometric cosine- G (NTC- G) family to generate flexible models. The new method is developed using the cosine function. A subcase of the NTC- G method termed as a new trigonometric cosine-Weibull (NTC-Weibull) is studied. Certain mathematical properties of the NTC-Weibull model are presented. For the NTC-Weibull model, the point estimators are acquired. Finally, the suitability of the NTC-Weibull distribution is demonstrated in light of a healthcare data set. Using the numerical and graphical approaches, the evaluation results of the NTC-Weibull distribution are compared with the Weibull and its four well-known existing modifications. [ABSTRACT FROM AUTHOR]

It is known that discrete analogs of differential operators play an important role in constructing optimal quadrature, cubature, and difference formulas. Using discrete analogs of differential operators, optimal interpolation, quadrature, and difference formulas exact for algebraic polynomials, trigonometric and exponential functions can be constructed. In this paper, we construct a discrete analogue D m (h β) of the differential operator d 2 m d x 2 m + 2 d m d x m + 1 in the Hilbert space W 2 (m , 0) . We develop an algorithm for constructing optimal quadrature formulas exact on exponential-trigonometric functions using a discrete operator. Based on this algorithm, in m = 2 , we give an optimal quadrature formula exact for trigonometric functions. Finally, we present the rate of convergence of the optimal quadrature formula in the Hilbert space W 2 (2 , 0) for the case m = 2 . [ABSTRACT FROM AUTHOR]

The concept of convexity is fundamental in order to produce various types of inequalities. Thus, convexity and integral inequality are closely related. The objectives of this paper are to present a new class of up and down convex fuzzy number valued functions known as up and down exponential trigonometric convex fuzzy number valued mappings ( U D E T - convex FNVMs) and, with the help of this newly defined class, Hermite–Hadamard-type inequalities (H–H-type inequalities) via fuzzy inclusion relation and fuzzy fractional integral operators having exponential kernels. This fuzzy inclusion relation is level-wise defined by the interval-based inclusion relation. Furthermore, we have shown that our findings apply to a significant class of both novel and well-known inequalities for U D E T - convex   F N V M s. The application of the theory developed in this study is illustrated with useful instances. Some very interesting examples are provided to discuss the validation of our main results. These results and other approaches may open up new avenues for modeling, interval-valued functions, and fuzzy optimization problems. [ABSTRACT FROM AUTHOR]

This paper is dedicated to finding the traveling wave solutions of the loaded Burgers equation. We show how to find the solutions via the (G′/G) - expansion method which is one of the most effective ways of finding solutions. When the parameters are taken as special values, the solitary waves are also derived from the traveling waves. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. This method is easy to implement using well-known software packages, which allows you to solve complex nonlinear evolution equations of mathematical physics. [ABSTRACT FROM AUTHOR]

Employing the Riccati–Bernoulli sub-ODE method (RBSM) and improved Weierstrass elliptic function method, we handle the proposed (2 + 1) -dimensional nonlinear fractional electrical transmission line equation (NFETLE) system in this paper. An infinite sequence of solutions and Weierstrass elliptic function solutions may be obtained through solving the NFETLE. These new exact and solitary wave solutions are derived in the forms of trigonometric function, Weierstrass elliptic function and hyperbolic function. The graphs of soliton solutions of the NFETLE describe the dynamics of the solutions in the figures. We also discuss elaborately the effects of fraction and arbitrary parameters on a part of obtained soliton solutions which are presented graphically. Moreover, we also draw meaningful conclusions via a comparison of our partially explored areas with other different fractional derivatives. From our perspectives, by rewriting the equation as Hamiltonian system, we study the phase portrait and bifurcation of the system about NFETLE and we also for the first time discuss sensitivity of the system and chaotic behaviors. To our best knowledge, we discover a variety of new solutions that have not been reported in existing articles V 1 , 2 ∗ ∗ , ... , V 7 , 8 ∗ ∗ , V 9 , 1 0 , ... , V 1 3 , 1 4 . The most important thing is that there are iterative ideas in finding the solution process, which have not been seen before from relevant articles such as [Tala-Tebue et al., 2014; Fendzi-Donfack et al., 2018; Ashraf et al., 2022; Ndzana et al., 2022; Halidou et al., 2022] in seeking for exact solutions about NFETLE. [ABSTRACT FROM AUTHOR]

This paper presents a variational model for the denoising of wrapped phase images. By enforcing the required Pythagorean trigonometric identity between the real and imaginary components of the signal, this model improves the signal-to-noise ratio of the restored signal. To preserve phase map discontinuities, the model is based on total variation. The existence and uniqueness of the model's solution are demonstrated using standard techniques. In addition, the convergence of a rapid fixed-point method to determine the numerical solution is demonstrated. Experiments on both synthetic and actual patterns validate the model's performance. [ABSTRACT FROM AUTHOR]