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Nguyen Minh Khoa has responded to the editor's letter regarding their article titled "On the polyconvolution operator with a trigonometric weight function for Hartley integral transforms and applications." They express gratitude for the feedback and address specific points raised by the anonymous reviewer. They confirm the accuracy of the factorization equality and Placherel's type theorem. However, they point out an error in the condition of Watson's type theorem and provide the corrected version. Nguyen Minh Khoa appreciates the comments and aims to improve the quality of their article. [Extracted from the article]

Motivated by a curiosity to explore the behavior of innovative arch structures enabled by the use of fiber-reinforced polymer (FRP) composites, this paper proposes a theoretical model built upon an enhanced formulation of the deflection method, broadening its scope to large-curvature problems. Traditionally, the deflection method approximates curvature as the second-order derivative of deflection, a simplification valid only for small curvatures. This limitation poses a challenge when applying the deflection method to problems involving large curvatures, a characteristic inherent in FRP-enabled arches where significant curvatures arise either initially or due to deformation. The enhanced formulation at the core of the proposed model addresses this challenge by incorporating a circular deflection function. This function posits that each deformed segment of the structural member can be represented by a circular arc, with its curvature and length related to the internal axial force and bending moment at the midpoint section of the segment. This feature facilitates the exact representation of curvature, offering the proposed model a unified approach capable of addressing both small- and large-curvature problems. The paper details the formulation and verification of the theoretical model, with an emphasis on its application to representative cases of FRP-enabled arches. [ABSTRACT FROM AUTHOR]

This article employs Kolb's Experiential Learning Theory (ELT) to address weaknesses in South Africa's Mathematics education, focusing on the use of concrete mathematical manipulatives for enhancing conceptual understanding in grade 12 learners. A Mathematical manipulative is defined as any material or object from the real world that children can move around to show a mathematical concept. Kolb's Experiential Theory links learning style to academic achievement. Kolb's ELT involves four learning modes: concrete experience, reflective observation, abstract conceptualization, and active experimentation. The use of manipulatives completes Kolb's learning cycle by providing a concrete experience, interaction and reflection with peers, abstract conceptualisation of triangles in the various planes and finally active experimentation. The study aims at improving the application of the sine rule, cosine rule, and area rule in solving 3-dimensional trigonometric problems. This qualitative study investigated the influence of mathematical models on teaching 3D trigonometric problems based on a sample of sixteen Grade 12 learners. Data was collected through observation, interviews, and written responses. The findings in this paper show that learners using manipulatives displayed heightened interests and improved understanding. [ABSTRACT FROM AUTHOR]

Purpose: Circular economy advocates innovations that upcycle wastes in the food supply chain to generate high added-value materials. These innovations are not only disruptive and green but also they are often initiated by startups, leading to the emergence of novel open-loop supply chains connecting actors in food and non-food sectors. While earlier research has highlighted the need to seek legitimacy for disruptive innovations to survive and grow, little is known about how these innovations occur and evolve across sectors. This paper aims to elaborate on this mechanism by exploring the function of the circular economy as a boundary object to facilitate legitimacy-seeking strategies. Design/methodology/approach: An exploratory multiple-case research design is adopted and features food waste innovation projects with multi-tier supply chains consisting of a food producer, a startup and a buying firm. The study is investigated from the legitimacy and boundary object lenses. Findings: The findings proposed a framework for the role of a boundary object in enabling legitimacy-seeking strategies for novel food waste innovations. First, the interpretative flexibility of the circular economy affords actors symbolic resources to conduct manipulation strategy to achieve cognitive legitimacy. Second, small-scale work arrangements enable creation strategy for the new supply chain to harness moral legitimacy. Finally, pragmatic legitimacy is granted via diffusion strategy enabled by scalable work arrangements. Originality/value: This paper provides novel insights into the emergence of food waste innovation from a multi-tier supply chain perspective. It also highlights the key role of the boundary object in the legitimacy-seeking process. [ABSTRACT FROM AUTHOR]

The primary focus of this paper is the determination of novel soliton solutions for the (2 + 1) -dimensional Heisenberg ferromagnetic spin chain (HFSC) problem. The suggested model may be thought of as a special case of the nonlinear Schrödinger equation. Exact solutions to the proposed equation have been found using efficient analytical approaches such as the singular manifold method and the tan (η 2) -expansion method. These proposed methods are significant mathematical tools to obtain the exact traveling wave solutions of nonlinear complex partial differential equations (PDEs). After decomposing the governing problem into its real and imaginary components, we analyze both the real and imaginary components separately. Soliton solutions are obtained using computational tools like Maple. Trigonometric function solutions, rational solutions and hyperbolic function solutions are found for Heisenberg ferromagnetic model by the application of the proposed techniques. Furthermore, the physical properties of the solutions, such as hyperbolic functions, are important. The hyperbolic tangent is used in the computation of magnetic moment and velocity in special relativity. As a result, we hypothesize that the obtained results have physically comparable meanings to those stated by these collocations. By selecting suitable values of arbitrary parameters, it has been observed graphically that the amplitude of obtained solutions can be changed. Graphical representations of few of the reported solutions are presented in both two and three dimensions by choosing appropriate values of parameters. The suggested approaches are very efficient and reliable for finding exact solutions of wide range of nonlinear PDEs in a complex medium. The advantage of these techniques is evident since they are not limited in their ability to locate such wave patterns. Back substitution in the original equation using Maple 18 validated all solutions found, and the physical significance of these results has been underlined. The constraint conditions for the existence of constructed solutions are also provided in this paper. [ABSTRACT FROM AUTHOR]

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]

In this paper, we introduce the difference equations of Bernoulli polynomials constructed using trigonometric functions and quantum numbers. Several types of difference equations have Bernoulli polynomials (QSB and QCB) as solutions and contain various properties. [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]

In this paper, the extended simplest equation method is utilized to construct the novel exact optical solitons solutions of the perturbed time fractional Chen–Lee–Liu equation with conformable fractional derivative. The acquired optical solitons and other solutions are expressed via the rational functions, the trigonometric functions, and the hyperbolic functions; in addition to guaranteeing the existence of the acquired results, the constrain conditions are provided. Furthermore, to elucidate the magnitude of the proposed equation, various obtained solutions are plotted via two-dimensional (2D) and three-dimensional (3D) graphs using appropriate values of parameters. It is observed that the present technique is a powerful tool and efficient for finding the analytical solution for nonlinear differential equations of integer and fractional orders. [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]

In this paper traveling wave solutions of the modified regularized and regularized long wave equations (MRLW & RLW equations) are simulated using modified trigonometric B-spline functions in collocation method. We validate the adaptability of the proposed method successfully by simulating single solitary wave motion, two and three solitary wave interactions. We also compute the three motion invariants numerically and these are found to be very very close with their analytical values. We also calculate 4 and Loo error norms for single solitary wave motion. Stability of the method is also discussed. Convergence rate for the proposed scheme is obtained numerically with respect to space step size. Computational complexity is analysed and it is found that the developed approach gives very good results with very less computational cost. in terms of the number of arithmetic operations, the computational complexity of the proposed method is found to be linear. [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 recent years, Sine Cosine Algorithm (SCA) is a kind of meta-heuristic optimization algorithm with simple structure, simple parameters and trigonometric function principle. It has been proved that it has good competitiveness among the existing optimization algorithms. However, the single mechanism of SCA leads to its insufficient utilization of the information of the whole population, insufficient ability to jump out of local optima and poor performance at solving complex objective function. Therefore, this paper introduces social learning strategy (SL) and elite opposition-based learning (EOBL) strategy to improve SCA, and proposes novel algorithm: enhancing Sine Cosine Algorithm based on elite opposition-based learning and social learning (ESLSCA). Social learning strategy takes full advantage of information from the entire population. The elite opposition-based learning strategy provides a possibility for the algorithm to jump out of local optima and increases the diversity of the population. To demonstrate the performance of ESLSCA, this paper uses 22 well-known benchmark test functions and CEC2019 test function set to evaluate ESLSCA. The comparisons show that the proposed ESLSCA has better performance than the standard SCA and it is very competitive among other excellent optimization algorithms. [ABSTRACT FROM AUTHOR]

Short-term traffic flow prediction is an essential component of intelligent transportation systems. Shallow and deep pattern learning methods have been widely applied to short-term traffic flow prediction. However, shallow learning methods struggle with highly volatile data and models are usually constant-coefficient. On the other hand, deep learning methods require significant computational resources and time. In this paper, we propose a new adaptive fluctuation grey model for short-term traffic flow prediction. We combine the fractional differential equation and fractional accumulation generation operator, and expand the GM(1,1) model using trigonometric functions. Furthermore, we improve the Harris hawks algorithm by optimizing the distribution of the initial population with Cauchy mutation operator and introducing boundary constraint handling techniques to enhance the model parameter search capability. Finally, we apply the model to short-term traffic flow parameter prediction and compare it with the benchmark model. Results indicate that the new model shows better accuracy performance and better extraction of fluctuation information. [ABSTRACT FROM AUTHOR]

This paper analyzes the coupled nonlinear (2+1)-dimensional complex modified Korteweg-de-Vries (cmKdV) equation, which appears in the fields of applied magnetism and nanophysics. By taking advantage of two mathematical integration approaches, namely, the modified generalized exponential rational function method and the extended tanh function method, a variety of exact optical soliton solutions are obtained for the governing cmKdV equation. These acquired soliton solutions are determined in terms of hyperbolic, exponential, and trigonometric function types. By choosing suitable values of parameters, some 3D, 2D, and contour plots are portrayed with the aid of symbolic computation in Mathematica to visualize the underlying dynamics of the generated solutions. These solutions include doubly soliton, multi-soliton, singular periodic soliton, anti-bell-shaped soliton, and hyperbolic structures. Moreover, the modulation instability of the governing equation is also investigated by using the linear stability analysis. The results presented in this paper are novel and are reported for the first time in the literature. Again, modulation instability analysis was carried out on the governing model for the first time. Thus, the results obtained demonstrate that the two new mathematical schemes are quite concise and effective and can be useful in understanding the dynamical behaviors of many other nonlinear physical models appearing in nonlinear optics, nanophysics, and so many other areas of nonlinear sciences. [ABSTRACT FROM AUTHOR]

The article discusses the use of Patty Paper in teaching the unit circle to students. It cites the identification of the x and y intercepts by the students. It indicates the interpretation of the coordinates in terms of 0°, 90°, 180°, 270°, and 360° angles. It reveals the contribution of the teaching method to work with related angles and solve trigonometric equations.

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This paper investigates the limit cycle bifurcation problem of the pendulum equation on the cylinder of the form x = y, y = --sin x under perturbations of polynomials of sin x, cos x and y of degree n with a switching line y = 0. We first prove that the corresponding first order Melnikov functions can be expressed as combinations of anti-trigonometric functions and the complete elliptic functions of first and second kind with polynomial coefficients in both the oscillatory and rotary regions for arbitrary n. We also verify the independence of coefficients of these polynomials. Then, the lower bounds for the number of limit cycles are obtained using their asymptotic expansions with n = 1, 2, 3. [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 discusses the wave propagation to the nonlinear Sharma–Tasso–Olver–Burgers (STOB) equation which is used as the governing model in different fields. Natural phenomena are typically complex and nonlinear, defying simple linear superposition. Researchers have been studying a wide range of natural phenomena in depth, and nonlinear science has gradually become a part of people's consciousness. One of the most significant research questions in nonlinear science centers around the nonlinear evolution equation and its precise solution. We have secured different shapes of the solitary wave solutions including kink-type, shock-type and combined solitary wave solutions with the assistance of recently developed integration tool, namely the new extended direct algebraic method (NEDAM). Additionally, the solutions for the hyperbolic, exponential and trigonometric functions are retrieved. Moreover, based on a comparison of our results to those that are well known, the study indicates that our solutions are innovative. Using proper parameters in numerical simulations and physical explanations, it is possible to demonstrate the significance of the results. The results of this research can improve the nonlinear dynamic behavior of a system and indicate that the methodology employed is adequate. It is proposed that the offered method can be utilized to support nonlinear dynamical models applicable to a wide variety of physical situations. We hope that a wide spectrum of engineering model professionals will find this study to be beneficial. [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 research paper, we implement the sine-Gordon expansion method to two governing models which are the (2+1)-dimensional Nizhnik–Novikov–Veselov equation and the Caudrey–Dodd–Gibbon–Sawada–Kotera equation. We use conformable derivative to transform these nonlinear partial differential models to ordinary differential equations. We find some wave solutions having trigonometric function, hyperbolic function. Under the strain conditions of these solutions obtained in this paper, various simulations are plotted. [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]

Significant number of procedures for solving of the finite degree-of-freedom forced nonlinear oscillator are developed. For all of them it is common that they are based on the exact solution of the corresponding linear oscillator. For technical reasons, the aim of this paper is to develop a simpler solving procedure. The rotating vector method, developed for the linear oscillator, is adopted for solving of the nonlinear finite degree-of-freedom oscillator. The solution is assumed in the form of trigonometric functions. Assuming that the nonlinearity is small all terms of the series expansion of the function higher than the first are omitted. The rotating vectors for each mass are presented in the complex plane. In the paper, the suggested rotating vector procedure is applied for solving of a three-degree-of-freedom periodically excited oscillator. The influence of the nonlinear stiffness of the flexible elastic beam, excited with a periodical force, on the resonant properties of the system in whole is investigated. It is obtained that the influence of nonlinearity on the amplitude and phase of vibration is more significant for smaller values of the excitation frequency than for higher ones. [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]

Megastable oscillations are a subject of significant research interest due to their broad range of potential applications. Typically, megastable systems are driven into oscillation by a forcing term. In this paper, we propose a novel megastable oscillator that utilizes a combination of Signum and trigonometric functions. To the best of our knowledge, no 3D megastable system has been found to exhibit hyperchaotic behavior without any forcing term. We demonstrate the megastability of our oscillator using phase portraits and basins of attraction and confirm the oscillations using the Hamiltonian energy method. We also conduct a stability analysis to explore the system’s nature and investigate the impact of parameters using a bifurcation diagram. Furthermore, we present a Lyapunov spectrum to identify regions of chaos, hyperchaos, and periodic oscillations. The results we obtain demonstrate the complexity of the system and its sensitivity to initial conditions, making it well-suited for applications such as random number generation and secure communication. [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]