SUPERSTABILITY OF A GENERALIZED TRIGONOMETRIC FUNCTIONAL EQUATION

In this work, we present an extremely efficient approach for a fast numerical evaluation of highly oscillatory spherical Bessel integrals occurring in the analytic expressions of the so-called molecular multi-center integrals over exponential type functions. The approach is based on the Slevinsky-Safouhi formulae for higher derivatives applied to spherical Bessel functions and on extrapolation methods combined with practical properties of sine and cosine functions. Recurrence relations are used for computing the approximations of the spherical Bessel integrals, allowing for a control of accuracy and the stability of the algorithm. The computer algebra system Maple was used in our development, mainly to prove the applicability of the extrapolation methods. Among molecular multi-center integrals, the three-center nuclear attraction and four-center two-electron Coulomb and exchange integrals are undoubtedly the most difficult ones to evaluate rapidly to a high pre-determined accuracy. These integrals are required for both density functional and ab initio calculations. Already for small molecules, many millions of them have to be computed. As the molecular system gets larger, the computation of these integrals becomes one of the most laborious and time consuming steps in molecular electronic structure calculation. Improvement of the computational methods of molecular integrals would be indispensable to a further development in computational studies of large molecular systems. Convergence properties are analyzed to show that the approach presented in this work is a valuable contribution to the existing literature on molecular integral calculations as well as on spherical Bessel integral calculations.

We generalize the super-stability result for cosine functional equation mappings with values in the field of complex numbers to the case of an arbitrary Hilbert space with the Hadamard product. On the other hand, by an example we prove that this generalization is not true for sine functional equation. Mathematics subject classification (2010): Primary: 39B72, 39B82; secondary: 46E40.

Application of the least-squares inversion method: Fourier series versus waveform inversion

This paper investigates some fundamental properties of hyperbolic trigonometric functions on the hyperbolic number plane. Hyperbolic numbers form a commutative ring with zero divisors, generated by two real numbers via the Cartesian basis. Within this algebraic framework, this paper systematically establish the system of hyperbolic trigonometric functions. By fully leveraging the decomposition properties of hyperbolic numbers and their associated trigonometric functions, this work effectively overcomes the mathematical difficulties arising from the presence of zero divisors in the hyperbolic number ring. On this basis, this paper for the first time in the context of a hyperbolic number ring containing zero divisors, systematically derived and rigorously proved the addition theorems for the hyperbolic sine, cosine, tangent, and cotangent functions. On this basis, we have successfully derived and rigorously proved the addition theorems for the hyperbolic sine, cosine, tangent, and cotangent functions, establishing a complete system of angle addition formulas and laying a solid foundation for the theory of hyperbolic functions. The addition theorems established in this research will provide an important theoretical foundation for the further development of hyperbolic analysis in function theory, while simultaneously injecting new research momentum into the further study of the properties and expansions of hyperbolic trigonometric series. These theoretical achievements are expected to play significant roles in the study of hyperbolic differential equations, geometric analysis, and related physical problems.

The prime goal of this paper is to establish sharp lower and upper bounds for useful functions such as the exponential functions, with a focus on exp(−x^2), the trigonometric functions (cosine and sine) and the hyperbolic functions (cosine and sine). The bounds obtained for hyperbolic cosine are very sharp. New proofs, refinements as well as new results are offered. Some graphical and numerical results illustrate the findings.

This paper presents a class of trigonometric Lagrange-mesh functions constructed from the Lagrange-Jacobi functions. These functions extend the set of trigonometric Lagrange-mesh functions to beyond only those constructed directly from the sine and cosine functions. Trigonometric Lagrange-Chebyshev functions obtained from the presented trigonometric Lagrange-Jacobi functions are equivalent to known trigonometric Lagrange-Chebyshev functions constructed directly from the sine and cosine functions and generate identical matrix elements for the kinetic energy operator.

In the present paper we study the generalized Hyers–Ulam stability of the generalized trigonometric functional equations f ( x y ) + μ ( y ) f ( x σ ( y ) ) = 2 f ( x ) g ( y ) + 2 h ( y ) , x , y ∈ S ; f ( x y ) + μ ( y ) f ( x σ ( y ) ) = 2 f ( y ) g ( x ) + 2 h ( x ) , x , y ∈ S , where S is a semigroup, σ : S ⟶ S is a involutive morphism, and μ : S ⟶ C is a multiplicative function such that μ ( x σ ( x ) ) = 1 for all x ∈ S . As an application, we establish the generalized Hyers–Ulam stability theorem on amenable monoids and when σ is an involutive automorphism of S.

One- and two-center overlap like quantum similarity integrals over ψ^((α)) functions are evaluated using the one-center expansion method. These integrals are expressed through expansion coefficients and usual two-center overlap integrals. The expressions derived in this work have no restrictions for the values of orbital parameters, quantum numbers and internuclear distances. Since these integrals of ψ^((α)) functions are calculated for the first time, the comparison is made with literature values obtained from Slater type functions. The computation results are in good agreement with literature values. The algorithm presented in this study could be useful when exponential type functions are employed in quantum similarity measures of atomic and molecular systems.

cosine and sine functions defined on a Banach space are useful tools in the study of wide classes of abstract evolution equations. In this paper, we introduce a definition of cosine and sine functions on time scales, which unify the continuous, discrete and the cases “in between.” Our definition includes several types of time scales such as real numbers set, integers numbers set, quantum scales, among others. We investigate the relationship between the cosine function on time scales and its infinitesimal generator, proving several properties concerning it. Also, we investigate the sine functions on time scales, presenting their main properties. Finally, we apply our theory to study the homogeneous and inhomogeneous abstract Cauchy problem on time scales in Banach spaces.

We propose a three-neuron heterogeneous cyclic Hopfield neural network (het-CHNN) utilizing three different activation functions: the hyperbolic tangent, sine, and cosine functions. The network’s globally uniformly ultimate boundedness is proved theoretically, and its chaotic dynamics are explored through numerical simulations and analog experiments. The numerical results demonstrate that the het-CHNN displays chaotic dynamics and multi-scroll chaotic attractors. Subsequently, the het-CHNN is implemented in an analog circuit, and hardware experiments are performed to verify the previous numerical results. Notably, the het-CHNN successfully resolves the issue of the absence of chaos in a three-neuron CHNN and currently appears to be the simplest three-neuron Hopfield neural network (HNN) that can generate chaos.

We utilize -fractional Caputo initial value problems of order to derive a -analogue for Gronwall-type inequality. Some particular cases are derived where -Mittag-Leffler functions and -exponential type functions are used. An example is given to illustrate the validity of the derived inequality.

Of the nuclear magnetic resonance (MMR), the nuclear shielding tensor is of a great interest. The relativistic calculation of the nuclear shielding tensor involves extremely challenging integrals of first and second order. Among the first order integrals are paramagnetic contribution integrals, which are extremely difficult to evaluate analytically and numerically, especially when using exponential type functions (ETFs). The main difficulty in the analytical development arises from the presence of 1/r5 in the operators. In the present contribution, we developed the Fourier transform of the operators of the paramagnetic contribution and we used the Fourier integral transformation to derive analytic expressions for the integrals under consideration over ETFs. The main difficulty in the numerical treatment of the obtained analytic expressions arises from the presence of highly oscillatory spherical Bessel integrals. Extrapolation methods and nonlinear transformations are used to develop highly accurate algorithms for the numerical evaluation of the integrals of the paramagnetic contribution in the relativistic calculation of the shielding tensor.

The nonlinear Schrodinger equation (NLSE) with the aid of three order dispersion terms is investigated to find the exact solutions via the extended $ (\frac{G'}{G^{2}})$ -expansion method and the first integral method. Many exact traveling wave solutions, such as trigonometric, hyperbolic, rational, soliton and complex function solutions, are characterized with some free parameters of the problem studied. It is corroborated that the proposed techniques are manageable, straightforward and powerful tools to find the exact solutions of nonlinear partial differential equations (PDEs). Some figures are plotted to describe the propagation of traveling wave solutions expressed by the hyperbolic functions, trigonometric functions and rational functions.

This study is situated within the framework of realistic mathematics education (RME) and presents a classroom proposal aimed at illuminating and modeling everyday periodic phenomena. Emphasis is placed on recognizing periodic functions as fundamental tools for characterizing the sine and cosine trigonometric functions. The investigation uses the concept of mathematization levels to delineate the transition from informal to formal knowledge through real-world problem-solving using the GeoGebra platform. This research was conducted with tenth-grade students at the San Vicente Educational Institution in the Buenaventura District. The outcomes reveal that students engaged in activities, and subsequent analyses demonstrate a nuanced understanding of periodic phenomena within realistic contexts. This comprehension encompasses proficiency in various representation systems, including tabular, graphical, and algebraic methods. Moreover, the process of progressing from informal to formal knowledge is marked by incremental levels of mathematization, encompassing both horizontal and vertical processes. The main focus of this research is to characterize the process of mathematization within a realistic context, particularly concerning the sine and cosine trigonometric functions, and evaluate the strengths and limitations of this approach. This study underscores the potential of RME for elucidating intricate mathematical modeling processes within real-world contexts. These insights underscore the value of integrating authentic scenarios to foster a comprehensive understanding of mathematical concepts. Challenges are also acknowledged, particularly in achieving the formalization of generalized models. Thus, this research contributes to discussions on effective mathematics education by showcasing the transformative potential of contextualized pedagogical strategies. Keywords: realistic mathematics education, trigonometric functions, levels of mathematization, context, levels of understanding. https://doi.org/10.55463/issn.1674-2974.50.7.24

Mathematics teachers’ ability to translate and make connections between representations of functions requires investigation. Consequently, this qualitative case study article focuses on pre-service mathematics teachers’ nature of understanding of the tangent function; a function bearing unique characteristics compared with the sine and cosine functions. Twenty-two finalist pre-service teachers were conveniently selected and assessed concerning the ability to translate a tangent function to the graphical representation. Likewise, participants’ abilities to correctly explain this function and provide appropriate justifications for espoused perspectives were investigated. Although the teachers learned higher mathematics, their assessment was school mathematicsbased. Descriptive analyses showed that only one teacher (5%) accurately completed the task. Eleven teachers (50%) did not provide graphs; suggesting a lack of knowledge required to change representation. Ten (45%) drew flawed graphs depicting a lack of understanding of discontinuity of the tangent function at certain angles and the role of a domain. Essentially, these demonstrated mere memorization of the appearance of the tangent graph. A purposive subsample of size six then participated in exploratory semi-structured interviews. The interviews allowed participants to elucidate their answers to the initial task. Content analysis of the transcripts corroborated the earlier finding as the interviewees could not coherently explain the tangent function, and failed to justify their reasoning. The teachers demonstrated a superficial understanding despite having studied advanced mathematics. This reinforces the view that studying advanced mathematics does not assure a relational understanding of school mathematics. Therefore, it is necessary for trainee teachers to explore school mathematics.