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Our result of this paper aims to indicate that the beta pricing formula could be applied in a long-term model setting as well.

In this paper, we show that the capital asset pricing model can be derived from a three-period general equilibrium model.

We show that our extended model yields a Pareto efficient outcome.

The capital asset pricing model (CAPM) model can be used for pricing long-lived assets.

Long-term modelling and sustainability can be modelled in our setting.

Our results were only known for two periods. The extension to 3 periods opens up a large scope of applicational possibilities in asset pricing, behavioural analysis and long-term efficiency.

This paper focuses on the unconditionally optimal error estimates of a linearized second-order scheme for a nonlocal nonlinear parabolic problem. The first step of the scheme is based on Crank–Nicholson method while the second step is the second-order BDF method.

A rigorous error analysis is done, and optimal L² error estimates are derived using the error splitting technique. Some numerical simulations are presented to confirm the study’s theoretical analysis.

Optimal L² error estimates and energy norm.

The goal of this research article is to present and establish the unconditionally optimal error estimates of a linearized second-order BDF finite element scheme for the reaction-diffusion problem. An optimal error estimate for the proposed methods is derived by using the temporal-spatial error splitting techniques, which split the error between the exact solution and the numerical solution into two parts, that is, the temporal error and the spatial error. Since the spatial error is not dependent on the time step, the boundedness of the numerical solution in L∞-norm follows an inverse inequality immediately without any restriction on the grid mesh.

The paper aims to build the relationship between an entire function of restricted hyper-order with its linear c-shift operator.

Standard methodology for papers in difference and shift operators and value distribution theory have been used.

The relation between an entire function of restricted hyper-order with its linear c-shift operator was found under the periphery of sharing a set of two small functions IM (ignoring multiplicities) when exponent of convergence of zeros is strictly less than its order. This research work is an improvement and extension of two previous papers.

This is an original research work.

The purpose of this two-part research was to investigate the effect of remote learning on student progress in elementary education. Part 2, presented in this paper, is a follow-up study to examine how student progression in the two pandemic-induced environments compared to the pre-pandemic conditions.

The authors expanded the quantitative, quasi-experimental factorial design of the authors' initial study with additional ex-post-facto standardized test score data from before the pandemic to enhance the group comparison with a control: the conventional pre-pandemic classroom environment. Thus, the authors were able to examine in which ways the two pandemic-induced learning environments (remote and hybrid) may have affected learner progress in the two subject areas: English Language (ELA) and Math. Additionally, the authors provided a grade-by-grade breakdown of analysis results.

Findings revealed significant group differences in grade levels at or below 6th grade. In the majority of analyzed comparisons, learner achievement in the hybrid group was significantly lower than those in either the remote or the classroom group, or both.

The additional findings further supported the authors' initial hypotheses: Differences in the consistency and continuity of educational approaches, as well as potential differences in learner predispositions and the availability of home support systems may have influenced observed results. Thus, this research also contributes to the general knowledge about learner needs in elementary education.

During the pandemic, remote learning became ubiquitous. However, in contrast to e-learning in postsecondary education, for which an abundance of research has been conducted, relatively little is known about the efficacy of such approaches in elementary education.

The initial value problem for a semi-linear high-order heat equation is investigated. In the focusing case, global well-posedness and exponential decay are obtained. In the focusing sign, global and non global existence of solutions are discussed via the potential well method.

Let (M, g) be a n-dimensional smooth Riemannian manifold. In the present paper, the authors introduce a new class of natural metrics denoted by g^f and called gradient Sasaki metric on the tangent bundle TM. The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, g^f) and several important results are obtained on curvature, scalar and sectional curvatures.

In this paper the authors introduce a new class of natural metrics called gradient Sasaki metric on tangent bundle.

The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.

The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.

The purpose of the study is to obtain and analyze vibro-acoustic characteristics.

A unified analysis model for the rotary composite laminated plate and conical–cylindrical double cavities coupled system is established. The related parameters of the unified model are determined by isoparametric transformation. The modified Fourier series are applied to construct the admissible displacement function and the sound pressure tolerance function of the coupled systems. The energy functional of the structure domain and acoustic field domain is established, respectively, and the structure–acoustic coupling potential energy is introduced to obtain the energy functional. Rayleigh–Ritz method was used to solve the energy functional.

The displacement and sound pressure response of the coupled systems are acquired by introducing the internal point sound source excitation, and the influence of relevant parameters of the coupled systems is researched. Through research, it is found that the impedance wall can reduce the amplitude of the sound pressure response and suppress the resonance of the coupled systems. Besides, the composite laminated plate has a good noise reduction effect.

This study can provide the theoretical guidance for vibration and noise reduction.

The authors discover a new identity concerning differentiable mappings defined on m-invex set via fractional integrals. By using the obtained identity as an auxiliary result, some fractional integral inequalities for generalized relative semi- m-(r;h1,h2)-preinvex mappings by involving generalized Mittag-Leffler function are presented. It is pointed out that some new special cases can be deduced from main results of the paper. Also these inequalities have some connections with known integral inequalities. At the end, some applications to special means for different positive real numbers are provided as well.