Einstein’s field equation is a highly general equation consisting of sixteen equations. However, the equation itself provides limited information about the universe unless it is solved with different boundary conditi...Einstein’s field equation is a highly general equation consisting of sixteen equations. However, the equation itself provides limited information about the universe unless it is solved with different boundary conditions. Multiple solutions have been utilized to predict cosmic scales, and among them, the Friedmann-Lemaître-Robertson-Walker solution that is the back-bone of the development into today standard model of modern cosmology: The Λ-CDM model. However, this is naturally not the only solution to Einstein’s field equation. We will investigate the extremal solutions of the Reissner-Nordström, Kerr, and Kerr-Newman metrics. Interestingly, in their extremal cases, these solutions yield identical predictions for horizons and escape velocity. These solutions can be employed to formulate a new cosmological model that resembles the Friedmann equation. However, a significant distinction arises in the extremal universe solution, which does not necessitate the ad hoc insertion of the cosmological constant;instead, it emerges naturally from the derivation itself. To the best of our knowledge, all other solutions relying on the cosmological constant do so by initially ad hoc inserting it into Einstein’s field equation. This clarification unveils the true nature of the cosmological constant, suggesting that it serves as a correction factor for strong gravitational fields, accurately predicting real-world cosmological phenomena only within the extremal solutions of the discussed metrics, all derived strictly from Einstein’s field equation.展开更多
This paper derives the Lindell formula based on the generalized variational principle.For the complex dielectric constant measurement of a small lossy dielectric rod with Rayleigh-Ritz method, an accurate variational ...This paper derives the Lindell formula based on the generalized variational principle.For the complex dielectric constant measurement of a small lossy dielectric rod with Rayleigh-Ritz method, an accurate variational analysis is given. The concept of complex frequency isintroduced in general, and the stability of the solution is discussed. Comparing with the resultof perturbation method, it is concluded that the deviation of perturbation algorithm should betaken into consideration.展开更多
A lot of methods, such as Jacobian elliptic function analysis, are used to look for the explicit exact solution of Duffing differential equation. The key of the analysis is to construct quotient trigonometric function...A lot of methods, such as Jacobian elliptic function analysis, are used to look for the explicit exact solution of Duffing differential equation. The key of the analysis is to construct quotient trigonometric function, and then nonlinear algebraic equation set theory and method are used for the solution of some kinds of nonlinear Duffing differential equation. In this paper, the exact solution of Duffing equation is obtained by using constant variation method, making use of the formula to solve cubic equations and general solution of the homogeneous equation of Duffing equation with appropriate Constant m and function f(t) .展开更多
In this article, we establish the existence of a sign-changing solution and two sign- constant solutions for nonlinear nonlocal problem involving the BO-ZK operator on bounded domain. Our main tool is constrained mini...In this article, we establish the existence of a sign-changing solution and two sign- constant solutions for nonlinear nonlocal problem involving the BO-ZK operator on bounded domain. Our main tool is constrained minimization on appropriate Nehari manifolds.展开更多
目的探讨和分析拉格朗日(Joseph Louis Lagrange,1736—1813)重新定义一阶偏微分方程完全积分概念的原因和背景。方法历史分析和文献考证。结果拉格朗日从欧拉的完全积分定义出发,在用常数变易法探讨一阶偏微分方程积分的过程中受到启发...目的探讨和分析拉格朗日(Joseph Louis Lagrange,1736—1813)重新定义一阶偏微分方程完全积分概念的原因和背景。方法历史分析和文献考证。结果拉格朗日从欧拉的完全积分定义出发,在用常数变易法探讨一阶偏微分方程积分的过程中受到启发,萌生了关于积分"完全性"的新思想。随后,他把这种新思想运用于常微分方程,成功解释了奇解现象,受此驱动,提出了一阶偏微分方程完全积分的新定义。结论拉格朗日的完全积分新定义是他追求方程一般性解法的体现和产物。展开更多
文摘Einstein’s field equation is a highly general equation consisting of sixteen equations. However, the equation itself provides limited information about the universe unless it is solved with different boundary conditions. Multiple solutions have been utilized to predict cosmic scales, and among them, the Friedmann-Lemaître-Robertson-Walker solution that is the back-bone of the development into today standard model of modern cosmology: The Λ-CDM model. However, this is naturally not the only solution to Einstein’s field equation. We will investigate the extremal solutions of the Reissner-Nordström, Kerr, and Kerr-Newman metrics. Interestingly, in their extremal cases, these solutions yield identical predictions for horizons and escape velocity. These solutions can be employed to formulate a new cosmological model that resembles the Friedmann equation. However, a significant distinction arises in the extremal universe solution, which does not necessitate the ad hoc insertion of the cosmological constant;instead, it emerges naturally from the derivation itself. To the best of our knowledge, all other solutions relying on the cosmological constant do so by initially ad hoc inserting it into Einstein’s field equation. This clarification unveils the true nature of the cosmological constant, suggesting that it serves as a correction factor for strong gravitational fields, accurately predicting real-world cosmological phenomena only within the extremal solutions of the discussed metrics, all derived strictly from Einstein’s field equation.
文摘This paper derives the Lindell formula based on the generalized variational principle.For the complex dielectric constant measurement of a small lossy dielectric rod with Rayleigh-Ritz method, an accurate variational analysis is given. The concept of complex frequency isintroduced in general, and the stability of the solution is discussed. Comparing with the resultof perturbation method, it is concluded that the deviation of perturbation algorithm should betaken into consideration.
文摘A lot of methods, such as Jacobian elliptic function analysis, are used to look for the explicit exact solution of Duffing differential equation. The key of the analysis is to construct quotient trigonometric function, and then nonlinear algebraic equation set theory and method are used for the solution of some kinds of nonlinear Duffing differential equation. In this paper, the exact solution of Duffing equation is obtained by using constant variation method, making use of the formula to solve cubic equations and general solution of the homogeneous equation of Duffing equation with appropriate Constant m and function f(t) .
文摘In this article, we establish the existence of a sign-changing solution and two sign- constant solutions for nonlinear nonlocal problem involving the BO-ZK operator on bounded domain. Our main tool is constrained minimization on appropriate Nehari manifolds.
文摘目的探讨和分析拉格朗日(Joseph Louis Lagrange,1736—1813)重新定义一阶偏微分方程完全积分概念的原因和背景。方法历史分析和文献考证。结果拉格朗日从欧拉的完全积分定义出发,在用常数变易法探讨一阶偏微分方程积分的过程中受到启发,萌生了关于积分"完全性"的新思想。随后,他把这种新思想运用于常微分方程,成功解释了奇解现象,受此驱动,提出了一阶偏微分方程完全积分的新定义。结论拉格朗日的完全积分新定义是他追求方程一般性解法的体现和产物。
