Unification for Gravity and Electromagnetic Field in Kerr Newman Solution

Solutions of unified theory equations of gravity and electromagnetism has to satisfy Einstein-Maxwellequation. Specially, solution of the unified theory is generally Kerr-Newman solution in vacuum. We finallyfound the revised Einstein gravity tensor equation with new term (2-order contravariant metric tensor two timesproduct and the constant matrix) is right in Kerr-Newman solution.

Unified Theory of Gravity and Electromagnetic Field in Reissner-Nodstrom Solution, Kerr-Newman Solution

Solutions of unified theory equations of gravity and electromagnetism satisfy Einstein-Maxwellequation. Hence, solutions of the unified theory is Reissner-Nordstrom solution in vacuum. We found inrevised Einstein gravity tensor equation, the condition is satisfied by 2-order contravariant metric tensor twotimes product, the constant matrix.

Numerical approximation of the scattering amplitude in elasticity

We propose a numerical method to approximate the scattering amplitudes for the elasticity system with a non-constant matrix potential in dimensions $$d=2$$ d = 2 and 3. This requires to approximate first the scattering field, for some incident waves, which can be written as the solution of a suitable Lippmann-Schwinger equation. In this work we adapt the method introduced by Vainikko (Res Rep A 387:3–18, 1997) to solve such equations when considering the Lamé operator. Convergence is proved for sufficiently smooth potentials. Implementation details and numerical examples are also given.

Sliding Mode Matrix-Projective Synchronization for Fractional-Order Neural Networks

This work generalizes the projection scaling factor to a general constant matrix and proposes the matrix-projection synchronization (MPS) for fractional-order neural networks (FNNs) based on sliding mode control firstly. This kind of scaling factor is far more complex than the constant scaling factor, and it is highly variable and difficult to predict in the process of realizing the synchronization for the driving and response systems, which can ensure high security and strong confidentiality. Then, the fractional-order integral sliding surface and sliding mode controller for FNNs are designed. Furthermore, the criterion for realizing MPS is proved, and the reachability and stability of the synchronization error system are analyzed, so that the global MPS is realized for FNNs. Finally, a numerical application is given to demonstrate the feasibility of theory analysis. MPS is more general, so it is reduced to antisynchronization, complete synchronization, projective synchronization (PS), and modified PS when selecting different projective matrices. This work will enrich the synchronization theory of FNNs and provide a feasible method to study the MPS of other fractional-order dynamical models.

Data Assimilation of Steam Flow Through a Control Valve Using Ensemble Kalman Filter

The present work concentrates on the simulation enhancement of steam flow through a control valve using novel data assimilation (DA) approach. Ensemble Kalman filter (EnKF) is applied to improve the performance of k-? shear stress transport (SST) model by optimizing its turbulence model constants. The selected measurement data at different operating conditions are used as observation, while the rest data are involved for validation. Firstly, four flow patterns, which arise on their respective operating conditions, are identified and analyzed to illustrate the basic characteristics of flow in the control valve. Then DA is performed based on the sample computation by perturbing the model constants and the EnKF process to determine the optimal constant matrix. This optimized constant matrix is subsequently used for the precomputation of the valve flow with significant improvement on the flow rate prediction. The velocity and turbulent kinetic energy fields with default and optimal model constants are also compared to illustrate the effect of DA. The results show that the DA enhanced model constants can significantly reduce the predicted volume flow rate error at all opening ratios presently concerned. With updated model constants, the velocity and turbulent kinetic energy distributions are greatly modified in the valve seat between main valve and control valve.

Construction of Stability Regions in the Parameter Space in a Penning Trap with a Rotating Electric Field

The dynamics of particles in a Penning trap with a rotating dipole electric field and a buffer gas is considered. A transition is made to a coordinate system that rotates together with the electric field, which makes it possible to reduce the system of ordinary differential equations with periodic coefficients to a linear differential system with a constant matrix. Using one of the modifications of the Hurwitz stability criterionthe Lienard-Chipart criterion, the stability analysis (according to Lyapunov) of particle motions in the trap is carried out and the stability regions in the trap parameter space are found.Calculations were carried out for a trap with “typical” main parameters. The biggest degree of stability was obtained at frequencies of rotation of the field close to “resonant”. Small relative deviations from these frequencies led to a significant decrease in the degree of stability and loss of stability at “small” values of the amplitude of the rotating field. At the same time, it was possible to partially compensate this by increasing the amplitude of the rotating field, but only to certain limits, after which stability was again lost.

Resolvent Approximations in L2-Norm for Elliptic Operators Acting in a Perforated Space

We study homogenization of a second-order elliptic differential operator Aε = - div a(x/ε)∇ acting in an ε-periodically perforated space, where ε is a small parameter. Coefficients of the operator Aε are measurable ε-periodic functions. The simplest case where coefficients of the operator are constant is also interesting for us. We find an approximation for the resolvent (Aε + 1)-1 with remainder term of order ε2 as ε → 0 in operator L2-norm on the perforated space. This approximation turns to be the sum of the resolvent (A0 + 1)-1 of the homogenized operator A0 = - div a0 ∇, a0 > 0 being a constant matrix, and some correcting operator εCε. The proof of this result is given by the modified method of the first approximation with the usage of the Steklov smoothing operator.

Robust Parametric Control of Lorenz System via State Feedback

This paper considers the parametric control to the Lorenz system by state feedback. Based on the solutions of the generalized Sylvester matrix equation (GSE), the unified explicit parametric expression of the state feedback gain matrix is proposed. The closed loop of the Lorenz system can be transformed into an arbitrary constant matrix with the desired eigenstructure (eigenvalues and eigenvectors). The freedom provided by the parametric control can be fully used to find a controller to satisfy the robustness criteria. A numerical simulation is developed to illustrate the effectiveness of the proposed approach.