Equivalence Conditions and Invariants for the General Form of Burgers’ Equations

Equivalence of differential equations is one of the most important concepts in the theory of differential equations. In this paper, the moving coframe method is applied to solve the local equivalence problem for the general form of Burgers’ equation, which has two independent variables under action of a pseudo-group of contact transformations. Using this method, we found the structure equations and invariants of these equations, as a result some conditions for equivalence of them will be given.

ZAKAT AS TAX REDUCTION: STUDY OF MUSLIM COMMUNITY PERCEPTION IN INDONESIA AND MALAYSIA

This study aims to examine the model adopted based on Muslim consumer perceptions of taxes through the zakat system. This research uses three stages of comprehensive technical analysis through demographic depiction of respondents based on distribution frequently, then tests the adopted factors using Exploratory Factor Analysis (EFA) to select and determine the number of factors and related items. In the final stage, data analysis is carried out in the form of the modeling technique using Structure Equations Model (SEM) to test the quality of the models and hypotheses produced. 152 respondents were collected who were sampled in this study, the majority of respondents are 77 Malaysian citizens and 75 Indonesian residents. At the testing stage of the model through the Structural Equation Model (SEM), based on the results of the formation factors in the test, it can be said that only the knowledge about tax, religious, and service variables have an impact on perception toward through zakat system positively and significantly, but through testing the serviceability of a model results in a determinant coefficient of 0.668, which it was relatively strong to explaining independent variable.

Further spherically symmetric solutions

We obtain the interior Schwarzschild solution; the stellar structure equations (Tolman-Oppenheimer-Volkoff); the Reissner-Nordstrom metric (charged black hole) and the de Sitter-Schwarzschild metric. These both illustrate how the field equation is tackled in non-vacuum cases, and bring out some of the physics of stars, electromagnetic fields and the cosmological constant.

The growth of structure

The growth of structure by gravitational collapse from initially small perturbations is described. The Jeans instability is calculated. The structure equations are obtained and solved in various cases (radiation-dominated, matter-dominated and others) via a linearized treatment. Hence the main features of the growth of density perturbations are obtained. The observed spectrum in the present is used to infer the primordial spectrum. The scale-invariant (Harrison-Zol’dovich) spectrum is described. The process of baryon acoustic oscillations is outlined and the sound horizon is defined. The chapter concludes with brief notes on galaxy formatiom.

Exact solutions in higher-dimensional Lovelock and AdS 5 Chern-Simons gravity

Lovelock gravity in D-dimensional space-times is considered adopting Cartan's structure equations. In this context, we find out exact solutions in cosmological and spherically symmetric backgrounds. In the latter case, we also derive horizons and the corresponding Bekenstein-Hawking entropies. Moreover, we focus on the topological Chern-Simons theory, providing exact solutions in 5 dimensions. Specifically, it is possible to show that Anti-de Sitter invariant Chern-Simons gravity can be framed within Lovelock-Zumino gravity in 5 dimensions, for particular choices of Lovelock parameters.

Neutron stars in Palatini $$R+\alpha R^2$$ and $$R+\alpha R^2+\beta Q$$ theories

We study solutions of the stellar structure equations for spherically symmetric objects in modified theories of gravity, where the Einstein-Hilbert Lagrangian is replaced by $$f(R)=R+\alpha R^2$$ f ( R ) = R + α R 2 and $$f(R,Q)=R+\alpha R^2+\beta Q$$ f ( R , Q ) = R + α R 2 + β Q , with R being the Ricci scalar curvature, $$Q=R_{\mu \nu }R^{\mu \nu }$$ Q = R μ ν R μ ν and $$R_{\mu \nu }$$ R μ ν the Ricci tensor. We work in the Palatini formalism, where the metric and the connection are assumed to be independent dynamical variables. We focus on stellar solutions in the mass-radius region associated to neutron stars. We illustrate the potential impact of the $$R^2$$ R 2 and Q terms by studying a range of viable values of $$\alpha $$ α and $$\beta $$ β . Similarly, we use different equations of state (SLy, FPS, HS(DD2) and HS(TMA)) as a simple way to account for the equation of state uncertainty. Our results show that for certain combinations of the $$\alpha $$ α and $$\beta $$ β parameters and equation of state, the effect of modifications of general relativity on the properties of stars is sizeable. Therefore, with increasing accuracy in the determination of the equation of state for neutron stars, astrophysical observations may serve as discriminators of modifications of General Relativity.

Study on an Integral Algorithm of Load Identification Based on Displacement Response

Load identification is a very important and challenging indirect load measurement method because load identification is an inverse problem solution with ill-conditioned characteristics. A new method of load identification is proposed here, in which a virtual function was introduced to establish integral structure equations of motion, and partial integration was applied to reduce the response types in the equations. The effects of loading duration, the type of basis function, and the number of basis function expansion items on the calculation efficiency and the accuracy of load identification were comprehensively taken into account. Numerical simulation and experimental results showed that our algorithm could not only effectively identify periodic and random loads, but there was also a trade-off between the calculation efficiency and identification accuracy. Additionally, our algorithm can improve the ill-conditionedness of the solution of load identification equations, has better robustness to noise, and has high computational efficiency.

Invariant quantities of scalar–tensor theories for stellar structure

We present the relativistic hydrostatic equilibrium equations for a wide class of gravitational theories possessing a scalar–tensor representation. It turns out that the stellar structure equations can be written with respect to the scalar–tensor invariants, allowing to interpret their physical role.