Symmetry Methods and Conservation Laws for the Nonlinear Generalized 2D Equal-Width Partial Differential Equation of Engineering

In this work, we study the generalized 2D equal-width equation which arises in various fields of science. With the aid of numerous methods which includes Lie symmetry analysis, power series expansion and Weierstrass method, we produce closed-form solutions of this model. The exact solutions obtained are the snoidal wave, cnoidal wave, Weierstrass elliptic function, Jacobi elliptic cosine function, solitary wave and exponential function solutions. Moreover, we give a graphical representation of the obtained solutions using certain parametric values. Furthermore, the conserved vectors of the underlying equation are constructed by utilizing two approaches: the multiplier method and Noether’s theorem. The multiplier method provided us with four local conservation laws, whereas Noether’s theorem yielded five nonlocal conservation laws. The conservation laws that are constructed contain the conservation of energy and momentum.

Thermodynamic Foundation of Generalized Variational Principle

One open question remains regarding the theory of the generalized variational principle, that is, why the stress-strain relation still be derived from the generalized variational principle while the Lagrangian multiplier method is applied in vain? This study shows that the generalized variational principle can only be understood and implemented correctly within the framework of thermodynamics. As long as the functional has one of the combination $A(\epsilon_{ij})-\sigma_{ij}\epsilon_{ij}$ or $B(\sigma_{ij})-\sigma_{ij}\epsilon_{ij}$, its corresponding variational principle will produce the stress-strain relation without the need to introduce extra constraints by the Lagrangian multiplier method. It is proved herein that the Hu-Washizu functional $\Pi_{HW}[u_i,\epsilon_{ij},\sigma_{ij}]$ and Hu-Washizu variational principle comprise a real three-field functional.

On the Distribution of Estates and the Distribution of Wealth: Evidence from the Dead

Detailed information about the distribution of estates left at death has commonly served as the basis for the estimation of wealth distributions among the living via the mortality multiplier method. The application of detailed mortality rates by demographics and other determinants of mortality is crucial for obtaining an unbiased representation of the wealth distribution of the living. Yet, in this paper we suggest that a simplified mortality multiplier method, derived using average mortality rates and aggregate tabulations by estate size, may be sufficient to derive compelling estimates of wealth concentration. We show that the application of homogeneous multipliers leads to estimates that are close in level and trend to the concentration of wealth derived in the existing literature with the detailed mortality multiplier method for a variety of countries. The use of mortality rates graduated by estate size does not confute this finding. We also derive the general formal conditions for the similarity between the distributions of wealth of the living and estates at death and discuss the main caveats. These findings may unlock a wide array of aggregate estate tabulations, previously thought to be unusable, for estimating historical trends of wealth concentration. (Stone Center on Socio-Economic Inequality Working Paper Series)

Methods for Estimating Population Size for Key Populations: A Global Scoping Review for HIV Research

Background: Estimating the population sizes of key populations is critical for understanding the overall HIV burden. This scoping review aims to synthesize existing methods for population size estimation among key populations (people who inject drugs, men who have sex with men (MSM), transgender persons, sex workers, and incarcerated individuals), and provide recommendations for future application of the existing methods.Main text: A scoping review was conducted and 39 of 688 studies met the inclusion criteria and were assessed. Estimation methods included five digital methods, one in-person method, and four hybrid methods. We summarized and organized the methods for population size estimation into the following five categories: methods based on independent samples (including capture-recapture method and multiplier method), methods based on population counting (including Delphi method and mapping method), methods based on the official report (including workbook method), methods based on social network (including respondent-driven sampling method and network scale-up method) and methods based on data-driven technologies (Bayesian estimation method, Stochastic simulation method, and LMS estimation method). 36 (92%) articles were published after 2010 and 23 (59%) used multiple methods. These include 11 in high-income countries and 28 in low-income countries. A total of 10 estimates the size of sex workers, 14 focused on MSM, and 10 focused on PWID. Conclusion: There was no gold standard for population size estimation. Among 120 studies that were related to population size estimation of key populations, the most commonly used population estimation method is the multiplier method (26/120 studies). Every method has its strengths. For example, some traditional methods are simple and easy to use for researchers. Some novel methods are time- and resources- saving. However, each method has its limitations and bias. For example, for the respondent-driven sampling method, stigma and discrimination may lead to the "hiddenness" of the key population; for the multiplier method, the quality of authentic data may also influence the accuracy of the estimation. In recent years, novel methods based on data-driven technologies such as Bayesian estimation have been developed and applied in many surveys.