We introduce a search algorithm that utilises differential operator realisations to find polynomial Casimir operators of Lie algebras. To demonstrate the algorithm, we look at two classes of examples: (1) the model filiform Lie algebras and (2) the Schrödinger Lie algebras. We find that an abstract form of dimensional analysis assists us in our algorithm, and greatly reduces the complexity of the problem.

In previous work, we introduced an algorithm that utilises differential operator realisations to find polynomial Casimir operators of Lie algebras. In this article we build on this work by applying the algorithm to several classes of finite dimensional conformal Galilei algebras with central extension. In these cases we highlight the utility of an algebra anti-automorphism, and give relevant details through key examples.

In this paper, a novel soft computing technique is designed to analyze the mathematical model of the steady thin film flow of Johnson–Segalman fluid on the surface of an infinitely long vertical cylinder used in the drainage system by using artificial neural networks (ANNs). The approximate series solutions are constructed by Legendre polynomials and a Legendre polynomial-based artificial neural networks architecture (LNN) to approximate solutions for drainage problems. The training of designed neurons in an LNN structure is carried out by a hybridizing generalized normal distribution optimization (GNDO) algorithm and sequential quadratic programming (SQP). To investigate the capabilities of the proposed LNN-GNDO-SQP algorithm, the effect of variations in various non-Newtonian parameters like Stokes number (St), Weissenberg number (We), slip parameters (a), and the ratio of viscosities (ϕ) on velocity profiles of the of steady thin film flow of non-Newtonian Johnson–Segalman fluid are investigated. The results establish that the velocity profile is directly affected by increasing Stokes and Weissenberg numbers while the ratio of viscosities and slip parameter inversely affects the fluid’s velocity profile. To validate the proposed technique’s efficiency, solutions and absolute errors are compared with reference solutions calculated by RK-4 (ode45) and the Genetic algorithm-Active set algorithm (GA-ASA). To study the stability, efficiency and accuracy of the LNN-GNDO-SQP algorithm, extensive graphical and statistical analyses are conducted based on absolute errors, mean, median, standard deviation, mean absolute deviation, Theil’s inequality coefficient (TIC), and error in Nash Sutcliffe efficiency (ENSE). Statistics of the performance indicators are approaching zero, which dictates the proposed algorithm’s worth and reliability.

In this paper, a new parametric compound G family of continuous probability distributions called the Poisson generalized exponential G (PGEG) family is derived and studied. Relevant mathematical properties are derived. Some new bivariate G families using the theorems of “Farlie-Gumbel-Morgenstern copula”, “the modified Farlie-Gumbel-Morgenstern copula”, “the Clayton copula”, and “the Renyi’s entropy copula” are presented. Many special members are derived, and a special attention is devoted to the exponential and the one parameter Pareto type II model. The maximum likelihood method is used to estimate the model parameters. A graphical simulation is performed to assess the finite sample behavior of the estimators of the maximum likelihood method. Two real-life data applications are proposed to illustrate the importance of the new family.