Massless spin 3/2 field, spherical solutions, eliminating of the gauge degrees of freedom

Relativistic system for a vector-bispinior describing a massless spin 3/2 field is studied in the spherical coordinates of Minkowski space. Presentation of the equation with the use of the covariant Levi-Civita tensor exhibits existence of the gauge solutions in the form of the covariant 4-gradient of an arbitrary bispinor. Substitution for 16-component field function is based on the use of Wigner functions, it assumes diagonalization of the operators of energy, square and third projection of the total angular momentum, and space reflection. We derive radial system for eight independent functions. General structure of the spherical gauge solutions is specified, and it is demonstrated that the gauge radial functions satisfy the derived system. It is proved that the general system reduces to two couples of independent 2-nd order and nonhomogeneous differential equations, their particular solutions may be found with the use of the gauge solutions. The corresponding homogeneous equations have one the same form, they have three regular singularities and one irregular of the rank 2. Frobenius types solutions for this equation have been constructed, and the structure of the involved power series with 4-term recurrent relations sre studied. Six remaining radial functions may be straightforwardly found by means of the simple algebraic relations. Thus, we have constructed two types of solutions with opposite parities which do not contain gauge constituents.

Optimal Centers’ Allocation in Smoothing or Interpolating with Radial Basis Functions

Function interpolation and approximation are classical problems of vital importance in many science/engineering areas and communities. In this paper, we propose a powerful methodology for the optimal placement of centers, when approximating or interpolating a curve or surface to a data set, using a base of functions of radial type. In fact, we chose a radial basis function under tension (RBFT), depending on a positive parameter, that also provides a convenient way to control the behavior of the corresponding interpolation or approximation method. We, therefore, propose a new technique, based on multi-objective genetic algorithms, to optimize both the number of centers of the base of radial functions and their optimal placement. To achieve this goal, we use a methodology based on an appropriate modification of a non-dominated genetic classification algorithm (of type NSGA-II). In our approach, the additional goal of maintaining the number of centers as small as possible was also taken into consideration. The good behavior and efficiency of the algorithm presented were tested using different experimental results, at least for functions of one independent variable.

Improving the accuracy of the neuroevolution machine learning potential for multi-component systems

In a previous paper [Fan Z et al. 2021 Phys. Rev. B, 104, 104309], we developed the neuroevolution potential (NEP), a framework of training neural network based machine-learning potentials using a natural evolution strategy and performing molecular dynamics (MD) simulations using the trained potentials. The atom-environment descriptor in NEP was constructed based on a set of radial and angular functions. For multi-component systems, all the radial functions between two atoms are multiplied by some fixed factors that depend on the types of the two atoms only. In this paper, we introduce an improved descriptor for multi-component systems, in which different radial functions are multiplied by different factors that are also optimized during the training process, and show that it can significantly improve the regression accuracy without increasing the computational cost in MD simulations.

Semi-Analytical Model to Predict the Elastic Post-buckling Response of Axially Compressed Cylindrical Shells with Tailored Distributed Stiffness

This study introduces an approximate analytical model to predict the post-buckling response of cylinders with tailored nonuniform distributed stiffness. The shell's wall thickness, and thus its stiffness, is tailored so as to obtain multiple controlled elastic local buckling events when the cylinder is subjected to uniform axial compression. The proposed model treats cylinder segments of different stiffness as individual panels and combines their response by considering them as connected linear or nonlinear springs. The governing equations for the panels are formulated using von Karman's theory and solved by Galerkin's approximate method for a predefined radial deformation. Radial deformation functions are used to improve the model's accuracy and results show that the model's accuracy increases significantly with the number of considered radial functions. The model's predicted axial response for different cylinders are compared to results from experiments on 3D printed samples. Results indicate that this model accurately predicts the order of the buckling events while the buckling forces from the model are higher than those measured experimentally.

Existence of radial solutions for a $p(x)$-Laplacian Dirichlet problem

In this paper, using variational methods, we prove the existence of at least one positive radial solution for the generalized $p(x)$ p ( x ) -Laplacian problem $$ -\Delta _{p(x)} u + R(x) u^{p(x)-2}u=a (x) \vert u \vert ^{q(x)-2} u- b(x) \vert u \vert ^{r(x)-2} u $$ − Δ p ( x ) u + R ( x ) u p ( x ) − 2 u = a ( x ) | u | q ( x ) − 2 u − b ( x ) | u | r ( x ) − 2 u with Dirichlet boundary condition in the unit ball in $\mathbb{R}^{N}$ R N (for $N \geq 3$ N ≥ 3 ), where a, b, R are radial functions.