Spinor fields in $f(\mathcal{Q})$-gravity

We present a tetrad--affine approach to $f(\mathcal{Q})$ gravity coupled to spinor fields of spin-$\frac{1}{2}$. After deriving the field equations, we derive the conservation law of the spin density, showing that the latter ensures the vanishing of the antisymmetric part of the Einstein--like equations, just as it happens in theories with torsion and metricity. We then focus on Bianchi type-I cosmological models proposing a general procedure to solve the corresponding field equations and providing analytical solutions in the case of gravitational Lagrangian functions of the kind $f(\mathcal{Q})=\alpha\mathcal{Q}^n$. At late time such solutions are seen to isotropize and, depending on the value of the exponent $n$, they can undergo an accelerated expansion of the spatial scale factors.

Lagrangian Formulation of Lorenz and Chen Systems

This paper presents the formulation of Lagrangian function for Lorenz, Modified Lorenz and Chen systems using Lagrangian functions depending on fractional derivatives of differentiable functions, and the estimation of the conserved quantity associated with the respective systems.

Lagrangian equations of motion. Conservation laws

This chapter addresses the invariance of the Lagrangian equations of motion under the coordinate to transformation, the transformation of the energy and generalised momenta under the coordinate transformation. The integrals of motion for a particle moving in the field with a given symmetry to the Noether’s theorem, the Lagrangian functions, and the Lagrangian equations of motion for the electromechanical system. The authors also discuss the influence of constraints and friction on the motion of a system, the virial theorem and its generalization in the presents of a magnetic field, and an additional integral of motion for a system of three interacting particles.

Lagrangian equations of motion. Conservation laws

This chapter addresses the invariance of the Lagrangian equations of motion under the coordinate to transformation, the transformation of the energy and generalised momenta under the coordinate transformation. The integrals of motion for a particle moving in the field with a given symmetry to the Noether’s theorem, the Lagrangian functions, and the Lagrangian equations of motion for the electromechanical system. The authors also discuss the influence of constraints and friction on the motion of a system, the virial theorem and its generalization in the presents of a magnetic field, and an additional integral of motion for a system of three interacting particles.

A Closer Look at the Minimum-Variance Portfolio Optimization Model

Recently, by imposing the regularization term to objective function or additional norm constraint to portfolio weights, a number of alternative portfolio strategies have been proposed to improve the empirical performance of the minimum-variance portfolio. In this paper, we firstly examine the relation between the weight norm-constrained method and the objective function regularization method in minimum-variance problems by analyzing the Karush–Kuhn–Tucker conditions of their Lagrangian functions. We give the range of parameters for the two models and the corresponding relationship of parameters. Given the range and manner of parameter selection, it will help researchers and practitioners better understand and apply the relevant portfolio models. We apply these models to construct optimal portfolios and test the proposed propositions by employing real market data.