In this paper we introduce the concept of DIRAC structures on (Hermiti an) modules and vector bundles and deduce some of their properties. Among other things we prove that there is a one to one correspondence between the set of all DIRAC structures on a (Hermitian) module and the group of all automorphisms of the module. This correspondence enables us to represent DIRAC structures on (Hermitian) modules and on vector bundles in a very suitable form and define induced DIRAC structures in a natural way.

The Lie derivation of multivector fields along multivector fields has been introduced by Schouten (see [10, 11]), and studdied for example in [5] and [12]. In the present paper we define the Lie derivation of differential forms along multivector fields, and we extend this concept to covariant derivation on tangent bundles and vector bundles, and find natural relations between them and other familiar concepts. Also in spinor bundles, we introduce a covariant derivation along multivector fields and call it the Clifford covariant derivation of that spinor bundle, which is related to its structure and has a natural relation to its DIRAC OPERATOR.

example of a fully relativistic equation with both negative and positive energy solutions. The effect of the negative energy states was mitigated by maximizing the energy with respect to a relevant parameter while at the same time minimizing it with respect to another parameter in the wave function. The Cornell potential and a power-law scalar and vector potentials were used in our calculations for the quark confinement. Cares were taken to avoid the Klein paradox by the dominance of the scalar component over the vector part.

The Schwarz inequality and Jensen’s one are fundamental in a Hilbert space. Regarding a sesquilinear map B(X, Y ) = Y*X as an OPERATORvalued inner product, we discuss OPERATOR versions for the above inequalities and give simple conditions that the equalities hold.

Distributions such as Bose-Einstein, Maxwell Boltzmann, Lag Normal, and exponential have been used to describe income of the low-and middle sections of society. The Pareto distribution also have been used to describe the income of the wealthy section of society. Due to the complexity of these models, it is difficult to estimate macroeconomic indicators. Also, one of other disadvantage of these models is non-uniformity for all income levels. In this study, the cumulative logistic and Fermi-DIRAC distribution were used for two categories of rural and urban data sets in the Iranian economy between 2009 to 2019. The results show that the proposed model was perfectefor all income levels in rural and urban areas at different times very well and covers all parts of the income distribution. Interpretation of DIRAC Fermi distribution parameters and the correlation and significant relationship of these parameters with macroeconomic indicators show the superiority of DIRAC Fermi distribution compared to logistics distribution.

In this paper, the relativistic DIRAC equation in one dimension is investigated for a particle in an external electromagnetic field, with the property of position-dependent effective mass approximation (PDEM), in the absence of vector potential. By removing the lower spinor component and combining the pair of equations, a Schrö dinger-like equation is obtained for the upper spinor component. Using canonical transformations and introducing two first-order Hermitian and anti-Hermitian differential OPERATORs, a formalism for pseudohermitic Hamiltonians with parity-time reversal symmetry (PT) has been obtained. Comparing the equation derived from pseudo-Hermitian Hamiltonian with the non-relativistic Schrö dinger equation leads to a general formalism for one-dimensional solvable imaginary non-Hermitian potentials with real energy spectra. Also, using this process, the complex potentials of Pö schl-Teller and Scarf II with real energy spectra in DIRAC equation with PDEM approximation and PT symmetry have been investigated and their application has been expressed. For some particular parameters we will see the phenomenon of energy-levels crossing. In fact, it means that energy levels disappear from the spectrum. Also, for the mentioned examples, potential figures are drawn.