International Journal of Geometric Methods in Modern Physics

In this short review, we discuss the approach of the commutator algebra of covariant derivative to analyze the gravitational theories, starting from the standard Einstein's general theory of relativity (GTR) and focusing on the Rastall theory. After that, we discuss the important role of the torsion in this mathematical framework.

In the appendix of the paper, we analyze the importance of the nascent gravitational wave (GW) astronomy as a tool to discriminate among the GTR and alternative theories of gravity.

Starting from integral balance laws, a model based on nonlinear ordinary differential equations (ODEs) describing the evolution of Phosphorus cycle in a lake is proposed. After showing that the usual homogeneous model is not compatible with the mixture theory, we prove that an ODEs model still holds but for the mean values of the state variables provided that the nonhomogeneous involved fields satisfy suitable conditions. In this model the trophic state of a lake is described by the mean densities of Phosphorus in water and sediments, and phytoplankton biomass. All the quantities appearing in the model can be experimentally evaluated. To propose restoration programs, the evolution of these state variables toward stable steady state conditions is analyzed. Moreover, the local stability analysis is performed with respect to all the model parameters. Some numerical simulations and a real application to lake Varese conclude the paper.

Recently, some authors have shown that a DNA molecule produces electromagnetic signals and communicates with other DNA molecules or other molecules. In fact, a DNA acts like a receiver or transmitter of radio waves. In this paper, we suggest a mathematical model for the DNA molecule and use of its communication to cure some diseases like cancer. In this model, first, by using concepts from string theory and M-theory, we calculate the energy of a DNA in terms of interactions between free electrons and bound electrons. We show that when a DNA is damaged, its energy changes and an extra current is produced. This extra current causes the electromagnetic signals of a damaged DNA molecule to be different when compared to the electromagnetic signals of a normal DNA molecule. The electromagnetic signals of a damaged DNA molecule induce an extra current in a normal DNA molecule and lead to its destruction. By sending crafted electromagnetic signals to normal DNA molecules and inducing an opposite current with respect to this extra current, we can prevent the destruction of normal DNA. Finally, we argue that the type of packing of DNA in chromosomes of men and women is different. This causes radiated waves from DNAs of men and women to have opposite signs and cancel the effect of each other in a pair. Using this property, we suggest another mechanism to cancel the effect of extra waves, which are produced by DNAs in cancer cells of a male or a female, by extra waves which are produced by DNAs in similar cells of a female or a male and prevent the progression of the disease.

We consider the Dirichlet to Neumann operator for Abelian Yang–Mills boundary conditions. The aim is constructing a complex structure for the symplectic space of boundary conditions of Euler–Lagrange solutions modulo gauge for space-time manifolds with smooth boundary. Thus we prepare a suitable scenario for geometric quantization within the reduced symplectic space of boundary conditions of Abelian gauge fields.

It is well known that the field equation of teleparallel theory which is equivalent to general relativity completely agrees with the field equations of general relativity. However, teleparallel equivalent of general relativity has six redundant degrees of freedom which spoil the true physics. These extra degrees are related to the local Lorentz transformation. In this study, we give three different tetrad fields having cylindrical symmetry and depend only on the radial coordinate. One of these tetrads contains an arbitrary function, which is responsible to reproduce the other solutions, which come from local Lorentz transformation. We show by explicate calculations that this arbitrary function spoils the calculations of the conserved charges. We formulate a skew-symmetric tensor whose vanishing value puts a constraint on this arbitrary function. This constraint fixed the redundant degrees of freedom which characterize the teleparallel equivalent of general relativity.

We show that a Morse-type potential simulates an analytic solution for the highly nonlinear global monopole field equation in three and higher-dimensional flat spacetimes. Owing to the fact that in the flat space limit the similar equation remains intact, we wish to borrow the curved space terminology of global monopole also in flat spacetime. This may provide a compelling example that can be used effectively in different nonlinear theories such as flat space ${\varphi }^4,$ as well as in curved spacetimes.

In this paper, we are going to study some $\beta$-changes of the special class of Finsler metrics which we refer to as the (almost) $\beta$-change and generalized (almost) $\beta$-conformal changes. We investigate Douglas and Weyl tensors under these changes. In particular, we find the necessary and sufficient conditions for the Douglas metrics to be invariant under these type of changes.

Motivated by considerations of Euclidean quantum gravity, we investigate a central question of spectral geometry, namely the question of reconstructability of compact Riemannian manifolds from the spectra of their Laplace operators. To this end, we study analytic paths of metrics that induce isospectral Laplace–Beltrami operators over oriented compact surfaces without boundary. Applying perturbation theory, we show that sets of conformally equivalent metrics on such surfaces contain no nontrivial convex subsets. This indicates that cases where the manifolds cannot be reconstructed from their spectra are highly constrained.

A spatially homogeneous anisotropic LRS Bianchi type-I cosmological model is studied in $f(R,T)$ gravity with a special form of Hubble's parameter, which leads to constant deceleration parameter. The parameters involved in the considered form of Hubble parameter can be tuned to match, our models with the $\mathrm{\Lambda }$CDM model. With the present observed value of the deceleration parameter, we have discussed physical and kinematical properties of a specific model. Moreover, we have discussed the cosmological distances for our model.

A detailed thermodynamical study has been presented for the inhomogeneous FLRW-type model of the Universe bounded by a horizon with three possible modifications of Bekenstein–Hawking formulation of thermodynamical parameters namely entropy and temperature. For the first choice of the thermodynamical system validity of both the first law of thermodynamics (FLT) and the generalized second law of thermodynamics (GSLT) are examined. Also, the integrability conditions for the exact one-forms in both the thermodynamical laws are analyzed and it is found that they are consistent with each other. On the other hand, for the other two choices of the thermodynamical system to hold the first law of thermodynamics, one must restrict the parameters (in the definition of the thermodynamical variables) in some specific integral form.

Given a matrix Lie algebra one can construct the 3-Lie algebra by means of the trace of a matrix. In the present paper, we show that this approach can be extended to the infinite-dimensional Lie algebra of vector fields on a manifold if instead of the trace of a matrix we consider a differential 1-form which satisfies certain conditions. Then we show that the same approach can be extended to matrix Lie superalgebras $𝔤𝔩(m,n)$ if instead of the trace of a matrix we make use of the supertrace of a matrix. It is proved that a graded triple commutator of matrices constructed with the help of the graded commutator and the supertrace satisfies a graded ternary Filippov–Jacobi identity. In two particular cases of $𝔤𝔩(1,2)$ and $𝔤𝔩(2,2)$, we show that the Pauli and Dirac matrices generate the matrix 3-Lie superalgebras, and we find the non-trivial graded triple commutators of these algebras. We propose a Clifford algebra approach to 3-Lie superalgebras induced by Lie superalgebras. We also discuss an application of matrix 3-Lie superalgebras in BRST-formalism.

In this paper, we first introduce the notion of almost quasi-Yamabe solitons and get some interesting formulas for them. Then, we explore conditions under which an almost quasi-Yamabe soliton is trivial and give some characterization results for it. Finally, we give a necessary and sufficient condition under which an arbitrary compact almost Yamabe soliton is necessarily gradiant.

In this paper, we first introduce $N$-Bishop frame for a normal direction curve which is defined as an integral curve of the principal normal of a curve. We express this new frame and its properties. Afterwards, we obtain new spherical images by translating $N$-Bishop frame vectors to the center of unit sphere ${𝕊}^2$ in ${𝔼}^3$. Then, these new spherical images are called $N$-Bishop spherical images. Additionally, we compute the Frénet–Serret equations of these new spherical images. Moreover, we show that integral curves of $N$-Bishop spherical images of slant helices are also slant helices. Finally, we present some illustrated examples.

Recently following the success of the density functional theory (DFT) in obtaining the structure and thermodynamics of homogeneous and inhomogeneous classical systems such as simple fluids, dipolar fluid and binary hard spheres, this theory was also applied to obtain the density profile of a molecular fluid in between hard planar walls by Kalpaxis and Rickayzen. In the theory of molecular fluids, the direct correlation function (DCF) can be used to calculate the equation of state, free energy, phase transition, elastic constants, etc. It is well known that the hard core molecular models play an important role in understanding complex liquids such as liquid crystals. In this paper, a classical fluid of nonspherical molecules is studied. The required homogeneous (DCF) is obtained by solving Orenstein–Zernike (OZ) integral equation numerically. Some of the molecules in the liquid crystals have a sphere shape and this kind of molecular fluid is considered here. The DCF sphere of the molecular fluid is calculated and it will be shown that the results are in good agreement with the pervious works and the results of computer simulation. Finally the electro-optical properties of ellipsoid liquid crystal using DCF of these molecules are calculated.

By defining a noncommutative symplectic structure, we study thermodynamics of Schwarzschild black hole in a Snyder noncommutative phase space for the first time. Since natural cutoffs are the results of compactness of symplectic manifolds in phase space, the physics of black holes in such a space would be affected mainly by these cutoffs. In this respect, this study provides a basis for more deeper understanding of the black hole thermodynamics in a pure mathematical viewpoint.

This work has focused on the violation of uncertainty relation, squeezing effect, photon antibunching and sub-Poissonian statistics for the Arik–Coon $q^{-1}$-oscillator coherent states associated with the noncommutative complex plane ${ℂ}_q$. It is shown that one has to use a generalized definition for the covariance between the operators $X$ and $P$. For $q>1$, Heisenberg's inequality violation with two different behaviors related to the role of the deformation parameter $q$ on the variances of the position and momentum operators is illustrated. We conclude that both weak and strong squeezing effects are exhibited by the $q$-coherent states. In particular, strong squeezing effect is a direct consequent of the violation of Heisenberg's inequality. Moreover, the photon antibunching and sub-Poissonian photon statistics are two features of the $q$-coherent states which are realized simultaneously with the squeezing effects. Clearly, the three later behaviors are different from their corresponding counterparts in the Arik–Coon $q^{-1}$-oscillator coherent states associated with a commutative complex plane.

In this paper, it is proved that the fiber manifold $M_2$ of a warped product manifold $M=M_1{\times }_f{M}_2$ inherits the Einstein-like class type of $M$ whereas the base manifold does under some conditions. Some related results are considered.

Exploiting an M-brane system whose structure and symmetries are inspired by those of graphene (what we call a graphene-brane), we propose here a similitude between two layers of graphene joined by a nanotube and wormholes scenarios in the brane world. By using the symmetries and mathematical properties of the M-brane system, we show here how to possibly increase its conductivity, to the point of making it as a superconductor. The questions of whether and under which condition this might point to the corresponding real graphene structures becoming superconducting are briefly outlined.

It is shown that anti-BRST invariance in quantum gauge theories can be considered as the quantized version of the symmetry of classical gauge theories with respect to different gauge fixing mechanisms.