International Journal of Geometric Methods in Modern Physics

We study the Lorentz symmetry breaking of the 3-form gauge theory down to its sub-group. A 3-form gauge theory is studied in such a Lorentz violating background and these symmetry violation effects will affect the aspects of such a gauge theory. Also, we study the gaugeon formalism and FFBRST of 3-form theory in such a background. It is seen that the generating functional gets modified. With this, we obtain a connection between covariant and noncovariant gauges of such a gauge theory. Furthermore, we study the Batalin–Vilkovisky (BV) formulation of such a gauge theory in such a Lorentz violating background.

The aim of this paper is to study special multiply Einstein warped products having an affine connection. Let $M=I{\times }_{f_1}{F}_1\times \cdots {\times }_{f_m}{F}_m$ be a multiply warped product such that $I\subset ℝ$ is an open interval, $dimI=1,$ $f_i:I\to (0,\infty ),$ $f_i\in {{𝒞}}^{\infty }(I),$ $dim{F}_i=k_i\ge 1$ for every $i\in \{1,\dots ,m\},$ $m\ge 1,$ $dimM=\overline{n}=1+{\sum }_{i=1}^m{k}_i$ and $\overline{\nabla }$ an affine connection on $M.$ We compute the warping functions that make $M$ an Einstein space in the following cases:

• ^(a)$\overline{\nabla }$ is a semi-symmetric metric/non-metric connection and all the fibers are Ricci flat.
• ^(b)$\overline{\nabla }$ is a quarter-symmetric metric/non-metric connection and all the fibers are Ricci flat.

Generalized Melvin solutions for rank-$3$ Lie algebras $A_3$, $B_3$ and $C_3$ are considered. Any solution contains metric, three Abelian 2-forms and three scalar fields. It is governed by three moduli functions $H_1(z),H_2(z),H_3(z)$ ($z={\rho }^2$ and $\rho$ is a radial variable), obeying three differential equations with certain boundary conditions imposed. These functions are polynomials with powers $(n_1,n_2,n_3)=(3,4,3),(6,10,6),(5,8,9)$ for Lie algebras $A_3$, $B_3$, $C_3$, respectively. The solutions depend upon integration constants $q_1,q_2,q_3\ne 0$. The power-law asymptotic relations for polynomials at large $z$ are governed by integer-valued $3\times 3$ matrix $\nu$, which coincides with twice the inverse Cartan matrix $2A^{-1}$ for Lie algebras $B_3$ and $C_3$, while in the $A_3$-case $\nu =A^{-1}(I+P)$, where $I$ is the identity matrix and $P$ is a permutation matrix, corresponding to a generator of the ${\mathbb{Z}}_2$-group of symmetry of the Dynkin diagram. The duality identities for polynomials and asymptotic relations for solutions at large distances are obtained. Two-form flux integrals over a two-dimensional disc of radius $R$ and corresponding Wilson loop factors over a circle of radius $R$ are presented.

In this paper, we construct a split octonionic mathematical approach to generalized electromagnetic and gravito-Heavisidian chirality of dyons by modification of the Drude–Born–Fedorov constitutive relations. In this context, we describe dual Euclidean space-times structure associated with $2\times 2$ Zorn’s vector matrix realization of split octonion. As such, using the Zorn’s vector matrix realization, an alternative form of generalized Proca–Maxwell equations of massive dyons is obtained in chiral media. It is well known that in weak unified gravito-Heavisidian field, the Einstein’s equations become Maxwell-like equations under the first approximation. Thus, we study the gravito-Heavisidian analogous theory to electromagnetic theory, and discuss the Drude–Born–Fedorov constitutive relations, gravito-Heavisidian field, Proca–Maxwell equations and gravito-Heavisidian wave equations for linear gravitational chiral field of gravito-dyons in flat split octonion space-time.

The present paper includes the study of symmetry analysis and conservation laws of the time-fractional Calogero–Degasperis–Ibragimov–Shabat (CDIS) equation. The Erdélyi–Kober fractional differential operator has been used here for reduction of time fractional CDIS equation into fractional ordinary differential equation. Also, the new conservation theorem has been used for the analysis of the conservation laws. Furthermore, the new conserved vectors have been constructed for time fractional CDIS equation by means of the new conservation theorem with formal Lagrangian.

In this paper, using the Lie symmetry method, we obtain optimal system, group invariants and exact solutions of $(2+1)$ dimensional Zabolotskaya–Khokhlov equation.

Exploring the geometry of mechanical systems subject to nonholonomic constraints and using various bundle and variational structures intrinsically present in the nonholonomic setting, we study the structure of the equations of motion in a way that aids the analysis and helps to isolate the important geometric objects that govern the motion of such systems. Furthermore, we show that considering different sets of transpositional relations corresponding to different transitivity choices provides different variational structures associated with nonholonomic dynamics, but the derived equations (being referred to as the generalized Hamel–Voronets equations) are equivalent to the Lagrange–d’Alembert equations. To illustrate results of this work and as some applications of the generalized Hamel–Voronets formalisms discussed in this paper, we conclude with considering the balanced Tennessee racer, as well as its modification being referred to as the generalized nonholonomic cart, and an $SE(2)$-snake with three wheeled planar platforms whose snake-like motion is induced by shape variations of the system.

A continuum theory of the mechanical behavior of solid materials is presented wherein fundamental geometric quantities such as the metric tensor and connection coefficients can depend on one or more director vectors, also called internal state vectors. This theory, referred to as generalized pseudo-Finsler geometric continuum mechanics, enables depiction of a very broad class of physical phenomena in deformable solid bodies. The general nonlinear theory is reported first, primarily summarizing prior work by the author. Next, a new application of the theory to torsional deformation of solids is presented, whereby a cylindrical sample of material may be simultaneously subjected to twisting, extension or compression along its axis, as well as possible radial confinement or contraction. The internal state variable, when normalized by a regularization length, is identified with an order parameter associated with inelastic deformation that may include slippage, fracture, and/or structural collapse. Evolution of the internal state follows a generalized Ginzburg–Landau type of kinetic equation. For axially homogeneous fields, a coupled system of nonlinear partial differential equations is obtained that can be integrated numerically. Results are first documented for generic solids representative of many crystals that exist in nature. Then solutions corresponding to realistic properties of crystals of boron carbide ceramic and ice are reported. Results for boron carbide predict a dominant effect of shearing over compression on the structural transformation process, in agreement with observations from atomistic simulations. Results for ice demonstrate periods of steady plastic flow under constant applied average shear stress as well as torsional rigidity varying with sample size. Existence of both phenomena agrees with experimental observations.

In the framework of the fermionic covariant prolongation structure theory (PST), we investigate the integrability of a new $(2+1)$-dimensional super nonlinear evolution equation (NEE). The prolongation structure of the super integrable system is presented. Moreover, we derive the Bäcklund transformation of the super integrable system.

A study is made of the LRS Bianchi type-I cosmological model in $f(R,T)$ modified gravity theory. Einstein’s field equations in $f(R,T)$ gravity are solved by taking $f(R,T)=R+2f(T)$ and the deceleration parameter $(q)$ to be a linear function of the Hubble parameter $(H)$. The universe begins with an initial singular state and changes with time from an early deceleration phase to a late time acceleration phase. We have found that the jerk parameter $(j)$ in the model approaches that of the $\mathrm{\Lambda }CDM$ model at late times. We also discuss the physical and geometrical properties of the model.

LRS Bianchi type I space time is considered to explore bulk viscous string model in modified $f(G)$ gravity. Field equations are solved with some physically viable condition. It is observed that the string phase of the universe disappears. The Gauss–Bonnet invariant significantly effects on galactic fluid. The universe is expanding and isotropic at large time. Some physical parameters are also discussed in detail.

The object of the present paper is to study almost pseudo-$Q$-symmetric manifolds $A{(\mathrm{\text{PQS}})}_n$. Some geometric properties have been studied which recover some known results of pseudo $Q$-symmetric manifolds. We obtain a necessary and sufficient condition for the ${𝒬}$-curvature tensor to be recurrent in $A{(\mathrm{\text{PQS}})}_n$. Also, we provide several interesting results. Among others, we prove that a Ricci symmetric $A{(\mathrm{\text{PQS}})}_n$ is an Einstein manifold under certain condition. Moreover we deal with ${𝒬}$-flat perfect fluid, dust fluid and radiation era perfect fluid spacetimes respectively. As a consequence, we obtain some important results. Finally, we consider $A{(\mathrm{\text{PQS}})}_4$-spacetimes.

Iacopini and Zavattini [Vacuum polarization effects in the ${({\mu }^{-4}\mathrm{\text{He}})}^{+}$ atom and the Born–Infeld electromagnetic theory, Nuovo Cimento B 78 (1983) 38–52] proposed a $(p,\tau )$-two-parameter modification of Born–Infeld electrodynamics, in which the classical self-energy for an electron takes a finite value for $p<1$. In this paper, we want to study a cylindrical capacitor from the viewpoint of Iacopini–Zavattini nonlinear electrodynamics analytically. The capacitance, the electrostatic potential energy, and the potential difference between the plates of a cylindrical capacitor are calculated in the framework of Iacopini–Zavattini electrodynamics for two specific values of $p=\frac{1}{2}$ and $p=\frac{3}{4}$. The study of the behavior of a nonlinear cylindrical capacitor in the weak electric fields shows that our results are compatible with the correspondence principle, i.e. we recover the results of Maxwell electrodynamics in the weak field regime. Finally, the invariance of Iacopini–Zavattini nonlinear electrodynamics under the duality transformation is investigated.

After reviewing Gribov ambiguity of non-Abelian gauge theories, a phenomenon related to the topology of the bundle of gauge connections, we show that there is a similar feature for noncommutative QED over Moyal space, despite the structure group being Abelian, and we exhibit an infinite number of solutions for the equation of Gribov copies. This is a genuine effect of noncommutative geometry which disappears when the noncommutative parameter vanishes.

We prove that if a Sasakian metric is a ∗-Ricci Soliton, then it is either positive Sasakian, or null-Sasakian. Next, we prove that if a complete Sasakian metric is an almost gradient ∗-Ricci Soliton, then it is positive-Sasakian and isometric to a unit sphere $S^{2n+1}$. Finally, we classify nontrivial ∗-Ricci Solitons on non-Sasakian $(\kappa ,\mu )$-contact manifolds.

In this paper, we introduce a mathematical model for the virus medical imaging (VMI). In this method, first, by proposing a mathematical model, we show that there are two types of viruses that each of them produce one type of signal. Some of these signals can be received by males and others by females. Then, we will show that in the VMI technique, viruses can communicate with cells, interior to human’s body via two ways. (1) Viruses can form a wire that passes the skin and reaches to a special cell. (2) Viruses can communicate with viruses interior to body in the wireless form and send some signals for controlling evolutions of cells interior to human’s body.

The paradox of a free falling radiating charged particle in a gravitational field is a well-known fascinating conceptual challenge that involves classical electrodynamics and general relativity (GR). We discuss this paradox considering the emission of radiation as a consequence of an explicit space/time symmetry breaking involving the electric field within the trajectory of the particle seen from an external observer. This occurs in certain particular cases when the relative motion of the charged particle does not follow a geodesic of the motion dictated by the explicit Lagrangian formulation of the problem and thus from the metric of spacetime. The problem is equivalent to the breaking of symmetry within the spatial configuration of a radiating system like an antenna: when the current is not conserved at a certain instant of time within a closed region, then emission of radiation occurs [D. Sinha and G. A. J. Amaratunga, Phys. Rev. Lett. 114(7) (2015) 147701]. Radiation from a system of charges is possible only when there is explicit breaking of symmetry in the electric field in space and time.

In this paper, the general procedure for obtaining the distribution for the superstatistics is presented. Besides, the new type of effective Boltzmann factor with non-vanishing $r$th moments for even $r$ is presented and some examples are discussed.

The purpose of this paper is to study the dynamics of non-interacting and interacting holographic dark energy (HDE) models in the framework of Brans–Dicke (BD) cosmology. As system’s infrared cutoff, we consider the future event horizon. The scalar function of BD theory is assumed to be a logarithmic form of scale factor, which is claimed to avoid a constant result for deceleration parameter. We investigate the cosmological implications of this model in detail. We obtain the time-dependent equation of state parameter and deceleration parameter which describe the phase transition of the Universe. We observe that the model explains the early time inflation and late time acceleration including matter-dominated phase. It is also observed that the equation of state parameter may cross phantom divide line in late time evolution. The cosmic coincidence problem is also discussed for both the models. We observe that this logarithmic form of Brans–Dicke scalar field is more appropriate to achieve a less acute coincidence problem in non-interacting model whereas a soft coincidence can be achieved if coupling parameter in interacting model has small value.