International Journal of Geometric Methods in Modern Physics

In this paper, we have investigated modified $f(R,T)$ cosmological models with observational constraints in Lyra’s geometry. We have studied the models in two cases: in the first one it is taken as constant displacement vector field and in second case it is taken as time-dependent. We have established a relationship among energy parameters ${\mathrm{\Omega }}_m,{\mathrm{\Omega }}_{\sigma },{\mathrm{\Omega }}_{\beta }$, respectively, called as matter, anisotropy and dark energy (DE) density parameters in Bianchi type-I space-time in Lyra’s geometry. We have compared our models with observational data sets with union 2.1 compilation SNe Ia data and $H_0$ data sets and have found best fit values of various energy parameters for $0\le z\le 2.3$. We have found that the model with constant displacement vector field is the best fit to the study of the whole evolution of the model and the gauge function $\beta$ of Lyra geometry behaves like a DE candidate. Also, the various physical and kinematical parameters have been studied in this study.

In this paper, we demonstrate that not only the heat kernel techniques are useful for computation of the parity anomaly, but also the parity anomaly turns out to be a powerful mean in studying the heat kernel. We show that the gravitational parity anomaly on four-dimensional manifolds with boundaries can be calculated using the general structure of the heat kernel coefficient $a_5$ for mixed boundary conditions, keeping all the weights of various geometric invariants as unknown numbers. The symmetry properties of the $\eta$-invariant allow to fix all the relevant unknowns. As a byproduct of this calculation, we get an efficient and independent crosscheck (and confirmation) of the correction of the general structure of $a_5$ for mixed boundary conditions, previously suggested in [I. G. Moss, Anomalies, boundaries and the in–in formalism, J. Phys. A 45 (2012) 374022, https://doi.org/10.1088/1751-8113/45/37/374022].

We show in this paper that many well-known theorems about the geometry of warped product submanifolds of Kaehler manifolds and itself nearly Kaehler manifolds can be generalized to CR-slant warped products in nearly Kaehler manifolds.

In amorphous semiconductors and insulators, the using conductivity formulas are semi-empirical and have no satisfying physical explanations. A conductivity equation has been derived by Debye for the response of ideal materials which is rarely observed in practice, but a general conductivity equation which includes the previous empirical equations via a correct choice of arbitrary parameters and moreover totally theoretical derivation had to be generated. Hence, to determine the motion of electrons in the amorphous environment, we defined the equation of motion including viscous forces as a function of coordinates, their derivatives and time variables. We developed a fractional form of this equation over these three variables and finally obtained the most generalized equation of motion, which counts the overall interactions by a fractional form as a variation of two variable. The improved formula, called the stretched Havriliak–Negami equation, has the same form and behavior as the semi-empirical equation and reducible to the Cole–Cole and Cole–Davidson-type of conductivity.

We consider interpolating sesqui-harmonic Legendre curves in Sasakian space forms. We find the necessary and sufficient conditions for Legendre curves in Sasakian space forms to be interpolating sesqui-harmonic. Finally, we obtain a proper example for an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.

A relevant problem in quantum computing concerns how fast a source state can be driven into a target state according to Schrödinger’s quantum mechanical evolution specified by a suitable driving Hamiltonian. In this paper, we study in detail the computational aspects necessary to calculate the transition probability from a source state to a target state in a continuous time quantum search problem defined by a multiparameter generalized time-independent Hamiltonian. In particular, quantifying the performance of a quantum search in terms of speed (minimum search time) and fidelity (maximum success probability), we consider a variety of special cases that emerge from the generalized Hamiltonian. In the context of optimal quantum search, we find it is possible to outperform, in terms of minimum search time, the well-known Farhi–Gutmann analog quantum search algorithm. In the context of nearly optimal quantum search, instead, we show it is possible to identify sub-optimal search algorithms capable of outperforming optimal search algorithms if only a sufficiently high success probability is sought. Finally, we briefly discuss the relevance of a tradeoff between speed and fidelity with emphasis on issues of both theoretical and practical importance to quantum information processing.

We show that the correspondence among ${\mathrm{\text{AdS}}}_3/{\mathrm{\text{CFT}}}_2$, the 1D Schwarzian Model, Sachdev–Ye–Kitaev model and $2+1\mathrm{\text{D}}$ Topological Quantum Gravity can be extended to the case of ${\mathrm{\text{dS}}}_3/{\mathrm{\text{CFT}}}_2$. The $R$-matrix, related to the gravitational scattering amplitude near the horizon of ${\mathrm{\text{dS}}}_3$ black hole, corresponds (on the side of the holographic projection) to a crossing kernel in the Schwarzian Model. The $R$-matrix is related to the 6j-symbol of SU$(1,1)$. We also find that in the Euclidean ${\mathrm{\text{dS}}}_3$ a new Kac–Moody symmetry of instantons emerges out. We dub these new solutions Kac–Moodions. A one-to-one correspondence of Kac–Moodion levels and SU$(2)$ spin representations is established. Every instanton then corresponds to spin representations deployed in Topological Quantum Gravity. The instantons are directly connected to the Black Hole entropy as punctures on its horizon. This strongly supports the recent proposal, in arXiv:1707.00347, that a Kac–Moody symmetry of gravitational instantons is related to the black hole information processing. We also comment on a further correspondence that can be established between the Schwarzian Model and noncommutative spacetimes in $2+1$D, passing through the equivalence with Topological Quantum Gravity with cosmological constant, in the limit when the latter vanishes.

Dynamical behavior and future singularities of $f(R,T,R_{\mu \nu }{T}^{\mu \nu })$ gravitational theory are investigated. This gravitational model is a more complete form of the $f(R,T)$ gravity which can offer new dynamics for the universe. We investigate this gravitational theory for the case $f=R+\alpha R_{\mu \nu }{T}^{\mu \nu }$ using the method of autonomous dynamical systems and by assuming an interaction between matter and dark energy. The fixed points are identified and the results are consistent with standard cosmology and show that for small $\alpha$, the radiation-dominated era is an unstable fixed point of the theory and the universe will continue its procedure toward matter era which is a saddle point of the theory and allows the evolution to dark energy-dominated universe. Finally, the dark energy-dominated epoch is a stable fixed point and will be the late time attractor for the universe. We also consider future singularities for the two $f=R+\alpha R_{\mu \nu }{T}^{\mu \nu }$ and $f=R+\alpha R{R}_{\mu \nu }{T}^{\mu \nu }$ cases and for $w=0,\frac{1}{3},1$ and $-1$. Our results show that for the case of $f=R+\alpha R_{\mu \nu }{T}^{\mu \nu }$, the future singularities of the universe will happen in the same condition as do for the Einstein–Hilbert FRW universe. However, a new type of singularity is obtained for $f=R+\alpha R{R}_{\mu \nu }{T}^{\mu \nu }$ that is captured by $t\to t_s;$ $a\to a_s;$ $\rho \to \infty ;$ and $\left|p\right|\to 0$.

The objective of this paper is to study contact CR-warped product submanifolds admitting Ricci soliton in the setting of generalized Sasakian space forms with a nearly trans-Sasakian structure. More precisely, we obtain some classifications for these warped product submanifolds by using Ricci curvature and Euler–Lagrange equation

In this paper, the Lie symmetry algebra admitted by the time fractional Black–Scholes equation is obtained by using the Lie group method. The constructed symmetry generators are investigated to construct a family of exact solutions and conservation laws for the studied equation. At the same time, the family of solutions is extended by using the invariant subspace method.

In this work, we have examined the behavior of Bianchi-I (axially symmetric) matter-dominated and the anisotropic Universe with the proposed dark energy, Tsallis holographic dark energy (THDE), with the Hubble horizon as infrared cut-off [Tavayef et al., Tsallis holographic dark energy, Phys. Lett. B 781 (2018) 195–200]. The Universe evolution from matter-dominated epoch to dark energy dominated epoch is described by our proposed THDE model. The EoS parameter in our THDE model explains the evolution of the Universe according to the value of nonextensive or Tsallis parameter $\delta$, phantom era (${\omega }_T\le -1$) or quintom (phantom line crossing) and the quintessence era (${\omega }_T\ge -1$), before reaching to completely dark energy-dominated era in the future. Additionally, we also plan to reconcile the dark energy by the method of reconstructing the evolution of the scalar field potential. For the analysis, we take into account the quintessence field and phantom scalar field for this reconstruction, which at present shows the accelerated expansion.

We recall the notion of Jacobi fields, as it was extended to systems of second-order ordinary differential equations. Two points along a base integral curve are conjugate if there exists a nontrivial Jacobi field along that curve that vanishes on both points. Based on arguments that involve the eigendistributions of the Jacobi endomorphism, we discuss conjugate points for a certain generalization (to the current setting) of locally symmetric spaces. Next, we study conjugate points along relative equilibria of Lagrangian systems with a symmetry Lie group. We end the paper with some examples and applications.

Under study in this paper is a time fractional generalized nonlinear diffusion equation which can be better to express diffusion phenomena than diffusion equation of integer order. Firstly, we apply the symmetry analysis method to find the symmetry of this considered equation. Then some conservation laws can also be constructed through the above obtained symmetry with the help of the Noether’s theorem. Next, we reduce this equation into an ordinary differential equation of fractional order in the symmetry with Erdélyi–Kober fractional differential operator under one-dimensional subalgebras optimal system framework. Finally, some exact solutions contain the invariant solutions have found for this given equation. The results give us a new interpretation of this type diffusion process.

In this study, we investigate the Bianchi type-V cosmological models with quark matter (QM) distribution and domain walls with observational constraints in $F(R,T)$ theory of gravity (Harko et al. in Phys. Rev. D 84 (2011) 024020), for a specific choice of the function $F(R,T)=R+2f(T)$ (where $R$ is Ricci scalar and $T$ is the trace of the energy–momentum tensor). We have used model-independent methods and hyperbolic expansion scale factors to achieve the deterministic solution of the modified Einstein field equations and these are also tested observationally. The behaviors of QM and domain walls give an idea of accelerated expansion of the Universe which agrees with recent observations of Type Ia Supernovae. We also discuss the physical implications of the derived models.

We discuss conformally flat plane wave solutions of Einstein equations depending on the plane wave phase $\xi =\omega \tau -qx$, where $\tau$ is the conformal time. We show that ideal fluid Einstein equations and scalar fields with exponential self-interaction have solutions of the form of traveling fluid plane waves. We consider in more detail the source depending on $\xi$ with $\omega =\left|q\right|$ describing models of a massless scalar field, electromagnetic field and relativistic particles with space-time depending mass density. We obtain explicit conformally flat metrics solving Einstein equations with a source of the energy–momentum moving with a speed of light.

The possibility that the vacuum energy density ${\rho }_{\mathrm{\Lambda }}$ is, indeed, varying in time, has been investigated lately from the running vacuum models perspective. Motivated by such models, in this work, we relate the decaying vacuum energy ${\rho }_{\mathrm{\Lambda }}(t)$ with a scalar field $\varphi$. We derive the equations of motion from such a premise and implement the first-order formalism in order to obtain analytical solutions to the cosmological parameters. We show that these are in agreement with recent Planck observational data. We discuss the physical consequences of having ${\rho }_{\mathrm{\Lambda }}(t)$ related to $\varphi$.