International Journal of Geometric Methods in Modern Physics

In this paper, we consider the action of Vect(S¹) by Lie derivative on the spaces of pseudodifferential operators $\mathrm{\Psi }{𝒟}{𝒪}$. We study the $𝔥$-trivial deformations of the standard embedding of the Lie algebra Vect(S¹) of smooth vector fields on the circle, into the Lie algebra of functions on the cotangent bundle $T^{\ast }{S}^1$. We classify the deformations of this action that become trivial once restricted to $𝔥$, where $𝔥=𝔞𝔣𝔣(1)$ or $𝔰𝔩(2)$. Necessary and sufficient conditions for integrability of infinitesimal deformations are given.

In this paper, some properties of the time fractional Boussinesq equation are presented. Group analysis of the time fractional Boussinesq equation with Riemann–Liouville derivative is performed and the corresponding optimal system of subgroups are determined. Next, we apply the obtained optimal systems for constructing reduced fractional ordinary differential equations (FODEs). Finally, we show how to derive exact solutions to time fractional Boussinesq equation via invariant subspace method.

For Hurwitz stable linear differential equation system with constant coefficients, we have proved continuity theorems which show how much change is permissible without disturbing the Hurwitz stability and the ${\kappa }^{\ast }$-Hurwitz stability. The results have been applied to the scalar–linear differential equations with order $k$ and some examples illustrating the efficiency of the theorems have been given.

In this paper, we deal with (non-abelian) extensions of a given hom-Lie superalgebra and find a cohomological obstacle to the existence of extensions of hom-Lie superalgebras. Moreover, the setting of covariant exterior derivatives, super connection, curvature and the Bianchi identity in differential geometry has been studied.

In the context of normal forms, we study a class of slow–fast Hamiltonian systems on general Poisson fiber bundles with symmetry. Our geometric approach is motivated by a link between the deformation theory for Poisson structures on fibered manifolds and the adiabatic perturbation theory. We present some normalization results which are based on the averaging theorem for horizontal 2-cocycles on Poisson fiber bundles.

Using Hodge diagram combinatorial data, we study qubit and fermionic Fock spaces from the point of view of type II superstring black holes based on complex compactifications. Concretely, we establish a one-to-one correspondence between qubits, fermionic spaces and extremal black holes in maximally supersymmetric supergravity obtained from type II superstring on complex toroidal and Calabi–Yau compactifications. We interpret the differential forms of the $n$-dimensional complex toroidal compactification as states of $n$-qubits encoding information on extremal black hole charges. We show that there are $2^n$ copies of $n$ qubit systems which can be split as $2^n=2^{n-1}+2^{n-1}$. More precisely, $2^{n-1}$ copies are associated with even $D$-brane charges in type IIA superstring and the other $2^{n-1}$ ones correspond to odd $D$-brane charges in IIB superstring. This correspondence is generalized to a class of Calabi–Yau manifolds. In connection with black hole charges in type IIA superstring, an $n$-qubit system has been obtained from a canonical line bundle of $n$ factors of one-dimensional projective space ${ℂ\mathbb{P}}^1.$

We discuss the warm inflation in the presence of shaft potential $\left(V(\varphi )=\frac{M_p^4{\varphi }^{2n-2}}{{({\varphi }^n+g^n)}^{2-\frac{2}{n}}}\right)$, tachyon scalar field and the generalized form of dissipative coefficient $\mathrm{\Gamma }=a_0\frac{T^q}{{\varphi }^{q-1}}$. In this respect, we investigate the inflationary parameters (slow-roll parameters, number of e-folds, scalar-tensor power spectra, spectral indices, tensor-to-scalar ratio and running of scalar spectral index) in both strong and weak dissipative regimes. It is interesting to mention that our inflationary parametric results (tensor-scalar ratio, spectral index and running of spectral) are consistent with the recent observational data such as BICEP$2$, WMAP$9$ and latest Planck data.

In this paper, we analyze the nonlinear dynamics of the modified Chua circuit system from the viewpoint of Kosambi–Cartan–Chern (KCC) theory. We reformulate the modified Chua circuit system as a set of two second-order nonlinear differential equations and obtain five KCC-invariants which express the intrinsic geometric properties. The deviation tensor and its eigenvalues are obtained, that determine the stability of the system. We also obtain the condition for Jacobi stability and discuss the behavior of deviation vector near equilibrium points.

An algebraic background in order to study the integrability properties of pseudo-null curve motions in a three-dimensional Lorentzian space form is developed. As an application, we delve into the relationship between the Burgers’ equation and the pseudo-null vortex filament equation. A recursion operator for the pseudo-null vortex filament equation is also provided.

The notion of singular generalized Finsler spacetime and singular generalized Berwald spacetime is introduced and their relevance for the description of classical gravity is discussed. A method to construct examples of such generalized Berwald spacetimes is sketched. The method is applied at two different levels of generality. First, a class of flat, singular generalized Berwald spacetimes is obtained. Then in an attempt of further generalization, a class of non-flat generalized Berwald spacetimes is presented and the associated Einstein field equations are discussed. In this context, an argument in favor of a small value of the cosmological constant is given. The physical significance of the models is briefly discussed in the last section.

The quantization scheme based on reduction of the physical states I. Y. Park, Hypersurface foliation approach to renormalization of ADM formulation of gravity, Eur. Phys. J. C 75(9) (2015) 459, arXiv:1404.5066 [hep-th] is extended to two gravity-matter systems and pure de Sitter (dS) gravity. For the gravity-matter systems, we focus on quantization in a flat background for simplicity, and renormalizability is established through gauge-fixing of matter degrees of freedom. Quantization of pure dS gravity has several new novel features. It is noted that the infrared divergence does not arise in the present scheme of quantization. The lapse function constraint plays a crucial role.

In this paper, $k$-inflation is analyzed in warm braneworld scenario. A general class of $k$-essence models with power-law kinetic term is investigated and weak and strong dissipation regimes are studied. Scalar perturbations and spectral index are derived. The results are discussed and applied to specific examples.

Every Walker $4$-manifold $M$, endowed with a canonical neutral metric $g$, admits a specific almost complex structure called proper. In this paper, we find the conditions under which a proper almost complex structure is a harmonic section or a harmonic map from $(M,g)$ to its hyperbolic twistor space.

An emergence of cosmic space has been suggested by Padmanabhan [Emergence and expansion of cosmic space as due to the quest for holographic equipartition, arXiv:hep-th/1206.4916] where he proposed that the expansion of the universe originates from a difference between the number of degrees of freedom on a holographic surface and the one in the emerged bulk. Now, a natural question that arises is how this proposal would explain the production of fermions and an emergence of the Pauli exclusion principle during the evolution of the universe? We try to address this issue in a system of $M0$-branes. In this model, there is a high symmetry and the system is composed of $M0$-branes to which only scalar fields are attached that represent scalar modes of the graviton. Then, when $M0$-branes join each other and hence form $M1$-branes, this symmetry is broken and gauge fields are formed. Therefore, these $M1$-branes interact with the anti-$M1$-branes and the force between them leads to a break of a symmetry such as the lower and upper parts of these branes are not the same. In these conditions, gauge fields which are localized on $M1$-branes and scalars which are attached to them symmetrically, decay to fermions with upper and lower spins which attach to the upper and lower parts of the $M1$-branes anti-symmetrically. The curvature produced by the coupling of identical spins has the opposite sign of the curvature produced by non-identical spins which lead to an attractive force between anti-parallel spins and a repelling force between parallel spins and hence an emergence of the Pauli exclusion principle. By approaching $M1$-branes to each other, the difference between curvatures of parallel spins and curvatures of anti-parallel spins increases, which leads to an inequality between the number of degrees of freedom on the surface and the one in the emerged bulk and hence lead to an occurrence of the cosmic expansion. By approaching $M1$-branes to each other, the square of the energy of the system becomes negative and hence tachyonic states arise. To remove these states, $M1$-branes compactify, the sign of gravity changes and anti-gravity emerges which leads to the branes moving away from each other. By joining $M1$-branes, $M3$-branes are produced which are similar to an initial system that oscillates between compacting and opening branches. Our universe is placed on one of these $M3$-branes and by changing the difference between the amount of couplings between identical and non-identical spins, it contracts or expands.

The object of the present paper is to study mixed quasi-Einstein manifolds. Some geometric properties of mixed quasi-Einstein manifolds have been studied. We also discuss $M{(\mathrm{\text{QE}})}_4$ spacetime with space-matter tensor and some properties related to it. Finally, we construct an example of a mixed quasi-Einstein spacetime.

The objective of this work enclosed with the study of spatially homogeneous anisotropic Bianchi type-I universe in $f(R,T)$ gravity (where $R$ is the Ricci scalar and $T$ is the trace of stress energy momentum tensor) in two different cases viz. $f(R,T)=R+2f(T)$ and $f(R,T)=f_1(R)+f_2(T)$ with bulk viscosity matter content. In this study, we consider a time varying deceleration parameter (DP), which generates an accelerating universe to obtain the exact solution of the field equations. The physical and kinematical properties of both the models are discussed in detail for the future evolution of the universe. We have explored the nature of WEC, DEC, SEC and energy density for both the cases. We have found that both the models, with bulk viscosity matter component, show an acceleration of the universe. We have also shown that the cosmic jerk parameter is compatible with the three kinematical data sets.

Inspired by the Hilbert–Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly nontrivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large $N$) region. The ordinates ${\lambda }_n$ are the positive imaginary parts of the nontrivial zeta zeroes in the critical line :$s_n=\frac{1}{2}+i{\lambda }_n$. The latter results are consistent with the validity of the Bohr–Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly nontrivial functional form of the potential is found via the Bohr–Sommerfeld quantization formula using the full-fledged Riemann–von Mangoldt counting formula (without any truncations) for the number $N(E)$ of zeroes in the critical strip with imaginary part greater than $0$ and less than or equal to $E$.