International Journal of Geometric Methods in Modern Physics

We investigate the multipartite entanglement in the phase space using the negativity of Wigner function (NWF) and in the Hilbert space using the negativity of partial transpose (NPT). We analyze comparatively these quantities and the different types of entanglements that are present in two major classes — GHZ and $W$ — made of coherent states. We show that the negativity of Wigner function can be used as a good measure of genuine entanglement in multipartite systems. However, the negativity of partial transpose is a good quantifier for only the bipartite entanglement in tripartite systems.

The cosmological scenario is built up within the framework of scalar field model possessing a noncanonical kinetic term in loop quantum gravity. The noncanonical scalar field is permitted to interact with dark matter field by assuming a specific form of coupling term. The equation of state parameter is set to be constant as well as variable (Chevallier–Polarski–Linder parametrization) and evaluated the behavior of universe through deceleration parameter and weak energy condition. These parameters favor the accelerated expansion of the universe for three values of equation of state parameter in both cases allowed by observational data. The squared speed of sound leads to the stability of the underlying models in both forms of equation of state parameter. Moreover, the validity of generalized second law of thermodynamics is analyzed by using first law of thermodynamics and assume the universe to be enclosed by apparent horizon. The Bekenstein, logarithmic and power-law entropy is being considered as entropy of horizon. The thermodynamic equilibrium condition is also discussed for all three cases of entropies. The generalized second law of thermodynamics and thermal equilibrium condition is satisfied for all the three types of entropies.

We investigate the effect of different metrizations of probability spaces on the information geometric complexity of entropic motion on curved statistical manifolds. Specifically, we provide a comparative analysis based upon Riemannian geometric properties and entropic dynamical features of a Gaussian probability space where the two distinct dissimilarity measures between probability distributions are the Fisher–Rao information metric and the $\alpha$-order entropy metric. In the former case, we observe an asymptotic linear temporal growth of the information geometric entropy (IGE) together with a fast convergence to the final state of the system. In the latter case, instead, we note an asymptotic logarithmic temporal growth of the IGE together with a slow convergence to the final state of the system. Finally, motivated by our findings, we provide some insights on a tradeoff between complexity and speed of convergence to the final state in our information geometric approach to problems of entropic inference.

Microtubules (MTs), the intracellular structures, are made-up of polar polymers that are composed of $\alpha$ and $\beta$ tubulins. The functions of MTs are shape the way for vesicles movement and asexual mitosis division. However, one of the main functions of MTs is stability of cells. Fewer geometrical methods are available in the literature to explore the molecular dynamics (MDs) of a MT, which is a difficult task due to its microscopic size and complex structure. A structural mechanics model with rather similar properties to MT can demonstrate the dynamics of MT. The first and most important step for this process is to obtain the interaction force between tubulins, and a mechanical model can be used to simulate the mechanical and dynamical properties of MTs by using meso- and macro-scale simulations. This work reports the interaction properties of $\beta$–$\alpha$ tubulin in MT. During this research, with the aid of the MD simulations, the interaction energy in $\beta$–$\alpha$ dimer is evaluated. The alpha-beta force–distance diagram is sketched with the aid of force and energy formulae. Thus, the graphical analysis supported the findings of this study.

In this paper, we obtain a new version of Serret-Frenet formulae for quaternionic curve in ${ℝ}^4$ by using a method similar to the method given by Bharathi and Nagaraj and called it Type 2-Quaternionic Frame. Also, we give an application of this new type of the quaternionic frame by an example.

This article presents nonlinear viscous cosmological model of the universe that arises in the modified gravity theory in which the Lagrangian of the gravitational action contains a general function $f(R,T)$, where $R$ and $T$ denote the curvature scalar and the trace of the energy–momentum tensor, respectively, in the flat Friedmann-Robertson-Walker background metric with isotropic fluid. We obtain cosmological solutions in $f(R,T)$ theory of gravity, specially for the choice $f(R,T)=f(R)+f(T)$ with nonlinear viscosity described by nonlinear Israel Stewart theory. The physical and geometrical properties of the models in $f(R,T)={\alpha }_{1}R+{\alpha }_{2}T$ gravity, where ${\alpha }_1$ and ${\alpha }_2$ are coupling parameters, with nonlinear Israel Stewart theory are studied in detail. The analysis of the variation of bulk viscous pressure, energy density, scale factor, Hubble parameter and deceleration parameter with cosmic evolution is done in the nonlinear Israel Stewart theory with $f(R,T)$ gravity. The stability analysis of the equilibrium points of the dynamical system associated with the exponential evolution of the universe in $f(R,T)$ gravity theory with nonlinear dissipative effects is also studied.

The objective of this paper is to study the curvature restricted geometric properties of anisotropic nonrelativistic scale invariant metrics, namely, Lifshitz and Schrödinger spacetime metrics. It is found that the Lifshitz spacetime metric admits two important pseudosymmetric type curvature conditions $C\cdot C=\frac{\mu (\mu -1)}{6}Q(g,C)$ and $R\cdot R-Q(S,R)=-2Q(g,C)$. Also, it is $2$-quasi Einstein and generalized Roter type manifold. Finally, Lifshitz spacetime is compared with Schrödinger spacetime.

In this paper, Lie group analysis method is applied to consider a vector field and symmetry reductions for time-fractional Sharma–Tasso–Olver equation, exact hyperbolic wave solutions, power series solutions and its convergence are investigated. Stability and boundedness analysis of trivial solution to the reduced ordinary differential equation is shown by constructing appropriate Lyapunov function. Conservation laws of the equation are well constructed with a detailed derivation making use of Noether’s operator.

Recent observations of the dark energy density have demonstrated the fine-tuning problem and the challenges faced by theoretical modeling. In this study, we apply the self-similar symmetry (SSS) model, describing the hierarchical structure of the universe based on the Dirac large numbers hypothesis, to Einstein’s cosmological term. We introduce a new similarity dimension, $D_B$, in the SSS model. Using the $D_B$ SSS model, the cosmological constant $\mathrm{\Lambda }$ is simply expressed as a function of the cosmic microwave background (CMB) temperature. The result shows that both the gravitational constant $G$ and $\mathrm{\Lambda }$ are coupled with the CMB temperature, which simplifies the solution of Einstein’s field equations for the variable $\mathrm{\Lambda }$–$G$ model.

Using a proper choice of the dynamical variables, we show that a non-autonomous dynamical system transforming to an autonomous dynamical system with a certain coordinate transformations can be obtained by solving a general nonlinear first-order partial differential equations. We examine some special cases and provide particular physical examples. Our framework constitutes general machineries in investigating other non-autonomous dynamical systems.

In this note, we support that the five-dimensional “ambient cosmology” does not appear to be necessary, neither is required for the scope of avoiding spacetime singularities. This scope can be reached with other, purely topological instruments, without postulating extra dimensions.

Our aim in this paper is to introduce a new kind of functions, Weakly Analytic Functions. The idea is to use this new kind of mathematical tools to describe many physical phenomena and to study the proprieties of some of the most important functions in mathematics.

In this paper, we give a local characterization for the Ricci tensor of an almost Kenmotsu $3$-manifold $(M,\varphi ,\xi ,\eta ,g)$ to be cyclic-parallel. As an application, we prove that if $M$ has cyclic-parallel Ricci tensor and satisfies ${\nabla }_{\xi }h=\delta h+2a\varphi h$, (where $2h$ is the Lie derivative of $\varphi$ along the Reeb flow and both $\delta$ and $a$ are smooth functions such that $a$ is invariant along the contact distribution), then $M$ is locally isometric to either the hyperbolic space ${ℍ}^3(-1)$ or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure.

This paper considers a kind of design of a ruled surface. The design interconnects some concepts from the fields of computer-aided geometric design (CAGD) and kinematics. Dual unit spherical Bézier-like curves on the dual unit sphere (DUS) are obtained by a novel method with respect to the control points. A dual unit spherical Bézier-like curve corresponds to a ruled surface by using Study’s transference principle and closed ruled surfaces are determined via control points and also, integral invariants of these surfaces are investigated. Finally, the results are illustrated by several examples and the motion interpolation was shown as an embodiment of this method.

We consider the restriction of a differential operator (DO) $D$ acting on the sections $C^{\infty }(\xi )$ of a vector bundle $\xi$ with base $B$, in the language of jet bundles. When the base of $\xi$ is restricted to a submanifold $\tilde{U}\subseteq B$, all information about derivatives in directions that are not tangent to $\tilde{U}$ is lost. To restrict $D$ to a DO $D_{\tilde{U}}$ acting on sections $C^{\infty }({\xi }_{\tilde{U}})$ of the restricted bundle ${\xi }_{\tilde{U}}=i^{\ast }\xi$ (with $i:\tilde{U}\hookrightarrow B$ the natural embedding), one must choose an auxiliary DO $M$ and express the derivatives non-tangent to $\tilde{U}$ from the kernel of $M$. This is equivalent to choosing a splitting of certain short exact sequence of jet bundles. A property of $M$ called formal integrability is crucial for restriction’s self-consistency. We give an explicit example illustrating what can go wrong if $M$ is not formally integrable.

As an important application of this methodology, we consider the dimensional reduction of DOs invariant with respect to the action of a connected Lie group $G$. The splitting relation comes from the Lie derivative of the action, which is formally integrable. The reduction of the action of another group is also considered.

An alternative approach introducing a 3-dimensional (3D) Ricci scalar curvature quantum operator given in terms of volume and area as well as new edge length operators is proposed. An example of monochromatic 4-valent node intertwiner state (equilateral tetrahedra) is studied and the scalar curvature measure for a regular tetrahedron shape is constructed. It is shown that all regular tetrahedron states are in the negative scalar curvature regime and for the semi-classical limit the spectrum is very close to the Euclidean regime.

Retraction: The 1/2 correction forms in geometric quantization of the symmetric free rigid body

[Int. J. Geom. Methods Mod. Phys., Vol. 9, No. 7 (2012) 1220011, https://doi.org/10.1142/S0219887812200113]

Ciprian Hedrea and Romeo Negrea

Department of Mathematics, Politehnica University of Timişoara, P-ta Victoriei 2, 300006 Timişoara, Romania

Ioan Zaharie

Department of Physics, Politehnica University of Timişoara, Bvd. V. Parvan 2, Timişoara, Romania

Received: 17 May 2019

Accepted: 21 May 2019

Published: 24 May 2019

The Romanian National Committee of Ethics has detected plagiarism in this article. Its publication status is now retracted with immediate effect from IJGMMP.

Publisher's Note:

As of 8^th June 2020, the case is under dispute and pending for final court decision.

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