We propose to substitute Newton’s constant $G_{N}$ for another constant $G_{2}$, as if the gravitational force would fall off with the $1/r$ law, instead of the $1/r^{2}$; so we describe a system of natural units with $G_{2} , c$ and $\hbar$. We adjust the value of $G_{2}$ so that the fundamental length $L = L_{\text{Pl}}$ is still the Planck’s length and so $G_{N} = L \times G_{2}$. We argue for this system as (1) it would express longitude, time and mass without square roots; (2) $G_{2}$ is in principle disentangled from gravitation, as in (2 + 1) dimensions there is no field outside the sources. So $G_{2}$ would be truly universal; (3) modern physics is not necessarily tied up to $(3 + 1)$-dim. scenarios and (4) extended objects with $p = 2$ (membranes) play an important role both in M-theory and in F-theory, which distinguishes three $(2, 1)$ dimensions.

As an alternative we consider also the clash between gravitation and quantum theory; the suggestion is that non-commutative geometry $[x_{i} , x_{j}] = \Lambda^{2} \theta_{ij}$ would cure some infinities and improve black hole evaporation. Then the new length 𝛬 shall determine, among other things, the gravitational constant $G_{N}$.

In analogy with the definition of resonant or quasi-bound states used in three-dimensional quantal scattering, we define the quasi-bound states that occur in one-dimensional transmission generated by twin symmetric potential barriers and evaluate their energies and widths using two typical examples: (i) twin rectangular barrier and (ii) twin Gaussian-type barrier. The energies at which reflectionless transmission occurs correspond to these states and the widths of the transmission peaks are also the same as those of quasi-bound states. We compare the behaviour of the magnitude of wave functions of quasi-bound states with those for bound states and with the above-barrier state wave function. We deduce a Breit–Wigner-type resonance formula which neatly describes the variation of transmission coefficient as a function of energy at below-barrier energies. Similar formula with additional empirical term explains approximately the peaks of transmission coefficients at above-barrier energies as well. Further, we study the variation of tunnelling time as a function of energy and compare the same with transmission, reflection time and Breit–Wigner delay time around a quasi-bound state energy. We also find that tunnelling time is of the same order of magnitude as lifetime of the quasi-bound state, but somewhat larger.

We discuss the relevance of random matrix theory for pseudo-Hermitian systems, and, for Hamiltonians that break parity 𝑃 and time-reversal invariance 𝑇. In an attempt to understand the random Ising model, we present the treatment of cyclic asymmetric matrices with blocks and show that the nearest-neighbour spacing distributions have the same form as obtained for the matrices with scalar entries. We also summarize the theory for random cyclic matrices with scalar entries. We have also found that for block matrices made of Hermitian and pseudo-Hermitian sub-blocks of the form appearing in Ising model depart from the known results for scalar entries. However, there is still similarity in trends even in log–log plots.

We considered coupled map lattices with long-range interactions to study the spatiotemporal behaviour of spatially extended dynamical systems. Coupled map lattices have been intensively investigated as models to understand many spatiotemporal phenomena observed in extended system, and consequently spatiotemporal chaos. We used the complex order parameter to quantify chaos synchronization for a one-dimensional chain of coupled logistic maps with a coupling strength which varies with the lattice in a power-law fashion. Depending on the range of the interactions, complete chaos synchronization and chaos suppression may be attained. Furthermore, we also calculated the Lyapunov dimension and the transversal distance to the synchronization manifold.

In this paper, we study the synchronization behaviour of two linearly coupled parametrically excited chaotic pendula. The stability of the synchronized state is examined using Lyapunov stability theory and linear matrix inequality (LMI); and some sufficient criteria for global asymptotic synchronization are derived from which an estimated critical coupling is determined. Numerical solutions are presented to verify the theoretical analysis. We also examined the transition to stable synchronous state and show that this corresponds to a boundary crisis of the chaotic attractor.

A $U_{e}(1)$-covariant $R_{\xi}$ gauge for the two-Higgs doublet model based on BRST (Becchi–Rouet–Stora–Tyutin) symmetry is introduced. This gauge allows one to remove a significant number of nonphysical vertices appearing in conventional linear gauges, which greatly simplifies the loop calculations, since the resultant theory satisfies QED-like Ward identities. The presence of four ghost interactions in these types of gauges and their connection with the BRST symmetry are stressed. The Feynman rules for those new vertices that arise in this gauge, as well as for those couplings already present in the linear $R_{\xi}$ gauge but that are modified by this gauge-fixing procedure, are presented.

Flavour equilibration for a thermally equilibrated but chemically non-equilibrated quark-gluon plasma is presented. Flavour equilibration is studied enforcing baryon number conservation. In addition to the usual processes like single additional gluon production $gg \rightleftharpoons ggg$ and its reverse and quark–antiquark pair production by gluon pair fusion $gg \rightleftharpoons q_{i}q_{i}^{-}$ and reverse thereof, processes like quark-flavour interchanging $q_{i}q_{i}^{-} \rightleftharpoons q_{j}q_{j}^{-}$ is also considered. The degree of equilibration is studied comparatively for various reactions/constraints that are being considered.

Taking Coulomb and proximity potentials as the interacting barrier we have studied the cold valley in the radioactive decay of ^244-260Fm isotopes. It is found that in addition to alpha particle minima, other minima occur at S, Ar and Ca clusters. We have computed the half-lives and other characteristics of different clusters emitted from these parents treating parent, daughter and emitted cluster as spheres. Our study reveals that most of these parents are unstable against alpha and heavy cluster (⁴⁶Ar, ^48,50Ca) emissions and stable against light cluster emission, except ⁸Be from ^244-248Fm isotopes. The most probable clusters from these parents are predicted to be ⁴⁶Ar, ^48,50Ca which indicate the role of doubly or near doubly magic clusters ($Z = 20$, $N = 28$) and also stress the role of doubly magic ²⁰⁸Pb daughter. The computed half-lives for alpha decay are in good agreement with the experimental data. It is found that the presence of neutron excess in the parent nuclei slows down the cluster decay process. The effect of quadrupole $(\beta_{2})$ and hexadecapole $(\beta_{4})$ deformations of parent and fragments on half-lives are also studied. It is found that inclusion of $\beta_{2}$ and $\beta_{4}$ reduces the height and shape of the barrier (increases barrier penetrability) and hence the half-life decreases.

A magnetized, low-𝛽 plasma in pure toroidal configuration is formed and extensively studied with ion mass as control parameter. Xenon, krypton and argon plasmas are formed at a fixed toroidal magnetic field of 0.024 T, with a peak density of $\sim 10^{11}$ cm^-3, $\sim 4 \times 10^{10}$ cm^-3 and $\sim 2 \times 10^{10}$ cm⁻³ respectively. The experimental investigation of time-averaged plasma parameter reveals that their profiles remain insensitive to ion mass and suggests that saturated slab equilibrium is obtained. Low-frequency (LF) coherent fluctuations ($\omega$ < $\omega_{\text{ci}}$) are observed and identified as flute modes. Here $\omega_{\text{ci}}$ represents ion cyclotron frequency. Our results indicate that these modes get reduced with ion mass. The frequency of the fluctuating mode decreases with increase in the ion mass. Further, an attempt has been made to discuss the theory of flute modes to understand the relevance of some of our experimental observations.

The EPR spectra for VO²⁺ in CaO–Al₂O₃–SiO₂ system are calculated using complete diagonalization method (CDM) and perturbation theory method (PTM). The calculated results are in good agreement with the observed values. By comparing the calculated results by CDM and PTM in a wide range of crystal field parameters, the validity of the PTM formulas has been studied. It is shown that, the PTM formulas are reasonable and reliable in a wide range of CF parameters.

In the present paper, we have investigated the static properties of the mixed ionic crystal NH₄Cl$_{1−x}$Br$_{x}$ using three-body potential model (TBPM) by the application of Vegard’s law. The results for the mixed crystal counterparts are also in fair agreement with the pseudo-experimental data generated from the application of Vegard’s law. The results for the end point members ($x = 0$ and 1) are in good agreement with the experimental data. The results on compressibility, molecular force constant, infrared absorption frequencies and Debye temperature are presented probably for the first time for these mixed crystal counterparts.

We study the interaction of magnons in dipolar spinor Bose–Einstein condensates in an optical lattice. By means of Holstein–Primakoff and Fourier transformations the energy spectra of the ground and the excited states is obtained analytically. Our results show that the collision of magnons is elastic which is expressed by the conservation of wave numbers in the process of collision. At last, we found that the interaction of magnons is attractive which tends to self-localization to form spin waves, i.e., a cluster of a macroscopic number of coherent magnons. Because of the attraction, the instability of spin wave brings about the existence of solitary wave.

We have studied the stability of the electrostatic ion cyclotron wave in a plasma consisting of isotropic hydrogen ions ($H^{+}$) and temperature-anisotropic positively ($O^{+}$) and negatively ($O^{−}$) charged oxygen ions, with the electrons drifting parallel to the magnetic field. Analytical expressions have been derived for the frequency and growth/damping rate of ion cyclotron waves around the first harmonic of both hydrogen and oxygen ion gyrofrequencies. We find that the frequencies and growth/damping rates are dependent on the densities and temperatures of all species of ions. A detailed numerical study, for parameters relevant to comet Halley, shows that the growth rate is dependent on the magnitude of the frequency. The ion cyclotron waves are driven by the electron drift parallel to the magnetic field; the temperature anisotropy of the oxygen ions only slightly enhance the growth rates for small values of temperature anisotropies. A simple explanation, in terms of wave exponentiation times, is offered for the absence of electrostatic ion cyclotron waves in the multi-ion plasma of comet Halley.