We introduce an inhomogeneous term, $f (t,x)$, into the right-hand side of the usual Burgers equation and examine the resulting equation for those functions which admit at least one Lie point symmetry. For those functions $f (t,x)$ which depend nontrivially on both $t$ and $x$, we find that there is just one symmetry. If $f$ is a function of only $x$, there are three symmetries with the algebra $sl(2,R)$. When $f$ is a function of only $t$ , there are five symmetries with the algebra $sl(2,R)\oplus_{s} 2A_1$. In all the cases, the Burgers equation is reduced to the equation for a linear oscillator with nonconstant coefficient.

Using the transfer matrix method, the effect of temperature on one-dimensional (1D) nanostructure photonic crystal with coupled defects has been investigated. One of the layers of this structure is silver. The complex refractive index of silver is dependent on temperature and wavelength. This structure is tunable with temperature and incident angle. It is found that the number of defect modes is equal to the number of coupled defects in all incident angles for both polarizations. Also by increasing the temperature, due to dissipation, the wavelength of the defect modes increases and the height of the defect modes decreases. The wavelengths of defect modes depend linearly on temperature for both polarizations in all incident angles.

The open shell molecules with even number of electrons have $\pi^2$ or $\pi^{2}_{g}$ ground-state electronic configuration. Several homonuclear diatomic molecules like $\rm{O_2, S_2, B_2}$ have $\pi^{2}_{g}$ ground state in the $D_{\infty h}$ point group and heteronuclear diatomic radicals like PH, NH, SO have $\pi^2$ ground state in the $C_{\infty v}$ point group. We have computed and presented here the rate coefficient of these open shell molecules $\rm{(O_2, S_2, B_2)}$ and radicals (PH, NH,SO) from the results of our previous studies using a well-established $\it {ab-initio}$ formalism: the $R$-matrix method. The rate coefficients for elastic and electron-excited processes are studied over a wide electron temperature range.

We derive the relativistic Hamiltonian of hydrogen atom in dynamical non-commutative spaces (DNCS or $\tau$ -space). Using this Hamiltonian we calculate the energy shift of the ground state as well the $2P_{1/2}$, $2S_{1/2}$levels. In all the cases, the energy shift depends on the dynamical non-commutative parameter $\tau$. Using the accuracy of the energy measurement, we obtain an upper bound for $\tau$. We also study the Lamb shift in DNCS. Both $2P_{1/2}$ and $2S_{1/2}$ levels receive corrections due to dynamical non-commutativity of space which is in contrast with the non-dynamical non-commutative spaces (NDNCS or $\theta$-space) in which the $2S_{1/2}$ level receives no correction.

The conductance of a single molecule transport junction comprising anthracene molecular junction (AMJ) with fullerene as alligator clips was investigated using $\it{ab-initio}$ density functional theory (DFT) in the Landauer–Imry regime of coherent tunnelling transport. In our previous research, we have already calculatedthe electrical transport properties of aromatic molecules with thiol, amine, hydroxyl and selenol end groups concluding the exceptional assistance in the formation of robust molecular junctions. In this article, we have presented the suitability of fullerene anchoring in coupling anthracene molecule with gold electrodes. AMJ with boron-20 (B-20) and C-20 alligator clips exhibited strongest conduction in contrast to nitrogen, oxygen, fluorine and neon alligator clips.

The upcoming 50 kt magnetized iron calorimeter (ICAL) detector at the India-based Neutrino Observatory (INO) is designed to study the atmospheric neutrinos and antineutrinos separately over a wide range of energies andpath lengths. The primary focus of this experiment is to explore the Earth matter effects by observing the energy and zenith angle dependence of the atmospheric neutrinos in the multi-GeV range. This study will be crucial toaddress some of the outstanding issues in neutrino oscillation physics, including the fundamental issue of neutrino mass hierarchy. In this document, we present the physics potential of the detector as obtained from realistic detector simulations.We describe the simulation framework, the neutrino interactions in the detector, and the expected responseof the detector to particles traversing it. The ICAL detector can determine the energy and direction of the muons to a high precision, and in addition, its sensitivity to multi-GeV hadrons increases its physics reach substantially. Itscharge identification capability, and hence its ability to distinguish neutrinos from antineutrinos, makes it an efficient detector for determining the neutrino mass hierarchy. In this report, we outline the analyses carried out for the determination of neutrino mass hierarchy and precision measurements of atmospheric neutrino mixing parameters at ICAL, and give the expected physics reach of the detector with 10 years of runtime. We also explore the potential of ICAL for probing new physics scenarios like CPT violation and the presence of magnetic monopoles.