Bianchi Type-I cosmological models containing perfect fluid with time varying 𝐺 and 𝛬 have been presented. The solutions obtained represent an expansion scalar 𝜃 bearing a constant ratio to the anisotropy in the direction of space-like unit vector $\lambda^{i}$. Of the two models obtained, one has negative vacuum energy density, which decays numerically. In this model, we obtain $\Lambda \sim H^{2}$, $\Lambda \sim R_{44}/R$ and $\Lambda \sim T^{-2}$ (𝑇 is the cosmic time) which is in accordance with the main dynamical laws for the decay of 𝛬. The second model reduces to a static solution with repulsive gravity.

One-dimensional Jacobian elliptic quasi-exactly solvable second-order differential equations are obtained by introducing the generalized third master functions. It is shown that the solutions of these differential equations are generating functions for a new set of polynomials in terms of energy with factorization property. The roots of these polynomials are the same as the eigenvalues of the differential equations. Some one-dimensional elliptic quasi-exactly quantum solvable models are obtained from these differential equations.

Using K-causal relation introduced by Sorkin and Woolgar [1], we generalize results of Garcia-Parrado and Senovilla [2,3] on causal maps. We also introduce causality conditions with respect to K-causality which are analogous to those in classical causality theory and prove their inter-relationships. We introduce a new causality condition following the work of Bombelli and Noldus [4] and show that this condition lies in between global hyperbolicity and causal simplicity. This approach is simpler and more general as compared to traditional causal approach [5,6] and it has been used by Penrose et al [7] in giving a new proof of positivity of mass theorem. $C^{0}$-space-time structures arise in many mathematical and physical situations like conical singularities, discontinuous matter distributions, phenomena of topology-change in quantum field theory etc.

We study the deep inelastic process $\nu_{\tau} + N \rightarrow \tau^{-} + X$ (with $N \equiv (n + p)/2$ an isoscalar nucleon), in the context of the two-Higgs doublet model Type II (2HDM(II)). We discuss the contribution to the total cross-section of diagrams, in which a charged Higgs boson is exchanged. We present results which show strong dependence of such contributions on tan 𝛽 and $M_{H^{\pm}}$. We show that for tan $\beta \simeq 150$ and $M_{H^{\pm}} \simeq 300$ GeV, the contribution of the charged Higgs boson exchange diagrams to the cross-section of the charged current inclusive $\nu_{\tau}N$ collision can become important. We find that this contribution for an inclusive dispersion generated through the collision of an ultra-high-energy tau-neutrino with $E_{\nu} \simeq 1021$ eV on a target nucleon can be as large as 40% of the value of the contribution of the $W^{\pm}$ exchange diagrams, provided $M_{H^{\pm}} \simeq 300$ GeV and tan $\beta \simeq 150$. Such enhancement and the induced variation on the mean inelasticity $\langle y \rangle^{CC}$ could lead to sizeable effects in the acceptance of cosmic tau-neutrino detectors at experiments such as HiRes, PAO, and the CRTNT, which are anchored to the ground, and at experiments such as EUSO and OWL, which are proposed to orbit around the Earth. We also compare the contribution to $\sigma_{H^{+}}^{\text{tot}}$ from the different allowed initial quarks and we show that the contribution from the bottom quark dominates by far. This means that the $H^{\pm}$ contribution practically always gives a top quark in the final state. Such a large component of the cross-section having a top quark event in the final state could have recognizable features in the EAS experiments.

Based on the concept of cold valley in cold fission and fusion, we have investigated the cluster decay process in ^248-254Cf isotopes. In addition to alpha particle minima, other deep minima occur for S, Ar and Ca clusters. It is found that inclusion of proximity potential does not change the position of minima but minima become deeper. Taking Coulomb and proximity potential as interacting barrier for post-scission region, we computed half-lives and other characteristics for various clusters from these parents. Our study reveals that these parents are stable against light clusters and unstable against heavy clusters. Computed half-lives for alpha decay agree with experimental values within two orders of magnitude. 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 in cluster radioactivity. Odd A clusters are found to be favorable for emission from odd A parents. Cluster decay model is extended to symmetric region and it is found that symmetric fission is also probable which stresses the role of doubly or near doubly magic ¹³²Sn nuclei. Geiger-Nuttal plots were studied for various clusters and are found to be linear with varying slopes and intercepts.

We have carried out photon attenuation measurements at several energies in the range from 49.38 keV to 57.96 keV around the K-absorption edges of the rare earth elements Sm, Eu, Gd, Tb, Dy and Er using 59.54 keV gamma rays from ²⁴¹Am source after Compton scattering from an aluminium target. Pellets of oxides of the rare earth elements were chosen as mixture absorbers in these investigations. A narrow beam good geometry set-up was used for the attenuation measurements. The scattered gamma rays were detected by an HPGe detector. The results are consistent with theoretical values derived from the XCOM package.

A quadratic model is suggested to describe the fringe shift occurred due to the phase variations of uncladded glass fiber introduced between the two plates of the liquid wedge interferometer. The fringe shift of the phase object is represented in the harmonic term which appears in the denominator of the Airy distribution formula of Fabry-Perot's interferometer. A computer program is written to plot the computed fringe shifts of the described model.

An experiment is conducted using liquid wedge interferometer where the fiber of a nearly quadratic thickness variation is introduced between the two plates of the interferometer. The obtained fringe shift shows a good agreement with the proposed quadratic model. Also, it is compared with the previous theoretical shift based on ray optics of semi-circular shape.

The influence of displacement of the pump source with respect to the crystal center on the thermal behavior of the laser crystal is studied analytically. We consider the pump energy to be deposited into the pump region which has been slightly displaced with respect to the crystal center. An analytical expression for temperature distribution for such off-central diode-pumped laser is investigated. The results are then applied to the Nd :YAG and Nd :YVO4 laser crystals and compared with the conventional diode-pumped lasers. We showed that in this special case, the temperature distribution equation in the off-central pumping convert to the conventional central pump scheme.

We work with a formulation of Noether-symmetry analysis which uses the properties of infinitesimal point transformations in the space-time variables to establish the association between symmetries and conservation laws of a dynamical system. Here symmetries are expressed in the form of generators. We have studied the variational or Noether symmetries of the damped harmonic oscillator representing it by an explicitly time-dependent Lagrangian and found that a five-parameter group of transformations leaves the action integral invariant. Amongst the associated conserved quantities only two are found to be functionally independent. These two conserved quantities determine the solution of the problem and correspond to a two-parameter Abelian subgroup.

We develop two measures to characterize the geometry of patterns exhibited by the state of spiral defect chaos, a weakly turbulent regime of Rayleigh-Bénard convection. These describe the packing of contiguous stripes within the pattern by quantifying their length and nearest-neighbor distributions. The distributions evolve towards unique distribution with increasing Rayleigh number that suggests power-law scaling for the dynamics in the limit of infinite system size. The techniques are generally applicable to patterns that are reducible to a binary representation.

The local, average and electronic structure of the semiconducting materials Si and Ge has been studied using multipole, maximum entropy method (MEM) and pair distribution function (PDF) analyses, using X-ray powder data. The covalent nature of bonding and the interaction between the atoms are clearly revealed by the two-dimensional MEM maps plotted on (1 0 0) and (1 1 0) planes and one-dimensional density along [1 0 0], [1 1 0] and [1 1 1] directions. The mid-bond electron densities between the atoms are 0.554 e/ų and 0.187 e/ų for Si and Ge respectively. In this work, the local structural information has also been obtained by analyzing the atomic pair distribution function. An attempt has been made in the present work to utilize the X-ray powder data sets to refine the structure and electron density distribution using the currently available versatile methods, MEM, multipole analysis and determination of pair distribution function for these two systems.

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Theoretical studies of spin-Hamiltonian (SH) parameters associated with Pr⁴⁺ in Sr₂CeO₄ single crystals have been made by using the complete diagonalizing energy matrix method (CDM) for the $4f^{1}$ electronic configuration. The calculated results are in excellent agreement with the experimental data. The negative signs of the anisotropic $g_{i}$-factors and hyperfine structure constants $A_{i}$ (where $i = ||$ or $\perp$) for the orthorhombic Pr⁴⁺ ion in Sr₂CeO₄ are suggested from the calculations. By comparing the results obtained by the CDM with the experimental data, one finds it is valid to interpret the SH parameters for $4f^{1}$ ions in crystals. The results are discussed.

The heavy fermion system (HFS) is described by the periodic Anderson model (PAM), treating the Coulomb correlation between the 𝑓-electrons in the mean-field Hartree-Fock approximation. Superconductivity is introduced by a BCS-type pairing term among the conduction electrons. Within this approximation the equation for the superconducting gap is derived, which depends on the effective position of the energy level of the 𝑓-electrons relative to the Fermi level. The latter in turn depends on the occupation probability $n^{f}$ of the 𝑓-electrons. The gap equation is solved self-consistently with the equation for $n^{f}$; and their temperature dependences are studied for different positions of the bare 𝑓-electron energy level, with respect to the Fermi level. The dependence of the superconducting gap on the hybridization leads to a re-entrant behaviour with increasing strength. The induced pairing between the 𝑓-electrons and the pairing of mixed conduction and 𝑓-electrons due to hybridization are also determined. The temperature dependence of the hybridization parameter, which characterizes the number of electrons with mixed character and represents the number of heavy electrons is studied. This number is shown to be small. The quasi-particle density of states (DOS) shows the existence of a pseudo-gap due to superconductivity and the signature of a hybridization gap at the Fermi level. For the choice of the model parameters, the DOS shows that the HFS is a metal and undergoes a transition to the gap-less superconducting state.

Speeds of sound and densities of three ternary liquid systems namely, toluene $+ n$-heptane $+ n$-hexane (I), cyclohexane $+ n$-heptane $+ n$-hexane (II) and 𝑛-hexane $+ n$- heptane $+ n$-decane (III) have been measured as a function of the composition at 298.15 K at atmospheric pressure. The experimental isothermal compressibility has been evaluated from measured values of speeds of sound and density. The isothermal compressibility of these mixtures has also been computed theoretically using different models for hard sphere equations of state and Flory's statistical theory. Computed values of isothermal compressibility have been compared with experimental findings. A satisfactory agreement has been observed. The superiority of Flory's statistical theory has been established quite reasonably over hard sphere models.

Microwave band gap structures exhibit certain stop band characteristics based on the periodicity, impedance contrast and effective refractive index contrast. These structures though formed in one-, two- and three-dimensional periodicity, are huge in size. In this paper, microstrip-based microwave band gap structures are formed by removing the substrate material in a periodic manner. This paper also demonstrates that these structures can serve as a non-destructive characterization tool for materials, a duplexor and frequency selective coupler. The paper presents both experimental results and theoretical simulation based on a commercially available finite element methodology for comparison.

K X-ray emission spectra of Ti, V, Cr and Mn generated by photon excitation have been studied with a crystal spectrometer. The measured energy shifts of K$_{\alpha}$ satellite relative to the diagram line are compared with values obtained by electron excitation and with different theoretical estimates. The present experimental values of K$_{\alpha}$L¹/K$_{\alpha}$L⁰ relative intensities are compared with values obtained by electron excitation.

The differential and total cross-sections have been investigated in the formation of H-atom in the 2s excited state of proton-lithium and proton-sodium scattering by using the Coulomb projected Born (CPB) approximation in the energy range from 50 to 10,000 keV. The results thus obtained are compared with the available results and found to be in reasonable agreement.