A variational approach to bäcklund and infinitesimal bäcklund transformation for Kadomtsev-Petviashvili equation

A new form of the Bäcklund transformation for the Kadomtsev-Petviashvili equation is obtained through the variational formalism. At no stage we use the equation of motion for the deduction of the Bäcklund transformation. So we follow the method of Steudel for the derivation of the infinite number of conservation laws using Noether's theorem in three dimensions and the corresponding infinitesimal Bäcklund transformation.     

Laser-produced copper ion energy spectrum employing Thomson scattering technique

Investigations were performed on laser-assisted ion emission when a copper target was irradiated with a Nd:YAG laser (1.064 μm 10 mJ and 1.1 MW). Charge states present in the ion core emitted from plasma were determined employing Faraday cup arrangement using a four-channel 500 MHz Digital Storage Oscilloscope YOKOGAWA DL 1740. The Thomson parabola scattering (TPS) technique was used to measure the energy of ions emitted from plasma. Singly charged positive ions were found to be in excessive amount. For the energy analysis, the ion beam was collimated using a pinhole arrangement. The collimated beam was then made to pass through a uniform magnetic field of 0.4 T. A solid-state nuclear track detector PM-355 was used to register the ion tracks. The energy of ions, found to be in the range of 9–50 keV. The ion energy spectra d ² N/dEdΩ show a decreasing trend with increasing ion energy. The empirical relation dN/dE ∞ E ^?n fits well with the experimental data at higher values of energy, i.e., greater than 20 keV.     

Warped product CR-submanifolds of cosymplectic manifolds

In the present paper, we study warped product CR-submanifolds of cosymplectic manifolds. It is shown that the warped product of the type ${N_\perp\times{_f}N_T}$ is trivial and obtain a characterization result for the warped product of the type ${N_T\times{_f}N_\perp}$ , where N _T and ${N_\perp}$ are invariant and anti-invariant submanifolds of a cosymplectic manifold ${\bar M}$ , respectively.     

Micellar effects on the alkaline hydrolysis of isatin and its derivatives

Herein we have investigated the hydrolysis of 1H-indol-2,3-dione (isatin, I) and its derivatives of different hydrophobicities, viz. N-dimethylaminomethyl indol-2,3-dione (II), N-morpholinomethyl indol-2,3-dione (III), N-pipridinomethyl indol-2,3-dione (IV), N-heptylaminomethyl indol-2,3-dione (V), N-dodecylaminomethyl indol-2,3-dione (VI), N-hexylanilinomethyl indol-2,3-dione (VII), N-decylanilinomethyl indol-2,3-dione (VIII), and N-hexadecylanilinomethyl indol-2,3-dione (IX), in the presence of an excess amount of sodium hydroxide. All the isatin derivatives were synthesized in the laboratory. The progress of the reactions was studied by exploiting UV-visible spectrophotometry. The observed rate constant, k(w), increases linearly on increasing the hydroxide ion concentration, indicating first-order dependence on [OH(-)]. The effects of surfactants, cationic (cetyltrimethylammonium chloride, CTACl), and anionic (sodium dodecyl sulfate, SDS) were also investigated. The rate of reaction increased on increasing the concentration of CTACl and, after reaching a maximum, it started decreasing. Conversely, anionic micelles of SDS inhibited the rate of hydrolysis of isatin and its derivatives. The results of the effect of CTACl were analyzed using a pseudophase ion-exchange model while the inhibition by SDS was analyzed using a simple Menger-Portnoy model. The effects of added salts, such as NaBr, NaCl, and (CH(3))(4)NBr, were also seen on the isatin hydrolysis. It was found that the addition of salts decreased the rate enhancement efficiency of the CTACl.     

A new biarylphosphine ligand for the Pd-catalyzed synthesis of diaryl ethers under mild conditions

A new bulky biarylphosphine ligand (L8) has been developed that allows the Pd-catalyzed C-O cross-coupling of a wide range of aryl halides and phenols under milder conditions than previously possible. A direct correlation between the size of the ligand substituents in the 2', 4', and 6' positions of the nonphosphine containing ring and the reactivity of the derived catalyst system was observed. Specifically, the rate of coupling increased with the size of these substituents.     

A study of the effect of the loss function on Bayes Estimate,posterior risk and hazard function for Lindley distribution

In this paper, mathematical properties of Lindley distribution via Bayesian approach are derived under different loss functions. These properties include: Bayes Estimators, posterior risks and failure rate function for simulation scheme. Elicitation of hyperparameters is also discussed. A real life application to waiting time data at the bank is also described for the developed procedures (also using WinBUGS). Results are compared on the basis of posterior risk.     

B.-Y. Chen’s Inequality for Bi-warped Products and Its Applications in Kenmotsu manifolds

In this paper, we establish a sharp inequality for the squared norm of the second fundamental form of bi-warped product submanifolds of Kenmotsu manifolds. The equality case is also considered. We also provide a non-trivial example and some applications of derived inequality.     

Heat and mass transfer analysis on the flow of a second grade fluid in the presence of chemical reaction

The laminar flow problem of convective heat transfer for a second grade fluid over a semi-infinite plate in the presence of species concentration and chemical reaction is investigated. The governing equations are transformed into a dimensionless system of three non-linear coupled partial differential equations. These equations have been solved analytically subject to the relevant boundary conditions by employing a homotopy analysis method (HAM). It is noted that for the arising system, the HAM performs extremely well in terms of efficiency and simplicity. The influence of dimensionless pertinent parameters on the velocity, temperature and concentration fields has been examined carefully.