A class of degenerate stochastic differential equations with non-Lipschitz coefficients

We obtain sufficient condition for SDEs to evolve in the positive orthant. We use arguments based on comparison theorems for SDEs to achieve this. As an application we prove the existence of a unique strong solution for a class of multidimensional degenerate SDEs with non-Lipschitz diffusion coefficients.     

Superconvergence of a Collocation-type Method for Hummerstein Equations

This paper considers the numerical solution of Hammerstein equationsof the form by a collocation method applied not to this equation, but ratherto an equivalent equation for z(t) :=g(t, y(t)). The desiredapproximation to y is then obtained by use of the (exact) equation In an earlier paper, questions of existence and optimal convergenceof the respective approximations to z and y were examined. Inthis sequel, collocation approximations to z are sought in certainpiecewise polynomial function spaces, and analogous of knownsuperconvergence results for the iterated collocation solutionof (linear) second-kind Fredhoim integral equations are statedand proved for the approximation to y.     

The cohomology of orbit spaces of certain free circle group actions

Suppose that \(G =\mathbb{S}^1\) acts freely on a finitistic space X whose (mod p) cohomology ring is isomorphic to that of a lens space \(L^{2m-1}(p;q_1,\ldots,q_m)\) or \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\). The mod p index of the action is defined to be the largest integer n such that α ⁿ ?≠?0, where \(\alpha \,\epsilon\, H^2(X/G;\mathbb{Z}_p)\) is the nonzero characteristic class of the \(\mathbb{S}^1\)-bundle \(\mathbb{S}^1\hookrightarrow X\rightarrow X/G\). We show that the mod p index of a free action of G on \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\) is p???1, when it is defined. Using this, we obtain a Borsuk–Ulam type theorem for a free G-action on \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\). It is note worthy that the mod p index for free G-actions on the cohomology lens space is not defined.     

Semigroups of transcendental entire functions and their dynamics

We investigate the dynamics of semigroups of transcendental entire functions using Fatou–Julia theory. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental semigroups. We provide some condition for connectivity of the Julia set of the transcendental semigroups. We also study finitely generated transcendental semigroups, abelian transcendental semigroups and limit functions of transcendental semigroups on its invariant Fatou components.     

On the elliptic flow for nearly symmetric collisions and nuclear equation of state

We present the results of elliptic flow for the collision of nearly symmetric nuclei (₁₀Ne²⁰+ ₁₃Al²⁷_{10}{\rm Ne}^{20}+\,_{13}{\rm Al}^{27}, ₁₈Ar⁴⁰+ ₂₁Sc⁴⁵_{18}{\rm Ar}^{40}+\,_{21}{\rm Sc}^{45}, ₃₀Zn⁶⁴+ ₂₈Ni⁵⁸_{30}{\rm Zn}^{64}+\,_{28}{\rm Ni}^{58}, ₃₆Kr⁸⁶+ ₄₁Nb⁹³_{36}{\rm Kr}^{86}+\,_{41}{\rm Nb}^{93}) using the quantum molecular dynamics (QMD) model. General features of elliptic flow are investigated with the help of theoretical simulations. The simulations are performed at beam energies between 45 and 105 MeV /nucleon. A significant change can be seen from in-plane to out-of-plane elliptic flow of different fragments with incident energy. A comparison with experimental data is also made. Further, we show that elliptic flow for different fragments follows power-law dependence as given by the function C(A_tot)^tC{(A_{\rm tot})^\tau}.     

One Pot Reduction of Imines Generated in-situ from Aldehydes and Amines by the NaBH4-InCl3 System

A combination of sodium borohydride and a catalytic amount of indium（Ⅲ） chloride in acetonitrile reduces imines formed in-situ from aldehydes and amines to the corresponding functionalised secondary and tertiary amines in moderate to good yields. Noteworthy is that highly chemoselective reactions were achieved in the presence of other functional groups such as halogens, carbon-carbon double bonds and hydroxyl groups.     

Complex dynamical invariants for two-dimensional complex potentials

Complex dynamical invariants are searched out for two-dimensional complex potentials using rationalization method within the framework of an extended complex phase space characterized by x?=?x ₁?+?i p ₃,?y?=?x ₂?+?i p ₄, ?p _x ?=?p ₁?+?i x ₃, ?p _y ?=?p ₂?+?i x ₄. It is found that the cubic oscillator and shifted harmonic oscillator admit quadratic complex invariants. The obtained invariants may be useful for studying non-Hermitian Hamiltonian systems.     

A study of nanosized zinc oxide and its nanofluid

The synthesis and characterization of nanosized zinc oxide and its nanofluid in a polyvinyl alcohol (PVA) matrix have been done in the present investigation. Crystalline zinc oxide nanoparticles are synthesized using single-step chemical method while the nanofluids are prepared by the dispersion of nanoparticles in PVA solution using an ultrasonicator. The prepared nanoparticles are characterized using X-ray diffraction, SEM?CEDX and UV?Cvisible spectrum. The particle size distribution measurement is carried out by acoustic particle sizer. The ultrasonic velocities are measured in the synthesized nanofluid under different physical conditions using an ultrasonic interferometer. It is found that the degree of crystallinity of nanoparticles depends on the evaporation rate during its synthesis and ultrasonic velocity has non-linear relation with temperature for the present nanofluid.     

Coupled Higgs field equation and Hamiltonian amplitude equation: Lie classical approach and (G′/G)-expansion method

In this paper, coupled Higgs field equation and Hamiltonian amplitude equation are studied using the Lie classical method. Symmetry reductions and exact solutions are reported for Higgs equation and Hamiltonian amplitude equation. We also establish the travelling wave solutions involving parameters of the coupled Higgs equation and Hamiltonian amplitude equation using (G??/G)-expansion method, where G?=?G(??) satisfies a second-order linear ordinary differential equation (ODE). The travelling wave solutions expressed by hyperbolic, trigonometric and the rational functions are obtained.