Molecular and crystal structure of 4′-cyanobenzo-15-crown-5

The crystal structure of 4′-cyanobenzo-15-crown-5 is determined by X-ray diffraction. The crystallographic parameters of single crystals are as follows: a = 40.479(6) Å, b = 8.821(2) Å, c = 8.484(2) Å, β = 93.94(2)°, V = 3022.2 ų, Z = 8, d _calcd = 1.27 g/cm³, and space group C2/c. The structure is solved by the direct method and refined by the least-squares procedure to R = 0.068. The conformation of the macrocycle in 4′-cyano-B15C5 is described by the a-a-g ^? -a-a-g-g ^? -a-a-g-a-g-a-a-s sequence of torsion angles.     

Synthesis of new polyfunctional additives to lubricating oils

Possibility of synthesizing new polyfunctional sulfur-, phosphorus-, and zinc-containing compounds with anti-wear and anti-scuff properties, promising as high-efficiency additives to lubricating oils, was studied.     

Catalytic cyclopropanation of fullerene[60] with diazomethane

The selective cyclopropanation of C₆₀-fullerene with diazomethane was performed under the catalysis with Pd(acac)₂, and individual 5,6-open and 6,6-closed cycloadducts were obtained.     

Symmetries,Lagrangian and Conservation Laws for the Maxwell Equations

The evolution equations of Maxwell’s equations has a Lagrangian written in terms of the electric E and magnetic H fields, but admit neither Lorentz nor conformal transformations. The additional equations ∇⋅E=0, ∇⋅H=0 guarantee the Lorentz and conformal invariance, but the resulting system is overdetermined, and hence does not have a Lagrangian. The aim of the present paper is to attain a harmony between these two contradictory properties and provide a correspondence between symmetries and conservation laws using the Lagrangian for the evolutionary part of Maxwell’s equations.     

Extension of Euler’s method to parabolic equations

Euler generalized d’Alembert’s solution to a wide class of linear hyperbolic equations with two independent variables. He introduced in 1769 the quantities that were rediscovered by Laplace in 1773 and became known as the Laplace invariants. The present paper is devoted to an extension of Euler’s method to linear parabolic equations with two independent variables. The new method allows one to derive an explicit formula for the general solution of a wide class of parabolic equations. In particular, the general solution of the Black–Scholes equation is obtained.     

Criteria for fourth-order ordinary differential equations to be linearizable by contact transformations

Solution of linearization problem of fourth-order ordinary differential equations via contact transformations is presented in the paper. We show that all fourth-order ordinary differential equations that are linearizable by contact transformations are contained in the class of equations which is at most quadratic in the third-order derivative. We provide the linearization test and describe the procedure for obtaining the linearizing transformations as well as the linearized equation. Moreover, we obtain the general form of ordinary differential equations of order greater than four linearizable via contact transformations.     

Rotationally symmetric internal gravity waves

Many mathematical models formulated in terms of non-linear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating non-linear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is to provide the group theoretical modeling of internal waves in the ocean. The approach is based on a new concept of conservation laws that is utilized to systematically derive the conservation laws of non-linear equations describing propagation of internal waves in the ocean. It was shown in our previous publication that uni-directional internal wave beams can be obtained as invariant solutions of non-linear equations of motion. The main goal of the present publication is to thoroughly analyze another physically significant exact solution, namely the rotationally symmetric solution and the energy carried by this solution. It is shown that the rotationally symmetric solution and its energy are presented by means of a bounded oscillating function.