New numerical methods and some applied aspects of the p-regularity theory

The theory of p-regularity is applied to optimization problems and to singular ordinary differential equations (ODE). The special variant of the method of the modified Lagrangian function proposed by Yu.G. Evtushenko for constrained optimization problems with inequality constraints is justified on the basis of the 2-factor transformation. An implicit function theorem is given for the singular case. This theorem is used to show the existence of solutions to a boundary value problem for a nonlinear differential equation in the resonance case. New numerical methods are proposed including the p-factor method for solving ODEs with a small parameter.     

Solutions of nonlinear differential equations

We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in the algebra of new generalized functions. The solution of such an equation is a new generalized function. In this article we formulate necessary and sufficient conditions for when the solution of the given equation in the algebra of new generalized functions is associated with an ordinary function. Moreover, a class of all possible associated functions is described.     

Kinetic peculiarities of <Emphasis Type="Italic">Quillaja Saponaria Molina</Emphasis> saponin sorption by chitosan

A relationship between adsorption layer structure at different stages of sorbent filling, hydrophilicity, and spatial orientation of saponin in chitosan phase is revealed by joint analysis of the kinetic curves of saponin sorption, IR spectra of chitosan and saponin samples, and computer-simulation data. The sorbent–sorbate complex is formed due to electrostatic interactions between protonated amino groups in chitosan and carboanions of glucuronic acid in saponin, as well as hydrogen bonding between NH₂ and OH groups in chitosan molecules and OH groups in the carbohydrate moiety of saponin molecules.     

Quantum-chemical approach to the justification of the mechanisms of sorption of the triterpenoid saponin by highly basic anion exchanger AB-17-2P in the chloride form

Computer simulation of interaction of the structural fragments of triterpenoid saponins with the anion exchanger AV-17-2P was carried out using the Gaussian 03 software. The results showed that a decrease in the energy of the system that includes the sum of the energies of electrostatic interactions and hydrogen bonding at the absorption by the anion exchanger of the polar part of the saponin molecule equals 18.9 kJ mol^?1 and at the absorption of oleanolic acid, 31.9 kJ mol^?1, that is, the sorption capacity of the aglycone was higher than that of glucuronic acid.     

A short elementary proof of the Lagrange multiplier theorem

We present a short elementary proof of the Lagrange multiplier theorem for equality-constrained optimization. Most proofs in the literature rely on advanced analysis concepts such as the implicit function theorem, whereas elementary proofs tend to be long and involved. By contrast, our proof uses only basic facts from linear algebra, the definition of differentiability, the critical-point condition for unconstrained minima, and the fact that a continuous function attains its minimum over a closed ball.     

New approach to optimality conditions for degenerate nonlinear programming problems

In the work, a new approach to constructing optimality conditions for degenerate smooth optimization problems with inequality constraints is proposed. The approach is based on the theory of p-regularity. A special case of degeneracy, when the first derivatives of some function-constraints are equal to zero up to some order, is considered. Optimality conditions for the general case of degeneracy with p = 2 are presented. Proposed constructions and optimality conditions are illustrated by an example. A general case of degeneracy is considered and optimality conditions for the case of p ≥ 2 are proposed.     

Kinetic and equilibrium characteristics of sorption of saponin of <Emphasis Type="Italic">Quillaja Saponaria Molina</Emphasis> on chitosan

The equilibrium and kinetic curves of the sorption of saponin of Quillaja saponaria molina on chitosan were analyzed. The inner diffusion was found to be limiting, and its coefficients were calculated. It was found that the form of the curves of the sorption isotherms of saponin is determined by the competing processes of association in solution and absorption by chitosan.