Profit maximization problem for Cobb–Douglas and CES production functions

Production functions are used to model the production activity of enterprises. In this article, we formulate the necessary and sufficient conditions of strict concavity for Cobb–Douglas and constant elasticity of substitution (CES) production functions. These conditions constitute the theoretical foundation for analyzing the profit maximization problem. An optimal solution is constructed in analytical form and some of its properties are described. Three approaches to solving the profit maximization problem are considered and their equivalence is established. For a Cobb–Douglas production function we investigate the dependence of the maximum profit on elasticity coefficients. A similar analysis is carried out also for the CES production function. The article presents a systematic and detailed discussion of the relevant topics. The topic is related to the investigation of innovation activity of enterprises. The theoretical results and the explicit analytical relationships provide a theoretical and algorithmic base for the “Planer” optimization software—a useful product for the analysis of the production activity of enterprises modeled by production function tools.     

Approximation by light maps and parametric Lelek maps

The class of metrizable spaces M with the following approximation property is introduced and investigated: M∈AP(n,0) if for every ε>0 and a map g:In→M there exists a 0-dimensional map g^′:In→M which is ε-homotopic to g. It is shown that this class has very nice properties. For example, if Mi∈AP(ni,0), i=1,2, then M₁×M₂∈AP(n₁+n₂,0). Moreover, M∈AP(n,0) if and only if each point of M has a local base of neighborhoods U with U∈AP(n,0). Using the properties of AP(n,0)-spaces, we generalize some results of Levin and Kato-Matsuhashi concerning the existence of residual sets of n-dimensional Lelek maps.     

Simple synthesis of new chiral oxindole derivative

A synthesis of chiral optically active 3-hydroxyoxindoles with 1,3-disubstituted 2,2-dimethyl-cyclobutane structural fragment by the Et₂NH-catalyzed aldol reaction of indoline-2,3-diones with (−)-(1S,3S)-pinonic acid derivatives is described. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 7, pp. 1540–1543, July, 2008.     

Instability of the finite‐difference split‐step method applied to the nonlinear Schrödinger equation. II. moving soliton

We analyze a mechanism and features of a numerical instability (NI) that can be observed in simulations of moving solitons of the nonlinear Schrödinger equation (NLS). This NI is completely different than the one for the standing soliton. We explain how this seeming violation of the Galilean invariance of the NLS is caused by the finite‐difference approximation of the spatial derivative. Our theory extends beyond the von Neumann analysis of numerical methods; in fact, it critically relies on the coefficients in the equation for the numerical error being spatially localized. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1024–1040, 2016     

Instability of the finite‐difference split‐step method applied to the nonlinear Schrödinger equation. I. standing soliton

We consider numerical instability that can be observed in simulations of solitons of the nonlinear Schrödinger equation (NLS) by a split‐step method (SSM) where the linear part of the evolution is solved by a finite‐difference discretization. The von Neumann analysis predicts that this method is unconditionally stable on the background of a constant‐amplitude plane wave. However, simulations show that the method can become unstable on the background of a soliton. We present an analysis explaining this instability. Both this analysis and the features and threshold of the instability are substantially different from those of the Fourier SSM, which computes the linear part of the NLS by a spectral discretization. For example, the modes responsible for the numerical instability are not similar to plane waves, as for the Fourier SSM or, more generally, in the von Neumann analysis. Instead, they are localized at the sides of the soliton. This also makes them different from “physical” (as opposed to numerical) unstable modes of nonlinear waves, which (the modes) are localized around the “core” of a solitary wave. Moreover, the instability threshold for thefinite‐difference split‐step method is considerably relaxed compared with that for the Fourier split‐step. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1002–1023, 2016     

Comparison of strange baryon production at LEP with various models

The strange baryon production rates measured at LEP are compared to several models: isospin, LPHD, QCM, Jetset, Herwig and MOPS. In particular, the parameters of the new MOPS model are adjusted in an attempt to reproduce the spin and strangeness dependence of the observed rates. Received: 8 April 1998 / Revised version: 27 August 1998 / Published online: 19 November 1998     

Separation of variables in anisotropic models and non-skew-symmetric elliptic <Emphasis Type="Italic">r</Emphasis>-matrix

We solve a problem of separation of variables for the classical integrable hamiltonian systems possessing Lax matrices satisfying linear Poisson brackets with the non-skew-symmetric, non-dynamical elliptic \(so(3)\otimes so(3)\)-valued classical r-matrix. Using the corresponding Lax matrices, we present a general form of the “separating functions” B(u) and A(u) that generate the coordinates and the momenta of separation for the associated models. We consider several examples and perform the separation of variables for the classical anisotropic Euler’s top, Steklov–Lyapunov model of the motion of anisotropic rigid body in the liquid, two-spin generalized Gaudin model and “spin” generalization of Steklov–Lyapunov model.     

Classifying homeomorphism groups of infinite graphs

In this paper, we classify topologically the homeomorphism groups H(Γ) of infinite graphs Γ with respect to the compact-open and the Whitney topologies.