Rational Approximations of Riemann--Liouville and Weyl Fractional Integrals

We obtain exact rational approximation orders for functions expressible as Riemann--Liouville and Weyl fractional integrals. New results and the strengthening and generalization of theorems due to Popov, Petrusheva, Pekarskii, Rusak, and the author, which are well known in the theory of rational approximation of differentiable functions, are obtained as consequences of theorems due to Pekarskii related to rational approximation of functions from the Hardy--Sobolev classes in the unit disk.     

Existence of Continuous Functions with a Given Order of Decrease of Least Deviations from Rational Approximations

Mathematical Notes - For a given strictly decreasing sequence {a n } =0 ∞ of real numbers convergent to zero, we construct a continuous function g on the closed interval [?1,1] such...     

Nonlocal fluid density distribution in the vicinity of the critical point

The differential equation for the component local density of a binary fluid was obtained as a result of an isoperimetric problem of the system free energy minimization under condition of particle number stability. The obtained expression can be used in the calculation of liquid's profiles as well as in the investigation of thermodynamic properties in the vicinity of the critical point. An isoperimetric problem solution for the model binary fluid of the system free energy minimization under condition of particle number stability was obtained in different cases: fluid within flat parallel layer under a gravitational field, under a gravitational field and wall potential as well as under a wall potential for another equation of state.     

Comparison of the rates of rational and polynomial approximations of differentiable functions

Translated from Matematicheskie Zametki, Vol. 44, No. 4, pp. 528–535, October, 1988.     

Small-size resonant photoacoustic cell with reduced window background for laser detection of gases

A photoacoustic resonant cell with the inner volume is ∼0.5 cm³ is presented. The shape of the cell cavity is chosen such as to minimize a background signal arising due to the absorption of laser beam in the cell windows. The experimental setup, the measurement procedure, and the design of the cell are described. The results of detecting ammonia in a nitrogen flow using the R(30) oscillating line of a CO₂ laser are represented. The minimal detectable absorption achieved in experiment is ∼3.2 × 10⁻⁸ cm⁻¹ W Hz^−1/2.     

Global Weak Solutions¶for the Two-Dimensional Motion¶of Several Rigid Bodies¶in an Incompressible Viscous Fluid

We consider the two-dimensional motion of several non-homogeneous rigid bodies immersed in an incompressible non-homogeneous viscous fluid. The fluid, and the rigid bodies are contained in a fixed open bounded set of ?². The motion of the fluid is governed by the Navier-Stokes equations for incompressible fluids and the standard conservation laws of linear and angular momentum rule the dynamics of the rigid bodies. The time variation of the fluid domain (due to the motion of the rigid bodies) is not known a priori, so we deal with a free boundary value problem. The main novelty here is thedemonstration of the global existence of weak solutions for this problem. More precisely, the global character of the solutions we obtain is due to the fact that we do not need any assumption concerning the lack of collisions between several rigid bodies or between a rigid body and the boundary. We give estimates of the velocity of the bodies when their mutual distance or the distance to the boundary tends to zero.     

IR Dichroism of Langmuir–Blodgett Films Based on Fatty Acids

We propose a method of measurement and quantitative estimation of dichroic ratios of submicron laminated Langmuir–Blodgett films by the IR spectra of multiple attenuated internal reflection. The method has been used in calculating the angles of orientation of molecular chains of stearic and palmitic acids in the process of their molecular deposition on an optical substrate. It has been found that in the process of structural self-organization of ensembles the degree of orientation of macromolecules is the higher, the greater the number of links entering into the aliphatic chain.     

A miniature prototype of a resonance photoacoustic cell for gas sensing

We present a miniature prototype of a resonance photoacoustic cell with a banana-shaped internal cavity with a volume of less than ～5 mm³. The cell design and the experimental setup used for testing operational characteristics of the prototype are described. The results of testing the prototype in the experiments on photoacoustic detection of radiation absorption by ammonia in a nitrogen flow by using a single-mode diode laser generating radiation near ～1.53 μm are presented. The amplitude-frequency characteristic of the photoacoustic response induced by absorption of radiation in the presence and absence of ammonia is analyzed. The experiment revealed that sensitivity of gas sensing is limited only by the microphone noise. The measured noise-limited minimal detectable absorption was found to be about 8.48 × 10^?8 cm^?1 W Hz^?1/2.