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We prove that there exists a one-parameter family of meromorphic solutions u ( τ ) vanishing at τ = 0 of the degenerate third Painlevé equation, u ″ τ = u ′ τ 2 u τ - u ′ τ τ + 1 τ - 8 ε u τ 2 + 2 a b + b 2 u τ , ε = ± 1 , ε b > 0 , for formal monodromy parameter a = ±i/2. We study number-theoretic properties of the coefficients of the Taylor-series expansion of u ( τ ) at τ = 0 and its asymptotic behaviour as τ → +∞. These asymptotics are visualized for generic initial data.

It is shown how to calculate asymptotics of integrals over the positive semi-axis of two functions related to the degenerate third Painlevé equation (dP3). As an example, the corresponding results for the meromorphic solution of dP3 vanishing at the origin are presented.

AbstractDifferential equations are derived for a symmetric transistor trigger and multivibrator. It is shown that the equations can be derived using the author’s expression for the current characteristic of a common-emitter transistor amplifier based on the Lambert W function (in spite of the fact that the equations can be formulated using different characteristics). The equilibrium state for the differential equations of the trigger is studied. Slow and fast motions are analyzed for the differential equations of the multivibrator. A numerical solution for the multivibrator is presented.

Schlesinger (S) transformations can be combined with a direct rational (R) pull-back of a hypergeometric 2 × 2 system of ODEs to obtain RS42 -pullback transformations to isomonodromic 2 × 2 Fuchsian systems with 4 singularities. The corresponding Painlevé VI solutions are algebraic functions, possibly in different orbits under Okamoto transformations. The paper demonstrates direct computations (involving polynomial syzygies) of Schlesinger transformations that affect several singular points at once, and presents an algebraic procedure of computing algebraic Painlevé VI solutions without deriving full RS-pullback transformations.

The environmental monitoring since 2009 in various water regime phases has been conducted by the Department of Ecology and Environmental Management for the water quality assessment of the Perm city’ small rivers. Such assessment of the water quality in the urban area of Mulyanka, Danilikha, Yegoshikha and Iva rivers catchments are presented in this study. The analysis is given for the background and mouth sections. The data of river’s water chemical composition was used for periods 2009-2010 and 2015-2018. A comparative analysis of water quality in terms of pollution indices is given.

We show that the emptiness formation probability of the six-vertex model with domain wall boundary conditions at its free-fermion point is a τ -function of the sixth Painlevé equation. Using this fact we derive asymptotics of the emptiness formation probability in the thermodynamic limit.

Copper is an important metal both in living organisms and in the industrial activity of humans, it is also a distributed water pollutant and a toxic agent capable of inducing acute and chronic health disorders. There are several fluorescent chemosensors for copper (II) determination in solutions; however, they are often difficult to synthesize and solvent-sensitive, requiring a non-aqueous medium. The present paper improves the known analytical technique for copper (II) ions, where the linear dependence between the ascorbic acid oxidation rate constant and copper (II) concentration is used. The limits of detection and quantification of the copper (II) analysis kinetic method are determined to be 82 nM and 275 nM, respectively. In addition, the selectivity of the chosen indicator reaction is shown: Cu2+ cations can be quantified in the presence of the 5–20 fold excess of Co2+, Ni2+, and Zn2+ ions. The La3+, Ce3+, and UO22+ ions also do not catalyze the ascorbic acid oxidation reaction. The effect of the concentration of the common background electrolytes is studied, the anomalous influence for chloride-containing salts is observed and discussed.

We study a multi-point correlation function of the six-vertex model on the square lattice with domain wall boundary conditions which is called the generalized emptiness formation probability. This function describes the probability of observing ferroelectric order around all vertices of any Ferrers diagram λ at the top left corner of the lattice. For the free fermion model, we derive and compare explicit formulas for this correlation function in two cases: when the diagram λ has a square or a triangular shape. We find a connection between our formulas and the τ-function of the sixth Painlevé equation. Bibliography: 25 titles.

A transition is made from piecemeal functions of the memristor model with threshold type switching to differentiable functions described by a single formula. Systems of equations are obtained and numerically solved for circuit sections in which the memristive device is connected in series with other discrete elements, a conventional resistor, diode, inductor, and capacitor. For the case of a serial connection of a memristor and a resistor, the calculated data are compared with the experiment. The case of series connection of a memristor and a semiconductor diode has been studied in detail. The assumptions concerning the mathematical description and physical interpretation of the influence of the electroforming process on the memristive system are presented.

The article proposes digital phase modulation and correlation processing of pulsed ultrasonic signals for the turbulent gas flows monitoring. Estimates of errors in measuring the relative time delay of ultrasonic pulses during multipath propagation and their interference in the receiver were made. The data obtained from the experiment show that the relative pulse delay measured in a turbulent gas flow has a resolution exceeding tenths of the carrier frequency period. As an example the possibilities of turbulence's quantitative control the air flow boundaries of a fan are shown. The perspectives to use the developed software and hardware in flow meters and other ultrasonic control and measuring devices are discussed.